mathematical foundations of quantum field theory
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Mathematical Foundations of Quantum Field Theory
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NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
World Scientific
Mathematical Foundations of Quantum Field Theory
Albert SchwarzUniversity of California at Davis, USA
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Library of Congress Cataloging-in-Publication DataNames: Shvar ts, A. S. (Alʹbert Solomonovich), author. Title: Mathematical foundations of quantum field theory / Albert Schwarz, University of California at Davis. Description: New Jersey : World Scientific, [2020] | Includes bibliographical references. Identifiers: LCCN 2019034999 | ISBN 9789813278639 (hardcover) Subjects: LCSH: Quantum field theory. Classification: LCC QC174.45 .S3295 2020 | DDC 530.14/3--dc23 LC record available at https://lccn.loc.gov/2019034999
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To the memory of my beloved wife Lucy
v
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Preface
This book is addressed to mathematicians and physicists, who are
interested in clear exposition in the foundations of quantum field
theory. I have tried very hard to satisfy both categories of readers.
I wanted the book to be accessible to a mathematician who does
not know quantum mechanics and interesting to a physicist who
specializes in quantum field theory. I aimed to have the rigor of the
proofs to be sufficient for a mathematician, but not so much that
it disturbed the reading for a physicist. I hope that this attempt to
satisfy these criteria is successful at least partly.
In this book, we talk almost exclusively about the results of
quantum field theory that do not depend on the assumption that the
theory is Lorentz-invariant (Lorentz-invariant theories are analyzed
only at the end of the book). This is the most essential difference that
sets this book apart from other books. Another important feature of
the book is the consideration of both the Hamiltonian and axiomatic
approaches to quantum field theory; we also establish the relation
between them.
In some existing books on quantum field theory, one can easily find
examples where the rules of the game (the main definitions) change
in the process of calculation. We can also see formal manipulations
with meaningless expressions, but in the result, we somehow obtain
a meaningful answer. This makes the study of quantum field theory
much more difficult for a mathematically inclined reader. Of course,
the reader understands that in changing the rules of the game,
physicists do not imitate the characters in the books of Lewis Caroll;
vii
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viii Mathematical Foundations of Quantum Field Theory
instead, they are guided by physical intuition, broad use of analogies,
and experiment. However, even if a mathematician recognizes that
physicists are doing the right things, this does not solve his problems.
I have tried to give an exposition of the main notions of quantum
field theory in such a way that without aiming for full mathematical
rigor, we instead obtain maximal clarity. However, in most cases, a
qualified mathematician should be able to fill in the details following
the proof outlines sketched in this book.
The first two chapters of the book and Section 3.1 contain a short
introduction to quantum mechanics, intended for mathematicians. In
Sections 3.1, 6.1, 6.2 and 6.3, we present some basic facts about Fock
space and operators on it; the rest of the book builds on these facts.
Chapter 4 is dedicated to the study of the operator of evolution
in the interaction representation and its adiabatic analog. Chapter 5
presents the theory of potential scattering in quantum mechanics.
Section 8.1 of Chapter 8, is devoted to translation-invariant
Hamiltonians and their operator realizations. The quantization of
classical translation-invariant system with an infinite number of
degrees of freedom is studied in Section 8.3 of Chapter 8.
Chapter 9 contains descriptions of different constructions of the
scattering matrix of a translation-invariant Hamiltonian. The proof
of the equivalence of these constructions is given in Chapter 11.
Chapters 10 and 12 introduce axiomatic scattering theory (in
Chapter 12, we consider Lorentz-invariant theories).
In Chapter 11, we study translation-invariant Hamiltonians in
the framework of perturbation theory; this chapter uses the results
of Chapter 10 on axiomatic scattering theory and the definition of
the canonical Faddeev transformation in Section 9.4 of Chapter 9.
A mathematician should begin to read these sections with Sec-
tion 11.5 of Chapter 11.
Chapter 13, added to the English edition, contains applications
of the methods of the preceding chapters to statistical physics. An
advanced reader can start with this chapter, returning in the case of
necessity to Chapter 10 and to Introduction.
The mathematically inclined reader, after the first five chap-
ters and Sections 6.1 and 6.2 of Chapter 6, can go straight to
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Preface ix
axiomatic scattering theory (Chapter 10 and Sections 12.1 and 12.2
of Chapter 12). After these, one may read Sections 8.1 and 8.3 of
Chapter 8, Section 12.3 of Chapter 12, Section 11.5 of Chapter 11,
and Chapter 13.
A physicist who wants to read the book on a rigorous level can
find the necessary mathematical definitions and theorems in the
appendix. If he is satisfied with a lower level of rigor, he should
begin his reading with the fourth chapter, since the material in the
first three chapters should be familiar. He can neglect the difference
between pre-Hilbert and Hilbert spaces, and the difference between
Hermitian operators and self-adjoint operators. A measurable func-
tion for a physicist is an arbitrary function and a measure space can
be understood as the n-dimensional Euclidean space (more precisely,
if functions on a set X can be integrated, then the set X can be
considered as a measure space). If the physicist encounters unfamiliar
mathematical notion, he can usually keep going without much harm.
The book generally does not contain references to the original
papers. (I have placed references to papers only when results are
formulated but not proved.)
I have received generous help in the production of this book.
I am grateful to Yu. Berezansky, F. Berezin, L. Faddeev,
V. Fateev, E. Fradkin, V. Galitsky, A. Povzner, M. Polivanov,
A. Rosly, V. Sushko, I. Todorov, Yu. Tyupkin, A. Vainshtein,
O. Zavyalov, and other mathematicians and physicists who kindly
devoted their attention to this book. I am also grateful to the
translator of the book, Dmitry Shemetov, for his diligent work and
for his patience.
I am also deeply indebted to my family for their support.
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Contents
Preface vii
Introduction xvii
1. Principles of Quantum Theory 1
1.1 Principles of quantum mechanics . . . . . . . . . . . 1
1.2 Evolution of state vectors . . . . . . . . . . . . . . . 1
1.3 Calculating the probabilities . . . . . . . . . . . . . 2
1.4 Heisenberg operators . . . . . . . . . . . . . . . . . 5
1.5 Integrals of motion and stationary states . . . . . . 6
2. Quantum Mechanics of Single-Particle
and Non-Identical Particle Systems 9
2.1 Quantum mechanics of a single scalar particle . . . 9
2.2 Quantum mechanics of particles with spin . . . . . 11
2.3 Quantum description of a system with
non-identical particles . . . . . . . . . . . . . . . . . 14
2.4 A particle in a box with periodic
boundary conditions . . . . . . . . . . . . . . . . . . 16
2.5 One-dimensional harmonic oscillator . . . . . . . . . 17
2.6 Multidimensional harmonic oscillator . . . . . . . . 20
xi
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xii Mathematical Foundations of Quantum Field Theory
3. Quantum Mechanics of a System of Identical
Particles 23
3.1 A system of n identical particles . . . . . . . . . . . 23
3.2 Fock space . . . . . . . . . . . . . . . . . . . . . . . 27
4. Operators of Time Evolution S(t, t0)
and Sα(t, t0) 39
4.1 Non-stationary perturbation theory . . . . . . . . . 39
4.2 Stationary states of Hamiltonians depending
on a parameter . . . . . . . . . . . . . . . . . . . . 43
4.3 Adiabatic variation of stationary state . . . . . . . 46
5. The Theory of Potential Scattering 51
5.1 Formal scattering theory . . . . . . . . . . . . . . . 51
5.2 Single-particle scattering . . . . . . . . . . . . . . . 57
5.3 Multi-particle scattering . . . . . . . . . . . . . . . 64
6. Operators on Fock Space 75
6.1 The representations of canonical and
anticommutation relations:
Fock representation . . . . . . . . . . . . . . . . . . 75
6.2 The simplest operators on Fock space . . . . . . . . 82
6.3 The normal form of an operator: Wick’s
theorem . . . . . . . . . . . . . . . . . . . . . . . . 88
6.4 Diagram techniques . . . . . . . . . . . . . . . . . . 98
7. Wightman and Green Functions 107
7.1 Wightman functions . . . . . . . . . . . . . . . . . . 107
7.2 Green functions . . . . . . . . . . . . . . . . . . . . 111
7.3 Kallen–Lehmann representation . . . . . . . . . . . 116
7.4 The equations for Wightman and Green
functions . . . . . . . . . . . . . . . . . . . . . . . . 120
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Contents xiii
8. Translation-Invariant Hamiltonians 123
8.1 Translation-invariant Hamiltonians
in Fock space . . . . . . . . . . . . . . . . . . . . . . 123
8.2 Reconstruction theorem . . . . . . . . . . . . . . . . 131
8.3 Interactions of the form V (φ) . . . . . . . . . . . . 141
9. The Scattering Matrix for Translation-
Invariant Hamiltonians 147
9.1 The scattering matrix for translation-invariant
Hamiltonians in Fock space . . . . . . . . . . . . . . 147
9.2 The definition of scattering matrix by means
of operator realization of a translation-invariant
Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 154
9.3 The adiabatic definition of scattering matrix . . . . 166
9.4 Faddeev’s transformation and equivalence
theorems . . . . . . . . . . . . . . . . . . . . . . . . 170
9.5 Semiclassical approximation . . . . . . . . . . . . . 180
10. Axiomatic Scattering Theory 191
10.1 Main assumptions and the construction
of the scattering matrix . . . . . . . . . . . . . . . . 191
10.2 Proof of lemmas . . . . . . . . . . . . . . . . . . . . 207
10.3 Asymptotic fields (in- and out-operators) . . . . . . 222
10.4 Dressing operators . . . . . . . . . . . . . . . . . . . 229
10.5 Generalizations . . . . . . . . . . . . . . . . . . . . 240
10.6 Adiabatic theorem in axiomatic scattering
theory . . . . . . . . . . . . . . . . . . . . . . . . . 253
11. Translation-Invariant Hamiltonians (Further
Investigations) 275
11.1 Connections between the axiomatic theory and the
Hamiltonian formalism . . . . . . . . . . . . . . . . 275
11.2 Heisenberg equations and canonical
transformations . . . . . . . . . . . . . . . . . . . . 279
11.3 Construction of an operator realization . . . . . . . 286
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xiv Mathematical Foundations of Quantum Field Theory
11.4 Dressing operators for translation-invariant
Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 294
11.5 Perturbation theory via the axiomatic
approach . . . . . . . . . . . . . . . . . . . . . . . . 301
12. Axiomatic Lorentz-Invariant
Quantum Field Theory 313
12.1 Axioms describing Lorentz-invariant scattering
matrices . . . . . . . . . . . . . . . . . . . . . . . . 313
12.2 Axiomatics of local quantum field theory . . . . . . 319
12.3 The problem of constructing a non-trivial
example . . . . . . . . . . . . . . . . . . . . . . . . 324
13. Methods of Quantum Field Theory
in Statistical Physics 331
13.1 Quantum statistical mechanics . . . . . . . . . . . . 331
13.1.1 Examples . . . . . . . . . . . . . . . . . . . 334
13.2 Equilibrium states of translation-invariant
Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 335
13.3 Algebraic approach to quantum theory . . . . . . . 336
13.3.1 Quantum field theory and statistical
physics in Rd . . . . . . . . . . . . . . . . . 338
13.3.2 Particles and quasiparticles . . . . . . . . . 340
13.3.3 Scattering . . . . . . . . . . . . . . . . . . . 341
13.3.4 Asymptotic behavior of 〈Q(x, t)Ψ,Ψ′〉 . . . 346
13.3.5 Scattering theory from asymptotic
commutativity . . . . . . . . . . . . . . . . 348
13.3.6 Green functions and scattering: LSZ . . . . 350
13.3.7 Generalized Green functions; the inclusive
scattering matrix . . . . . . . . . . . . . . . 353
13.4 L-functionals . . . . . . . . . . . . . . . . . . . . . . 355
13.4.1 Translation-invariant Hamiltonians in the
formalism of L-functionals;
one-particle states . . . . . . . . . . . . . . 360
13.4.2 Quadratic Hamiltonians . . . . . . . . . . . 361
13.4.3 Perturbation theory . . . . . . . . . . . . . 362
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Contents xv
13.4.4 GGreen functions . . . . . . . . . . . . . . . 365
13.4.5 Adiabatic S-matrix . . . . . . . . . . . . . . 367
13.4.6 Scattering of (quasi-)particles; inclusive
cross-section . . . . . . . . . . . . . . . . . 368
Appendix 375
A.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . 375
A.2 Systems of vectors in a pre-Hilbert
vector space . . . . . . . . . . . . . . . . . . . . . . 376
A.3 Examples of function spaces . . . . . . . . . . . . . 377
A.4 Operations with Hilbert spaces . . . . . . . . . . . . 381
A.5 Operators on Hilbert spaces . . . . . . . . . . . . . 383
A.6 Locally convex linear spaces . . . . . . . . . . . . . 392
A.7 Generalized functions (distributions) . . . . . . . . 393
A.8 Eigenvectors and generalized eigenvectors . . . . . . 403
A.9 Group representations . . . . . . . . . . . . . . . . . 404
Bibliography 409
Index 413
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Introduction
The discovery of quantum field theory began with a rudimentary
form of quantum electrodynamics, the basic equations of which
were written at the end of the 1920s (Dirac, Heisenberg, Pauli).
These equations contain a small parameter e2
~c ≈1
137 , so it was
natural to try to find a solution in the framework of perturbation
theory with respect to this parameter. It turned out, however, that
only in the lowest orders did the perturbation theory provide finite
solutions consistent with experimental results. In higher orders, the
perturbation theory resulted in divergent integrals.
Significant progress was made only two decades later, beginning
with the work of Bethe (1947), Schwinger (1958) and Tomonaga
(1946) and finally culminating with Feynman’s (2005) method, which
made it possible to extract finite results from the divergent integrals
of higher orders of the perturbation theory with striking agree-
ment to experimental results (covariant theory of renormalization).
Feynman’s method turned out to be applicable to whole classes of
theories (to the so-called renormalizable theories); it was developed
in many works, among which it is necessary to mention the article
by Dyson (1949). The method was put on a solid mathematical
foundation in the article by Bogolyubov and Parasyuk (1955). These
groundbreaking papers started a stormy period of development of
QFT in many directions. These developments did not come without
delays and disappointments. Moreover, in these years, none of the
fundamental problems of QFT have been solved.
xvii
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xviii Mathematical Foundations of Quantum Field Theory
Important steps forward were the discovery of non-Abelian
gauge fields, quantum chromodynamics, and the Higgs mechanism.
These developments led to the creation of the standard model that
describes electromagnetic, weak and strong interactions in very good
agreement with experiment. (The standard model assumes that
neutrinos are massless; hence one should modify it to take into
account the neutrino mass. This is the only deviation of standard
model from the experiment that has been discovered by now.)
However, physicists believe that the standard model comes from more
fundamental theory. They think that it comes from the theory where
electromagnetic, weak and strong interactions are on equal footing
(grand unification) or from theory that also describes gravity (like
string theory). Furthermore, we now significantly better understand
the mathematical structure of QFT. I would like to first mention the
algebraic approach to quantum field theory that is closely related to
the axiomatic quantum field theory (see, for example, Bogolyubov
et al. (1956), Lehmann et al. (1955), Wightman (1956), Haag (1958),
Araki and Haag (1967), Haag and Kastler (1964), Ruelle (1962),
Hepp (1965)). In particular, one should mention the construction
of a scattering theory in the axiomatic framework. In the case
of renormalizable theories, one can construct objects obeying the
axioms of relativistic QFT in the framework of perturbation theory,
but it is difficult to give a rigorous construction outside of this
framework. Intensive development in this direction received the name
of constructive QFT (see, for example, Jaffe (2000) and Velo and
Wightman (2012)); it was successful in dimensions < 4, but realistic
four-dimensional theories are still out of reach. Remarkable progress
in the study of conformal field theories and supersymmetric theories
(theories having symmetries mixing bosons and fermions) has led to
a much better understanding of QFT. Very important information
about QFT comes from string theory (one can obtain QFT from
string theory in some limit). Today, QFT presents an extensive field
of activity not only in physics but also in mathematics, being a source
of numerous clearly defined mathematical problems.
This book gives an introduction to QFT. It is in many ways
different from other books. In particular, we completely separate
the treatment of renormalization and divergences. It became clear
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Introduction xix
long ago that even in the absence of ultraviolet divergences, it is
impossible to use the standard quantum mechanical definition of
scattering matrix. The reason is that in QFT, there exists no natural
representation of full Hamiltonian in the form H0 + gV , where the
first term is a free (quadratic) Hamiltonian and the second term is
the interaction that can be considered as perturbation. This means
that the particles described by H0 (bare particles) are not the same
as the particles described by the full Hamiltonian (dressed particles).
This necessitates the change of the definition of the scattering matrix
and the renormalization of Feynman diagrams. This renormalization
is not related to ultraviolet divergences, a fact that is not emphasized
in many textbooks.
In this book, we do not assume Lorentz invariance in the study of
QFT; we discuss relativistic local theories only at the end. Therefore,
one can apply many of the statements to statistical physics with
minimal changes. This is the reason why I have added a new
chapter devoted to the applications of methods of QFT to statistical
physics. In particular, the new chapter contains the description
of the formalism based on the consideration of states as positive
functionals on Weyl or Clifford algebra (L-functionals) (Schwarz,
1967; Tyupkin, 1973; Schwarz, 2019b). This formalism allows us to
derive the diagram techniques of thermo-field dynamics (TFD) that
coincide in the case at hand with Keldysh diagrams (see Chu and
Umezawa (1994) for the review of TFD and Keldysh formalism).
Some of notions and results of the last chapter are based on recent
papers (Schwarz, 2019b,c). I would like to mention the notion of
inclusive scattering matrix and its expression in terms of generalized
Green functions. (Inclusive scattering matrix is closely related to
the inclusive cross-section that is necessary in the consideration of
scattering in the case when the theory does not have a particle
interpretation, in particular, for the consideration of collisions of
thermal quasiparticles.)
The exposition in the book was influenced by the algebraic
approach to quantum theory. However, the starting point is the
standard Hilbert space formulation. I am including in this edition a
short review of some questions of quantum theory based on algebraic
approach. For some readers, it would be reasonable to start with
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xx Mathematical Foundations of Quantum Field Theory
this review, while others could read it after finishing the book or in
parallel with reading the book. An advanced reader can jump to the
last chapter after reading the review.
Review of algebraic approach to quantum theory
First of all some terminological conventions.
Talking about algebra, we always have in mind unital (having unit
element) associative algebra over complex numbers with involution
denoted by ∗. (An involution is an antilinear map A → A∗ obeying
A∗∗=A, (AB)∗=B∗A∗.) We assume that the algebra is a topological
space and all operations are continuous, but for many of our
statements, these requirements are not sufficient (one should assume
that we have a Banach algebra or C∗-algebra or impose some other
conditions). In our terminology, an automorphism preserves not only
operations in the algebra but also the involution.
We say that h is a derivation of the algebra A if h(AB) =
h(A)B + Ah(B); we also assume that the derivation is compatible
with involution.
We say that a derivation is an infinitesimal automorphism if the
equation idAdt = h(A(t)) has a solution for all initial data A(0); then
a map A(0)→ A(t) specifies an automorphism αt. It is easy to check
that αt+s = αtαs (an infinitesimal automorphism generates a one-
parameter group of automorphisms). One can consider infinitesimal
automorphisms as elements of the Lie algebra of the group of
automorphisms. (This Lie algebra should be defined on the vector
space of tangent vectors to one-parameter families of automorphisms
at the unit element of the group. In the infinite-dimensional case, it
is not clear whether such a vector can also be considered as a tangent
vector to a one-parameter group of automorphisms. In what follows,
we disregard these subtleties.)
Hamiltonian formalism
The equations of motion of a three-dimensional non-relativistic
particle in a potential field U(~x) have the form
d~p
dt= −∇U,
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Introduction xxi
where ~p = m~dxdt stands for the momentum of the particle. To solve
these equations (i.e. to find the trajectory of the particle), we should
know the initial data: the coordinates and the momenta at some
moment of time. One says that the coordinates and the momenta
specify the state of our particle at the given moment and that the
equations of motion allow us to find the state of the particle at any
moment if we know it at one of the moments. The equations of motion
can be written in the form
d~p
dt= −∂H
∂~x,
~dx
dt=∂H
∂~p,
where H = p2
m + U(~x) is called Hamiltonian function (one can
say the Hamiltonian function is the energy expressed in terms
of momenta and coordinates). Similar equations are valid for any
mechanical system, but the number of degrees of freedom (the
number of coordinates and momenta) and the Hamiltonian function
can be arbitrary. This gives the so-called Hamiltonian formalism
of mechanics. (In Lagrangian formalism, the state is specified by
coordinates and velocities.)
In Hamiltonian formalism, the (pure) state of a classical mechan-
ical system (at the time t) is characterized by 2n numbers: p =
(p1, . . . , pn) (generalized momenta) and q = (q1, . . . , qn) (gener-
alized coordinates). (Together, these numbers specify a point of
2n-dimensional space called the phase space of the system).
More generally, we can define a state as a probability distribution
on the phase space. The set D of probability distributions is a convex
set, the pure states can be identified with extreme points of this set.
Every state can be considered as a mixture of pure states. (The
mixture of states ω1, . . . , ωn with probabilities p1, . . . , pn is the state
p1ω1+· · ·+pnωn. If states are labeled by continuous parameter λ ∈ Λ,
one defines the mixture of the states as an integral∫ω(λ)ρ(λ)dλ
where ρ(λ) stands for the density of the probability distribution
on Λ. Note that the definition of mixture can be used for any
convex set.)
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xxii Mathematical Foundations of Quantum Field Theory
The evolution of a pure state is governed by Hamiltonian equa-
tions
dq
dt=∂H
∂p,dp
dt= −∂H
∂q, (I.1)
where the function H(p, q, t) is called the Hamiltonian.
To write down the equation of motion for general state, we
introduce the notion of Poisson bracket of two functions on the phase
space by the formula
f, g = −∂f∂p
∂g
∂q+∂f
∂q
∂g
∂p. (I.2)
It is easy to check that the Poisson bracket is antisymmetric and
satisfies Jacobi identity, hence it specifies a structure of Lie algebra
on functions on phase space.1
Denoting by ρ(p, q, t) the density of the probability distribution
on the phase space at the moment t, we obtain the equation
d
dtρ(p, q, t) = H, ρ(p, q, t), (I.3)
governing the evolution of state (Liouville equation). If U(t) denotes
the evolution operator (the operator transforming the state at the
moment 0 into the state at the moment t) we can write (I.3) in the
form
d
dtU(t) = LU(t),
where Lρ = H, ρ. Note that ρ is in general a generalized function
on phase space. To verify (I.3), it is sufficient to check that for pure
states (represented by δ-functions) it is equivalent to (I.1).
1The multiplication and the Poisson bracket specify the structure of Poissonalgebra on the space of functions on the phase space (see the definition of Poissonalgebra in Section 1.3). This means that the phase space is a Poisson manifold.Moreover, it is a symplectic manifold, i.e. the Poisson structure is non-degenerate.We have considered the Hamiltonian formalism on a flat symplectic manifold, butit can be considered on any symplectic manifold.
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Introduction xxiii
A physical quantity (an observable) can be considered as a real
function f(p, q) on the phase space. It follows from the chain rule
that
d
dtf(p(t), q(t)) = −∂f
∂p
∂H
∂q+∂f
∂q
∂H
∂p. (I.4)
One can rewrite (I.4) in the form
d
dtf(p(t), q(t)) = f,H. (I.5)
It follows that in the case f,H = 0, the expression f(p(t), q(t))
does not depend on time (in other words, f is an integral of motion).
In particular, if H does not depend on time, the function H(p, q)
is an integral of motion. It can be identified with the energy of the
system.
Let us denote by A the set of all complex continuous functions
on the phase space considered as an algebra with respect to the
conventional addition and multiplication of functions. Formula (I.5)
gives an equation for the evolution in the algebra A.
Note that every state ω (considered as a measure on the phase
space) specifies a linear functional on A by the formula ω(f) =∫fω.
This functional obeys the positivity condition: ω(f) ≥ 0 if f ≥ 0. The
evolution of states and the evolution in A are related by the formula
(ω(t))(f) = ω(f(t)).
Quantum mechanics: An algebraic approach
The picture of the preceding section can be modified to describe
quantum mechanics. The main idea is to allow non-commuting
physical quantities.
The starting point is a unital associative algebra A over complex
numbers equipped with an antilinear involution A → A∗ (generaliz-
ing complex conjugation in the algebra A of the preceding section).
States are identified with positive linear functionals on A (linear
functional ω is positive if ω(A∗A) ≥ 0). We assume that states are
normalized, i.e. ω(1) = 1. The set D of normalized states is convex.
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xxiv Mathematical Foundations of Quantum Field Theory
The extreme points of this set are called pure states. Every state is
a mixture of pure states.
Let us denote by Aut the group of automorphisms of the algebra
A commuting with involution. This group naturally acts on states.
In quantum system, the state depends on time and this dependence
can be described by the evolution operator U(t) transforming ω(0)
into ω(t). This is the so-called Schrodinger picture; it is equivalent
to Heisenberg picture where the elements of A depend on time, but
the states do not: ω(t)(A) = ω(A(t)).
The evolution operator satisfies the equation
idU
dt= H(t)U(t), (I.6)
which is equivalent to the equation of motion idωdt = H(t)ω(t). Here,
H(t) stands for an element of Lie algebra of the group Aut (for
infinitesimal automorphism). It plays the role of the Hamiltonian
of the quantum system. If H does not depend on t, the evolution
operators obey U(t + τ) = U(t)U(τ) (constitute a one-parameter
subgroup).
To specify a quantum system, we should fix an algebra with
involution A and an infinitesimal automorphism H (or a family of
infinitesimal automorphisms H(t)). Then the evolution is governed
by (I.6).
In what follows, we assume that H does not depend on t unless
the dependence on t is explicitly mentioned.
Note that an infinitesimal automorphism can be considered as
a derivation of the algebra A. However, a derivation specifies a
quantum system only if it can be integrated to a one-parameter group
of automorphisms.
The textbook form of quantum mechanics corresponds to the case
when A is the algebra of bounded linear operators in Hilbert space
E with involution defined as Hermitian conjugation. The states are
specified by density matrices (positive operators with unit trace)
by the formula ωK(A) = TrKA. Pure states correspond to vectors
Ψ ∈ E ; proportional vectors specify the same state. If the vector Ψ
is normalized, the corresponding state is the functional 〈AΨ,Ψ〉.
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Introduction xxv
Every unitary operator W determines an automorphism of the
algebra with involution A by the formula A → W−1AW . This
correspondence allows us to identify the group Aut with the group of
unitary operators and infinitesimal automorphisms with self-adjoint
operators. The equation of motion for the density matrix has the form
idKdt = HK, where H is an operator on the space L of trace class
operators in E defined by the formula HK = −HK + KH and the
equation of motion for the vector Ψ ∈ E representing the pure state
is idΨdt = HΨ where H is a self-adjoint operator (Hamiltonian). The
evolution operator in E will be denoted by U(t); it obeys idUdt = HU .
The evolution operator in L will be denoted by U(t); it is easy to
check that U(t)K = U−1(t)KU(t).
Note that in the case at hand pure states have very simple
description, therefore, very often it is convenient to work with pure
states and to consider other states as mixtures of pure states. In
principle, we can work only with pure states for any algebra A, but
in general, this is not convenient because the description of pure
states is complicated.
For any algebra A and any state ω, we can construct a pre-Hilbert
space E and a representation of A by operators in this space such
that for some cyclic vector Φ ∈ E , we have ω(A) = 〈AΦ,Φ〉. (An
element A ∈ A is represented by operator A; we assume that the
map A → A is an algebra homomorphism and is compatible with
involution: A∗ = (A)∗. The vector Φ is cyclic in the following sense:
every other vector can be represented in the form AΦ where A ∈ A.)
This construction (Gelfand–Naimark–Segal (GNS) construction) is
essentially unique (up to equivalence).
Let us sketch the proof of this theorem. Assume that the
representation we need is constructed. Let us introduce in A an inner
product by the formula 〈A,B〉 = ω(B∗A). It is easy to see that the
map ν : A → E sending A to AΦ preserves this inner product. It
follows from cyclicity of Φ that this map is surjective; this allows us
to identify E with the quotient of A with respect to zero vectors.
(Recall that a zero vector is a vector that is orthogonal to all other
vectors.) The obvious relation ν(BA) = Bν(A) allows us to describe
our representation in terms of the algebra A and state ω. Namely,
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xxvi Mathematical Foundations of Quantum Field Theory
we construct E as A factorized with respect to zero vectors of the
inner product 〈A,B〉 = ω(B∗A), the operation of the multiplication
from the left by A descents to the operator A, the unit element of Acorresponds to the vector Φ.
We worked with pre-Hilbert spaces, but we can take a completion
of E to obtain a representation of A by operators acting in Hilbert
space.
We see that every state of A can be represented by a vector in
Hilbert space. However, we cannot consider all states as elements of
the same Hilbert space.
If ω is a stationary state (a state invariant with respect to time
evolution), then the group U(t) descends to a group U(t) of unitary
transformations of corresponding space E . The generator H of U(t)
plays the role of Hamiltonian.
We say that the stationary state ω is a ground state if the
spectrum of H is non-negative.
This definition agrees with the definition of the ground state in
Hilbert space formulation of quantum mechanics. Let us apply the
GNS construction to the algebra of bounded operators in Hilbert
space E and to a state represented by a vector Φ ∈ E where Φ is
an eigenvector of the Hamiltonian: HΦ = EΦ. Then the space given
by GNS construction can be identified with E and the generator of
U(t) is equal to H − E. The condition that Φ is the eigenstate with
minimal eigenvalue is equivalent to the positivity of H − E.
The representation containing the ground state will be called
ground state representation. This representation is especially impor-
tant in quantum field theory: we will consider particles as elementary
excitations of ground state.
Let us define the correlation functions in a stationary state ω as
functions
wn(t1, . . . , tn) = ω(A1(t1) . . . An(tn)),
where A1, . . . , An ∈ A.
The Green functions in the state ω are defined by the formula
Gn(t1, . . . , tn) = ω(T (A1(t1) . . . An(tn))),
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Introduction xxvii
where T stands for time ordering. It is easy to express them in terms
of correlation functions.
The correlation functions in the ground state are called Wightman
functions. The properties of particles in quantum field theory can be
expressed in terms of these functions and/or corresponding Green
functions; the same is true for scattering matrix.
Classical and quantum
To relate quantum and classical mechanics, we consider a family
of algebras A~ depending smoothly on the parameter ~ assuming
that for ~ = 0, we have a commutative algebra with the product
that will be denoted A ·B. More precisely, we assume that all these
algebras are defined on the same vector space, in other words, the
addition and multiplication by a number do not depend on ~, but the
multiplication of elements A,B of the algebra (denoted by A ·~ B)
smoothly depends on ~. The commutator [A,B]~ = A ·~ B − B ·~ Avanishes for ~ = 0, therefore, we can introduce a new operation
A,B (Poisson bracket) using the formula
[A,B]~ = iA,B~ +O(~2).
It is easy to verify that A0 (the algebra with commutative
multiplication A ·B that we have for ~ = 0) with the new operation
is Poisson algebra, i.e. the new operation satisfies the axioms of Lie
algebra and
A ·B,C = A,C ·B +A · B,C.
If there exists an involution A→ A∗ compatible with multiplica-
tion in all algebras A~, then it is also compatible with the Poisson
bracket (i.e. the Poisson bracket of two self-adjoint elements is again
a self-adjoint element). This statement is one of the reasons why a
factor i is included in the definition of the Poisson bracket.
Quantization; Weyl algebra
We have found that the classical mechanics can be obtained from
quantum mechanics in the limit ~ → 0. Conversely, quantum
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xxviii Mathematical Foundations of Quantum Field Theory
mechanics can be obtained as a deformation of classical mechanics.
We can start with Poisson algebra and deform it (this means we
would like to construct a family A~ that gives this Poisson algebra
in the limit ~→ 0).
If a Poisson algebra A is an algebra of polynomial functions on a
vector space with coordinates (u1, . . . , un), then the Poisson bracket
can be written in the form
A,B =1
2σkl(u)
∂A
∂uk∂B
∂ul. (I.7)
It specifies a structure of Poisson manifold on the vector space. (Here,
uk, ul = σkl(u).)
Let us consider the case when the Poisson bracket on polynomial
functions on vector space is defined by the formula (I.7) with constant
coefficients σkl. Moreover, we assume that the matrix σkl is non-
degenerate, then the dimension of the vector space is necessarily
even. (This is the situation in Section 1.1.) Then we can define the
algebra A~ as a unital associative algebra with generators uk and
relations ukul − uluk = iσkl. This algebra is called Weyl algebra.
If the coordinates uk are considered as real numbers, we introduce
an involution in Weyl algebra requiring that the generators uk are
self-adjoint. If the Poisson bracket is written in the form (I.2) (we
can always write it in this form changing coordinates), then the Weyl
algebra is generated by self-adjoint elements (p1, . . . , pn, q1, . . . , qn)
with relations pkpl = plpk, qkql = qlqk, pkq
l − qlpk = ~i δlk. These
relations are called canonical commutation relations (CCR). Instead
of self-adjoint generators (p1, . . . , pn, q1, . . . , qn), one can consider
generators ak, a∗k where ak = 1√
2(qk + ipk), a
∗k = 1√
2(qk− ipk). These
generators satisfy relations akal = alak, a∗ka∗l = a∗l a
∗k, aka
∗l − a∗l ak =
~δkl. These relations are also called CCR. To say that the family of
algebras A~ can be considered as a deformation of the commutative
polynomial algebra, we should realize their elements as polynomials.
This can be done in many different ways. For example, we can note
that using CCR we are able to move all operators qk to the left
and all operators pk to the right; “removing hats” in the expression,
we obtain a polynomial called (q − p) symbol of the element of
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Introduction xxix
Weyl algebra. This representation of elements of Weyl algebra by
polynomials does not agree with involution ((q − p)-symbol of self-
adjoint element is not necessarily real), however, it is easy to modify
the construction to avoid this drawback. For example, one can write
down an element of Weyl algebra in terms of generators ak, a∗k and use
CCR to move a∗k to the left and ak to the right. The expression we get
is called normal form of the element of Weyl algebra. Considering ak,
a∗k in the normal form as complex variables, we obtain a polynomial
called Wick symbol.
We can go in opposite direction and obtain an element of Weyl
algebra from a polynomial. This operation is called quantization.
Quantization allows us, for example, to obtain quantum Hamiltonian
from classical Hamiltonian. It is important to note that the quantiza-
tion depends on the choice of symbol (we have “ordering ambiguity”).
However, for some classical Hamiltonians, there exists a natural
choice of corresponding quantum Hamiltonians. In particular, this
is true for Hamiltonians represented as a sum of kinetic energy
expressed as a function of momenta and potential energy depending
on coordinates (no ordering ambiguity).
Stationary states
A state that does not depend on time is called stationary state. In
what follows, we work in the formalism where the states are described
by density matrices in Hilbert space E . The state represented by
a density matrix K is stationary if K obeys HK = 0, i.e. if
K commutes with the Hamiltonian H. (Recall that H acts as a
commutator with H.) An important particular case of a stationary
state is the Gibbs stateK = Z−1e−βH where Z = Tre−βH . This state
corresponds to the equilibrium state with the temperature T = β−1.
It tends to the ground state as T → 0.
Let us assume that the operator H has discrete spectrum with
orthonormal basis φn of eigenvectors with eigenvalues En. Then it
is convenient to work in representation where H is represented by
a diagonal matrix with entries En (it is called H-representation).
In this representation, the eigenvectors of the operator H in the
space L are matrices ψmn having only one non-zero entry equal
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xxx Mathematical Foundations of Quantum Field Theory
to 1 in the position (m,n). Alternatively, one can define ψmn as
an operator acting by the formula ψmnx = 〈x, φn〉φm where x ∈ E .
Corresponding eigenvalues are Em −En. We see that the vectors φnand the corresponding density matrices ψnn are stationary states.
It follows that all diagonal density matrices in H-representation are
stationary states. Moreover, if the spectrum is simple, all stationary
states are represented by diagonal matrices.
A density matrix K0 represents a ground state if Tr HK0 ≤Tr HK for any density matrix K. If the ground state is unique, it
is necessarily pure and corresponds to the eigenfunction of H with
lowest eigenvalue.
Adiabatic approximation
Let us consider the case of slowly varying Hamiltonian H(t). We
will assume that the energy levels En(t) are distinct and vary
continuously with t, corresponding eigenvectors will be denoted by
φn(t). We assume that these eigenvectors constitute an orthonormal
system. Then it is easy to prove that in the first approxima-
tion the evolution operator U(t) transforms eigenvector to eigen-
vector
U(t)φn(0) = e−iαn(t)φn(t),dαn(t)
dt= En(t). (I.8)
To verify (I.8), we check that the RHS satisfies the equation of motion
up to terms that are small for slow varying H(t) (see Section 4.3 for
more details).
Let us introduce the operators ψmn(t) by the formula ψmn(t)x =
〈x, φn(t)〉φm(t). (These operators are eigenvectors of the operator
H(t) in the space L.) Applying (I.8) or analyzing directly the
evolution operator U(t) in L, we obtain that U(t) transforms
eigenvector into eigenvector
U(t)ψmn(0) = e−iβmn(t)ψmn(t),dβmn(t)
dt= Em(t)− En(t) (I.9)
(this equation is true up to terms that can be neglected for slowly
varying Hamiltonian). Note that βmm does not depend on t.
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Introduction xxxi
Decoherence
Let us consider a quantum system (atom, molecule, etc.) described
by a Hamiltonian H with simple discrete spectrum. We assume that
the system is “almost closed” in the following sense: the interaction
with the environment can be described as an adiabatic change of
the Hamiltonian H. Let us consider the evolution operator U(T )
assuming that H(0) = H(T ). If the Hamiltonian is time-independent,
then for the eigenvector φn with the eigenvalue En, we can say that
U(T )φn = Cn(T )φn where Cn(T ) = e−iEnT . If the Hamiltonian is
slowly changing, we have the same formula with Cn(T ) calculated
from (I.8). Hence, if we started with pure stationary state, we remain
in the same state. Similarly, U(T )ψmn = Cmn(T )ψmn and the phase
factor is constant for m = n.
It is natural to assume that the environment is random (the
time-dependent Hamiltonian depends on some parameters λ ∈ Λ
with some probability distribution on Λ), then for m 6= n, we have
a random phase factor Cmn(λ, T ). If we start with density matrix
K =∑kmnψmn (with matrix entries kmn in H-representation), then
the density matrix Kλ(T ) is equal to∑Cmn(λ, T )kmnψmn, i.e. the
matrix entries acquire phase factor Cmn(λ, T ). Now, we should take
the mixture K(T ) of states Kλ(T ) (this means that we should take
the average of phase factors). It is obvious that non-diagonal entries
of K(T ) are smaller by absolute value than corresponding entries of
K. Imposing some mild conditions on the probability distribution on
Λ, one can prove that the non-diagonal entries of K(T ) tend to zero
as T → ∞. In other words, the matrix K(T ) tends to a diagonal
matrix K having the same diagonal entries as K. (See Schwarz and
Tyupkin (1987) and Schwarz (2019a) for more details.)
The matrix K can be considered as a mixture of pure states
corresponding to the vectors φn with probabilities knn.
This phenomenon is known as decoherence.
Observables and probabilities
Until now, we did not relate the formalism of quantum mechanics
to experiment. We know that (at least in some cases) quantum
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xxxii Mathematical Foundations of Quantum Field Theory
mechanics can be considered as a deformation of classical mechanics,
therefore, one can conjecture that classical observables correspond
to quantum observables. In particular, remembering that the energy,
which is represented by Hamiltonian function, is an integral of motion
if the system is invariant with respect to time shift should have
quantum analog with similar properties. This is not quite true: a
quantum system does not have a definite energy and measuring its
energy by means of macroscopic device we obtain different values
of energy with some probabilities. (A similar thing happens when
we are trying to use a thermometer to measure the temperature of
a system that is out of equilibrium. Such a system does not have
a temperature, but still we can measure it; thermometer readings
will be different, but we will obtain some probability distribution
of these readings.) The interaction with macroscopic device leads to
decoherence (the non-diagonal matrix entries of the density matrix
K in H-representation die) and we obtain a mixture of pure states
with probabilities knn. If the density matrix K corresponds to a
pure state described by a vector φ in E , then in general, decoherence
leads to mixture of pure states corresponding to the vectors φn with
probabilities pn = |〈φ, φn〉|2. However, if φ coincides with one of
eigenfunction φk, all probabilities pn vanish except pk = 1. Therefore,
we say that an eigenfunction of H with eigenvalue E has definite
value of energy equal to E. For any other normalized vector φ, we
can speak only about a probability to get some value E measuring
the energy. This probability is non-zero only for eigenvalues En of
H; it is equal to pn = |〈φ, φn〉|2. For a state represented by density
matrix K, the probability is given by the formula pn = 〈Kφn, φn〉.Other observables are represented by self-adjoint operators; the
formulas we have written for energy remain valid for any observable.
Again, if the physical quantity A is represented by self-adjoint
operator A in E , the quantity A has definite value a if a state is
represented by an eigenvector of A with eigenvalue a. Otherwise, we
can talk only about probabilities. It is convenient to write the density
matrix K representing a state in A-representation; if the operator
A has simple discrete spectrum, the probabilities are equal to the
diagonal entries of K in A-representation.
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Introduction xxxiii
In the general case, we can specify the probability distribution of
the observable A by the formula
f(A) = TrKf(A), (I.10)
where f stands for any piecewise continuous function and f(A) =∫f(a)ρ(a)da denotes the mean value of f(A) with respect to the
probability distribution ρ(a)da. (It is also called the expectation
value of f(A).) If the operator A has simple discrete spectrum, (I.10)
is equivalent to the formulas for probabilities we gave in this case.
If A1, . . . , An are commuting self-adjoint operators, one can define
the joint probability distribution of corresponding physical quantities
A1, . . . , An using the formula
f(A1, . . . , An) = TrKf(A1, . . . , An). (I.11)
More generally, if quantum mechanical system is specified by
C∗-algebra A and a state by a positive functional ω, then the joint
probability distribution of physical quantities A1, . . . , An is given by
the formula
f(A1, . . . , An) = ω(f(A1, . . . , An)). (I.12)
(Physical quantities Ai correspond to self-adjoint elements Ai of A.
The assumption that A is a C∗-algebra guarantees that the notion
of a function of a family of commuting self-adjoint elements makes
sense.)
Integrals of motion
One can work either in the Schrodinger picture where states are
time-dependent, but observables do not depend on time, or in the
Heisenberg picture, where states do not depend on time, but the
observables do. These pictures are equivalent:
TrK(t)f(A) = TrKf(A(t))
if A(t) obeys the Heisenberg equation
idA
dt= [H, A].
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xxxiv Mathematical Foundations of Quantum Field Theory
This implies that the observable A commuting with the Hamiltonian
H is an integral of motion (corresponding probabilities do not depend
on time).
Let us suppose that we have a one-parameter family U(t)
of symmetries of the system. (A symmetry is an automorphism
preserving the equations of motion. In our case, we consider it as
a unitary operator, commuting with the Hamiltonian.) The operator
idUdt is an integral of motion — a self-adjoint operator commuting
with the Hamiltonian.
The Hamiltonian itself is an integral of motion corresponding to
the time translation. We have already noted that the corresponding
observable is energy.
The integral of motion corresponding to invariance with respect
to spatial translations is a component of momentum.
The integral of motion corresponding to the invariance with
respect to rotation around some axis is a component of angular
momentum.
Weyl and Clifford algebras
In quantum theory in algebraic approach, we are starting with a
unital associative algebra with involution. We have seen already that
a natural candidate for this algebra is Weyl algebra. In this section,
we will study this algebra and its cousin, Clifford algebra. Recall
that the Weyl algebra is generated by self-adjoint elements pi, qi
with relations
pkpl = plpk, qkql = qlqk, pkql − qlpk =
1
iδlk. (I.13)
These relations are called canonical commutation relations (CCR).
Instead of self-adjoint generators pi, . . . , qi, one can consider gen-
erators ak, a∗k where ak = 1√
2(qk + ipk), a
∗k = 1√
2(qk − ipk). These
generators satisfy relations
akal = alak, a∗ka∗l = a∗l a
∗k, aka
∗l − a∗l ak = δkl. (I.14)
Both (I.13) and (I.14) are called CCR.
For finite number of degrees of freedom, there exists only one
irreducible representation of CCR. The relations (I.13) can be
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Introduction xxxv
represented in the space L2(Rn) by operators of multiplication and
differentiation: qkψ(q1, . . . , qn) = qkψ(q1, . . . , qn), pkψ(q1, . . . , qn) =1i∂∂qk
ψ(q1, . . . , qn). In non-relativistic quantum mechanics, this rep-
resentation is very convenient. If the classical Hamiltonian has the
form T (p) + V (q) where the kinetic energy T (p) is a quadratic
function of momenta and the potential energy V (q) has a non-
degenerate minimum, then in a neighborhood of this minimum, the
classical system can be approximated by multidimensional harmonic
oscillator. Similarly, for the quantized Hamiltonian H = T (p)+V (q),
we can approximate low energy levels as energy levels of quantum
oscillator. The Hamiltonian of one-dimensional quantum oscillator
can be written in the form
H =p2
2+q2
2= a∗a+
1
2.
Noting that Ha∗ = a∗(H + 1), Ha = a(H − 1), we obtain that the
operator a∗ transforms an eigenfunction φ with eigenvalue E in an
eigenfunction with eigenvalue E + 1. Similarly, the operator a either
sends φ to zero (if φ is the ground state) or to an eigenfunction with
eigenvalue E−1. Using this fact, we can check that the ground state
θ has energy 12 and the states 1√
n!(a∗)nθ constitute an orthonormal
basis consisting of eigenfunctions of H with eigenvalues n+ 12 .
In appropriate coordinates, the Hamiltonian of multidimensional
quantum oscillator can be considered as a sum of non-interacting
one-dimensional oscillators:
H =∑(
ωka∗kak +
ωk2
);
again applying many times operators a∗k to the ground state θ, we
obtain a basis of eigenfunctions.
We see that it is convenient to use the operators a∗k, ak in the
analysis of excitations of the ground state. In quantum field theory,
we are interested first of all in the excitations of ground state; this
is one of the many reasons why these operators are so useful.
Let us consider now the Weyl algebra with infinite number of
generators. This notion can be made precise in different ways. We
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xxxvi Mathematical Foundations of Quantum Field Theory
can consider simply an algebra with infinite number of generators
obeying (I.14), but it is more convenient to start with some (pre)-
Hilbert space B and consider an algebra with generators a(f), a∗(f)
and relations
a(λf + µg) = λa(f) + µa(g), a∗(f) = (a(f))∗,
a(f)a(g) = a(g)a(f), a∗(f)a∗(g) = a∗(g)a∗(f),
a(f)a∗(g)− a∗(g)a(f) = 〈f, g〉.(I.15)
A representation of Weyl algebra (= representation of CCR) is
a family of operators in a (pre)-Hilbert space obeying (I.15). (Note
that we use the same notation for elements of Weil algebra and for
operators.)
If B = l2, then a(f) =∑fkak, a
∗(f) =∑fka∗k where ak, a
∗k
obey (I.14). If B is some space of test functions on Rn (for example,
the Schwartz space), we can say that the formal expression a(f) =∫f(x)a(x)dx specifies a(x) as a generalized operator function on Rn.
We can also use this notation and terminology in cases when B is a
space of functions on some set.
The theory of Weyl algebra is very similar to the theory of
Clifford algebra (an algebra that is defined by canonical anti-
commutation relations (CAR) where commutators are replaced by
anticommutators). The representations of Clifford algebra are also
called representations of CAR.
In the simplest form, Clifford algebra can be defined as an algebra
with generators obeying
akal = −alak, a∗ka∗l = −a∗l a∗k, aka∗l + a∗l ak = δkl. (I.16)
More generally, to define the Clifford algebra, we start with some
(pre)-Hilbert space B and consider an algebra with generators
a(f), a∗(f) and relations
a(λf + µg) = λa(f) + µa(g), a∗(f) = (a(f))∗,
a(f)a(g) = −a(g)a(f), a∗(f)a∗(g) = −a∗(g)a∗(f),
a(f)a∗(g) + a∗(g)a(f) = 〈f, g〉.
(I.17)
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Introduction xxxvii
Note that both Weyl and Clifford algebras can be extended in
various ways (for example, one can introduce some norm and consider
a completion with respect to this norm).
Weyl algebra or Clifford algebra (or tensor product of these
algebras) can play the role of the algebra with involution A in the
algebraic description of quantum system.
In this description, we also need an infinitesimal automorphism
H of the algebra. The simplest way to construct it is to take a
self-adjoint element h of the algebra and to define a derivation
by the formula HA = [h,A]. (This construction works for any
algebra; derivations obtained this way are called inner derivations. In
C∗-algebra, one can prove that an inner derivation is an infinitesimal
automorphism considering eith.) If the algebra is generated by ak, a∗k
obeying (I.14) or (I.16), the element h can be represented in normal
form (where ak are from the right) by means of finite sum:
h =∑mn
∑k1,...,km,l1,...,ln
Γk1,...,km,l1,...,lna∗k1 . . . a
∗kmal1 . . . aln . (I.18)
Here, Γk1,...,km,l1,...,ln = Γln,...,l1,km,...,k1 (this condition guarantees that
h is self-adjoint).
In the case of infinite number of generators, one can modify this
construction to obtain other infinitesimal automorphisms. Namely,
we can consider (I.18) as a formal expression; if the expressions
[ak, h], [a∗k, h] can be regarded as elements ofA, these formulas specify
a derivation of algebra. This happens if for every k, there exists
only a finite number of summands in (I.18) where one of the indices
is equal to k. (Then only a finite number of terms survives in the
commutator.) To check that the derivation specifies a one-parameter
family of automorphism, we should verify that the equations of
motion idakdt = [ak, h], ia∗kdt = [a∗k, h] have a solution. If h is a quadratic
hamiltonian, the equations of motion are linear and the solution is
the same as in classical theory. If h is a sum of quadratic Hamiltonian
and a summand multiplied by a parameter g, then it is easy to
prove that the equations of motion can be solved in the framework
of perturbation theory with respect to g.
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xxxviii Mathematical Foundations of Quantum Field Theory
For example, we can take h =∑εka∗kak. Then we obtain
an infinitesimal automorphism leading to well-defined equations of
motion:
idakdt
= −εkak, ida∗kdt
= εka∗k.
If εk ≥ 0, the ground state representation is in this case the
Fock representation (the representation containing a cyclic vector
θ obeying akθ = 0). To prove this fact, we note that in the space
of Fock representation (Fock space), we have an orthogonal basis
of eigenvectors (a∗1)n1 . . . (a∗k)nk . . . θ with eigenvalues
∑εknk. Here,
nk = 0, 1, 2, . . . in the case of CCR and nk = 0, 1 in the case of CAR,
only finite number of nk does not vanish.
If εk < 0, for some k, then in the case of CCR, the ground state
does not exist. Let us consider the case of CAR. Let us denote by N
the set of indices k where εk < 0. If N is finite, one can find ground
state Φ in the same Fock space: it obeys the conditions akΦ = 0
for k /∈ N , and a∗kΦ = 0 for k ∈ N. If N is infinite, we should
introduce a new system of generators satisfying the same relations
(make canonical transformation): bk = ak, b∗k = a∗k for k /∈ N , bk =
a∗k, b∗k = ak for k ∈ N . Then the ground state θ′ lies in the Fock
space constructed by means of bk, b∗k and obeys bkθ
′ = 0.
The above statements can be easily generalized in various
directions. In particular, one can work with algebras defined by
relations (I.15) or (I.17). Then the Fock representation can be defined
as a representation in a (pre)-Hilbert space F that contains a cyclic
vector θ obeying a(f)θ = 0. If B consists of functions on M, we
can write formally a(f) =∫f(k)a(k)dk (integration over M) and
consider a(k), a∗(k) as generalized functions on M.
The equations of motion
ida(k)
dt= −ε(k)a(k), i
da∗(k)
dt= ε(k)a∗(k) (I.19)
are coming from the formal Hamiltonian∫ε(k)a∗(k)a(k)dk.
Again in the case ε(k) ≥ 0, the Fock representation is the ground
state representation and the vector θ is the ground state. The formal
Hamiltonian becomes a self-adjoint operator in Fock space.
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Introduction xxxix
More generally, any quadratic Hamiltonian leads to linear classical
equations of motion that can also be considered as quantum equa-
tions of motion. This is true, for example, for Klein–Gordon equation
and for Dirac equation. To construct the ground state representation,
one should change variables to write the equations in the form (I.19)
where a, a∗ obey CCR or CAR.
If a formal Hamiltonian is represented as H0 + V where H0
is a quadratic Hamiltonian, then in good situations, not only H0
but also the full Hamiltonian is a well-defined self-adjoint operator
in the ground state representation of H0. If we started with a
translation-invariant Hamiltonian without ultraviolet divergences,
then usually we obtain a “good” Hamiltonian after the volume cutoff
(infrared cutoff). We can calculate Wightman functions of “good”
Hamiltonian (at least in the framework of perturbation theory).
Then taking the limit of Wightman functions as the volume tends to
infinity, we obtain Wightman functions of the original Hamiltonian
(these construction can be considered as a definition of Wightman
functions). Finally, using Wightman functions and an analog of GNS
construction, we can construct the ground state representation of
the original formal Hamiltonian and of Heisenberg operators in this
space. The Heisenberg operators satisfy the equations of motion
coming from the formal Hamiltonian. We say that we have obtained
an operator realization of formal Hamiltonian (see Section 11.3 for
more details).
Quantum field theory via the algebraic approach
Quantum field theory can be considered as a particular case of
quantum mechanics.
To define quantum field theory on d-dimensional space (on (d+ 1)-
dimensional space-time), we assume that the group of space-time
translations acts on the algebra of observables. We can say that quan-
tum field theory is quantum mechanics with the action of commutative
Lie group on the algebra of observables A. In other words, we assume
that operators α(x, t) where x ∈ Rd, t ∈ R are automorphisms of Apreserving the involution and that α(x, t)α(x′, t′) = α(x + x′, t + t′).
We will use the notation A(x, t) for α(x, t)A where A ∈ A.
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xl Mathematical Foundations of Quantum Field Theory
The action of translation group on A induces the action of this
group on the space of states.
Let us now consider a state ω that is invariant with respect to
translation group. We will define (quasi-)particles as “elementary
excitations” of ω. (If ω is the ground state, one uses the word
“particles”; if ω is an equilibrium state, we are talking about thermal
quasiparticles.) To consider collisions of (quasi-)particles, we require
that ω satisfies the cluster property in some sense. Instead of cluster
property, one can impose a condition of asymptotic commutativity
of the algebra A(ω) that consists of operators A corresponding to
the elements A ∈ A in GNS construction. In other words, one can
require that the commutator [A(x, t), B] where A,B ∈ A is small for
x→∞.
The weakest form of cluster property is the following condition:
ω(A(x, t)B) = ω(A)ω(B) + ρ(x, t), (I.20)
where A,B ∈ A and ρ in some sense is small for x→∞. For example,
we can impose the condition that∫|ρ(x, t)|dx < c(t) where c(t)
has at most polynomial growth. Note that (I.20) implies asymptotic
commutativity in some sense: ω([A(x, t), B]) is small for x→∞.
To formulate more general cluster property, we introduce the
notion of correlation functions in the state ω :
wn(x1, t1, . . .xn, tn) = ω(A1(x1, t1) · · ·An(xn, tn)),
where Ai ∈ A. They generalize Wightman functions of relativistic
quantum field theory. We consider corresponding truncated correla-
tion functions wTn (x1, t1, . . .xn, tn) (see Section 10.1 for definition).
We have assumed that the state ω is translation-invariant; it
follows that both correlation functions and truncated correlation
functions depend on differences xi − xj , ti − tj . We say that the
state ω has cluster property if the truncated correlation functions
are small for xi − xj → ∞. A strong version of cluster property is
the assumption that the truncated correlation function tends to zero
faster than any power of ||xi − xj ||. Then its Fourier transform with
respect to variables xi has the form νn(p2, . . . ,pn, t1, . . . , tn)δ(p1 +
· · ·+ pn) where the function νn is smooth.
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Introduction xli
Let us show how one can define one-particle excitations of the
state ω and the scattering of (quasi-)particles.
The action of translation group on A generates unitary represen-
tation of this group on the (pre-)Hilbert space H constructed from
ω. Generators of this representation P and −H are identified with
momentum operator and Hamiltonian. The vector in the space Hthat corresponds to ω will be denoted by Φ. If Φ is a ground state,
we say that it is the physical vacuum. If ω obeys KMS-condition
ω(A(t)B) = ω(BA(t+ iβ)), (I.21)
we say that ω is an equilibrium state with the temperature T = 1β .
(It is assumed that A(t) can be analytically continued to the strip
0 ≤ Im t ≤ β.)
We say that a state σ is an excitation of ω if it coincides with ω
at infinity. More precisely, we should require that σ(A(x, t))→ ω(A)
as x → ∞ for every A ∈ A. Note that the state corresponding to
any vector AΦ where A ∈ A is an excitation of ω; this follows from
cluster property.
One can define one-particle state (one-particle excitation of the
state ω) as a generalized H-valued function Φ(p) obeying PΦ(p) =
pΦ(p), HΦ(p) = ε(p)Φ(p). (More precisely, for some class of test
functions f(p), we should have a linear map f → Φ(f) of this
class into H obeying PΦ(f) = Φ(pf), HΦ(f) = Φ(ε(p)f) where
ε(p) is a real-valued function. For definiteness, we can assume that
test functions belong to the Schwartz space S(Rd).) We require that
there exist an element B ∈ A such that BΦ = Φ(φ) where φ is a
non-vanishing function. (The symbol B denotes the operator in Hcorresponding to the element B ∈ A.) We also assume that there
exists an element A ∈ A and a function g(x, t) ∈ S(Rd+1) such that
B =∫g(x, t)A(x, t)dxdt (it follows from this assumption that B(x, t)
is a smooth function of x, t).
We assume that Φ(f) is normalized (i.e. 〈Φ(f),Φ(f ′)〉 = 〈f, f ′〉).Of course, it is possible that there are several one-particle states;
to simplify the notations, we assume that there exists only one kind
of particles.
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xlii Mathematical Foundations of Quantum Field Theory
Note that in relativistic theory, one-particle states can be iden-
tified with irreducible representations of the Poincare group; our
assumption means that we consider a scalar particle.
Let us fix a function f(p). Define a function f(x, t) as the
Fourier transform of the function f(p)e−iε(p)t with respect to p.
Let us introduce the notation B(f, t) =∫f(x, t)B(x, t)dx. (We
assume that this expression specifies an element of A such that
B(f, t) =∫f(x, t)B(x, t)dx.) It is easy to check that
B(f, t)Φ =
∫f(p)φ(p)Φ(p)dp (I.22)
does not depend on t.
Let us consider the vectors
Ψ(f1, . . . , fn|t) = B(f1, t) · · ·B(fn, t)Φ, (I.23)
where f1, . . . , fn satisfy the following condition: if fi(p) 6= 0, fj(p′) 6=
0, i 6= j, then ∇ε(p) 6= ∇ε(p′). (More precisely, we should assume
that the distance between ∇ε(p) and ∇ε(p′) is bounded from below
by a positive number.) We impose the additional requirement that
ε(p) is a smooth strictly convex function. Then one can derive from
cluster property or from asymptotic commutativity of A(ω) that
these vectors have limits in H as t→ ±∞; these limits will be denoted
by Ψ(f1, . . . , fn| ±∞).
Let us assume the vectors Ψ(f1, . . . , fn|−∞) span a dense subset
of H. (One can hope that this condition is satisfied when ω is the
ground state. In other cases, one should consider inclusive cross-
sections and inclusive scattering matrix; see Chapter 13 and Schwarz
(2019c) for more detail.) Then we can define in-operators a+in, ain on
H. In particular, to define a+in(f) =
∫(f(p)a+
in(p)dp, we can use the
formula
a+in(f φ)Ψ(f1, . . . , fn| −∞) = Ψ(f, f1, . . . , fn| −∞),
The definition of out-operators is similar. Using these notions, we
can define the scattering amplitudes by the formula
Smn(p1, . . . ,pm|q1, . . . ,qn)
= 〈a+in(q1) . . . a+
in(qn)Φ, a+out(p1) . . . a+
out(pm)Φ〉.
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Introduction xliii
Concluding remarks
In conclusion, a couple of general words about quantum theory.
A quantum mechanical system is specified by a Hamiltonian
(infinitesimal generator of time evolution) and a QFT is specified
by a Hamiltonian and a momentum operator (generator of spatial
translations). Knowing these data, we can define a notion of particle
in QFT; a particle can be interpreted as an elementary excitation of
ground state. Under certain conditions, we can define the scattering
of particles.
Note that string field theory can be regarded as a quantum field
theory in the sense of this definition.
Quantum mechanics is a deterministic theory. This means that
knowing the state at some moment and the Hamiltonian governing
the evolution of the state, we can in principle predict the state at
any moment — precisely as in classical mechanics. The probabilities
in quantum mechanics can be explained in the same way as in
classical statistical physics — they come from a random environment
(see Sections “Decoherence” and “Observables and probabilities” on
page xxxi for an explanation of how probabilities can be obtained
from decoherence).2
The notion of particle in quantum field theory is a secondary
notion. There exists no physical difference between elementary and
composite particles. (There are theories that can be represented in
two different ways: elementary particles of one approach or com-
posite particles of another approach.) Probably, the most revealing
illustration of properties of quantum particles is given by analogy
with nonlinear scattering in classical field theory. Such a theory can
have particle-like solutions, for example, solitons. (A soliton is a finite
energy solution of the form s(x−vt). It can be visualized as a bump
moving with constant speed that does not change its form. If we have
a bump moving with constant average speed, but the form of the
bump changes with time, we can talk about generalized soliton or, in
different terminology, about particle-like solution.) It seems that for
2We do not consider subtle questions of measurement theory. The abovestatement means only that the standard formulas for probabilities can be obtainedfrom decoherence.
March 30, 2020 11:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-00-fm page xliv
xliv Mathematical Foundations of Quantum Field Theory
large class of theories, the asymptotic behavior of any finite energy
solution as t → ±∞ can be described as superposition of particle-
like solutions and almost linear “tail” (see, for example, Tiupkin
et al., 1975; Soffer, 2006; Tao, 2009). (For integrable theories with one
spatial dimension, this is a rigorous theorem; in higher dimensions,
we only have experimental confirmation.) This means, in particular,
that we can talk about scattering of solitons. Two solitons collide
and we see a field configuration that does not resemble any particle-
like solution. After that, solitons miraculously reappear (with the
same velocities in integrable theories, with different velocities in non-
integrable ones.) This is precisely the picture we can see considering
a collision of quantum particles. It seems that for a given asymptotic
behavior, one can rigorously construct a solution of equations of
motion having this behavior. (In quantum field theory, this was
done in Haag–Ruelle–Araki scattering theory in the relativistic case.
Non-relativistic generalization of this theory was given in Fateev
and Shvarts (1973); see also Chapters 10 and 13.) Both in classical
and quantum integrable theories in two-dimensional space-time,
one can analyze asymptotic behavior of any solution of equations
of motion and rigorously justify the above picture. This can be
done also in non-relativistic quantum mechanics: one can prove
that asymptotically every solution behaves as a superposition of
elementary and composite particles (bound states); see, for example,
Hunziker and Sigal (2000). It is not known whether this statement is
correct for more general theories (and it is definitely wrong for many
relativistic conformal field theories).
We see that scattering of solitons eerily resembles quantum scat-
tering. This similarity becomes even more complete if the classical
theory has topological integrals of motion (i.e. the space of finite
energy solutions is disconnected). Magnetic charge is an example of
topological integral of motion, the solitons with minimal (by absolute
value) magnetic charge are magnetic monopoles (if the charge is
positive) or antimonopoles (if the charge is negative). Monopoles can
annihilate with antimonopoles.
Quantum mechanics is a consistent theory that does not need an
interpretation in the framework of classical field theory. I do not think
that quantum scattering can be interpreted as classical scattering
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Introduction xlv
of solitons in random environment (although this possibility cannot
be excluded). My point is that nonlinear classical scattering is very
similar to quantum scattering. This idea is supported by remark that
in the limit ~ → 0, the scattering of solitons can be obtained from
the scattering of quantum particles.
Notation
A ∪B the union of sets A and B;
A ∩B the intersection of sets A and B;
A×B the cross product of sets A and B (the set of all (a, b)
with a ∈ A and b ∈ B);
En the n-dimensional Euclidean space;
L2(X) the Hilbert space of square-integrable complex
functions on the measure space X;
〈x, y〉 the scalar multiplication of elements of (pre-)Hilbert
space (for Euclidean space elements x,y ∈ En, this
quantity will sometimes be denoted xy);
S(En) the space of smooth functions of n variables that
rapidly decay; by smooth, we mean infinitely
differentiable and by rapidly decaying, we mean
faster than any power;
Ω the cube with length L edges in E3, satisfying the
equations 0 ≤ x ≤ L, 0 ≤ y ≤ L, 0 ≤ z ≤ L;
TΩ a set of vectors with k = 2πnL , where n is an integer
vector;
φk the orthonormal basis in the space L2(Ω), formed by
the functions φk(x) = eixk;
H1 +H2 a direct sum of Hilbert spaces;
H1 + · · ·+Hn + · · ·
a direct sum of an infinite sequence of Hilbert spaces;
F (H) the Fock space built with the Hilbert space H (if
H = L2(X), then F (L2(X)) = F0 + F1 + · · ·+Fn + · · · , where Fn consists of square-integrable
symmetric functions in n variables in the case of
bosons; in the case of fermions, the functions are
antisymmetric);
March 30, 2020 11:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-00-fm page xlvi
xlvi Mathematical Foundations of Quantum Field Theory
[A,B] =
AB −BAthe commutator of A and B;
[A,B]+ =
AB +BA
the anticommutator of A and B;
Two operators A and B on a Hilbert space are (Hermitian)
conjugate to each other if 〈Ax, y〉 = 〈x,By〉 for all x, y;
A∗ (the adjoint operator) the conjugate operator to A
with the maximal domain of definition;
A+ the conjugate operator to A where the domain of
definition coincides with A (for operators defined
on the whole Hilbert space A∗ = A+; in this case,
we will use A∗);
slimAn the strong limit of a sequence of operators An (it is
required that all the operators have the same
domain). One says that A = slimAn if
Ax = limAnx for all x in the domain D;
wlimAn the weak limit of a sequence of operators An. One
says that A = wlimAn if 〈Ax, y〉 = lim〈Anx, y〉 for
all x, y in the domain D;
δk,l = δkl the Kronecker symbol (δk,l = 0 if k 6= l, else it is 1);
δ(ξ, η) delta function (here, ξ and η belong to a measure
space); if ξ, η ∈ En, then δ(ξ, η) = δ(ξ − η);
θ(t) = 0 the Heaviside step function; θ(t) = 0 if t < 0, and
θ(t) = 1 with t ≥ 0;1
w+i0 the formula limε→0+1
w+iε = 1i
∫θ(t) exp(iwt)dt;
the d’Alembert operator∂2
∂t2− ∂2
∂x2− ∂2
∂y2− ∂2
∂z2= ∂2
∂t2−∆;
supp f the support of the function f defined as the closure
of the set of arguments where f is non-zero.
For the momentum variables, we use the notations k, p, l, q; for
the energy variable, we usually use ω. For coordinates, we use x, ξ,
y, η, and time is denoted by t or τ .
We use the system of units where the Planck constant ~ and the
speed of light c are equal to 1.
March 26, 2020 11:53 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch01 page 1
Chapter 1
Principles of Quantum Theory
1.1 Principles of quantum mechanics
Almost all modern physics theories begin with the following scheme.
First, one specifies a mathematical object ψt that describes the
state of an observed physical system at time t. Second, one obtains
equations that govern the object’s change with time (usually taking
the form ∂ψt
∂t = A(ψt)). Finally, one defines ways to calculate
experimentally observed quantities from the object ψt.
Accordingly, we will formulate the postulates of quantum mechan-
ics in this manner.
The state of a quantum system at a fixed moment in time is
specified by a non-zero vector ψ taken from a complex Hilbert space
R; two vectors ψ and ψ′ = Cψ, differing only by a non-zero scalar
factor C, represent the same state. By choosing C = ‖ψ‖−1, we can
normalize the vector ψ′ and hence a state can be represented as a
normalized vector (a vector with norm equal to 1). If we don’t say
otherwise, we consider quantum states to be normalized vectors in
the rest of the book.
1.2 Evolution of state vectors
Knowing the state of a quantum system at time t0 (denoted by ψt0),
one can calculate the state ψt for any moment in time t, that is,
there exists a linear operator U(t, t0), the evolution operator, that
transforms the state vector ψt0 into the vector ψt.
The evolution operator U(t, t0) satisfies the group relations
U(t2, t1)·U(t1, t0) = U(t2, t0), U(t0, t0) = 1. It obeys the Schrodinger
1
March 26, 2020 11:53 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch01 page 2
2 Mathematical Foundations of Quantum Field Theory
equation:
i∂U(t, t0)
∂t= H(t)U(t, t0), (1.1)
with the initial condition U(t0, t0) = 1. Here, H(t) is a self-adjoint
operator called the Hamiltonian operator of the quantum system.
The Schrodinger equation can also be written in the form
idψtdt
= H(t)ψt, (1.2)
though less precisely, since the operator H(t) may not be defined on
the full Hilbert space R, while U(t, t0) is defined everywhere.
It is easy to check that U∗(t, t0) = U(t0, t); using the group
property, we can see that the operator is unitary (to be rigorous, we
should assume that the evolution operator is unitary and prove that
the Hamiltonian is self-adjoint using the unitarity of the evolution
operator).
If the Hamiltonian does not depend on time explicitly, we can
write U(t, t0) in the form U(t, t0) = exp(−iH(t− t0)).For a non-relativistic one-dimensional particle, the state space R
can be taken to be L2(E1), the space of square-integrable functions
ψ(x) of one variable x, where −∞ < x < ∞. The Hamiltonian
operator in this case can be written as H(t) = − 12m
d2
dx2+ V(x, t)
where V(x, t) is the multiplication operator by the function V(x, t)
(t plays the role of a parameter here).1 This Hamiltonian describes
the non-relativistic particle with mass m, moving in the field with
time-dependent potential V(x, t).
1.3 Calculating the probabilities
An observable in quantum mechanics corresponds to a self-adjoint
operator A on the space R. We will consider the correspondence
between classical and quantum observables later in the book.
1It is easy to check that V(x, t) = V(x, t), where x is the operator ofmultiplication by x.
March 26, 2020 11:53 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch01 page 3
Principles of Quantum Theory 3
An important general remark: the energy corresponds to the
Hamiltonian operator H.
If we know the operator A corresponding to the observable a and
a state of the quantum system, written as a normalized vector ψ ∈ R,
we can obtain the probability distribution of the values obtained by
measuring the observable a in the state ψ. Let us denote by µ(α)
the probability that after measurement we obtain the value a to
be less than or equal to α. In quantum mechanics, we postulate that
µ(α) = 〈eα(A)ψ,ψ〉, where eα(λ) is a function equal to 1 when λ ≤ αand 0 otherwise.
This postulate can be formulated another way: for any function
f , the mean value f(a) =∫f(α)dµ(α) of f(a) in the state ψ is given
by the formula f(a) = 〈f(A)ψ,ψ〉. (The probability µ(a) is equal to
the mean value of eα(a), hence the first form of the postulate follows
from the second form. To derive the second form from the first one,
we note that any function f can be represented as a limit of linear
combinations of the functions eα.)
If the observables a1, . . . , an correspond to commuting operators
A1, . . . , An, then a1, . . . , an are simultaneously measurable. This
means that for any state ψ we can find the joint probability
distribution for the values obtained when measuring a1, . . . , ansimultaneously. Namely, let us denote by µ(α1, . . . , αn) the prob-
ability that in the state ψ we obtain a1 ≤ α1, . . . , an ≤ αn. We
postulate this probability to be 〈eα1,...,αn(A1, . . . , An)ψ,ψ〉 where
eα1,...,αn(λ1, . . . , λn) is a function equal to 1 when λ1 ≤ α1, . . . , λn ≤αn and 0 otherwise.
As in the case of a single observable, the above postulate can
be reformulated by demanding that for any function f(a1, . . . , an)
the mean value f(a1, . . . , an) of f(a1, . . . , an) in the state ψ can be
obtained by the formula f(a1, . . . , an) = 〈f(A1, . . . , An)ψ,ψ〉.For a single one-dimensional particle, the position observable x
(the x-coordinate of the particle) corresponds to the multiplication
operator xψ = xψ. The momentum of the particle is represented by
the operator p = 1iddx , with the operator p2
2m = − 12m
d2
dx2corresponding
to the kinetic energy. The operator corresponding to the potential
energy is the multiplication operator by V(x, t). The momentum
March 26, 2020 11:53 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch01 page 4
4 Mathematical Foundations of Quantum Field Theory
and the kinetic energy are simultaneously measurable and so are
the position and the potential energy. However, it is impossible to
measure the position and the momentum simultaneously.
If A is a self-adjoint operator with discrete spectrum, then there
exists an orthonormal basis of eigenvectors ri in R for the operator
A with corresponding eigenvalues ai.
In the basis ri, the matrices for the operators A and eλ(A) are
diagonal, with the diagonal elements equal to ai for the operator A
and equal to eλ(ai) for the operator eλ(A). Let us suppose that a
normalized state vector is represented in the basis ri as ψ =∑
i ciri,
then
〈eλ(A)ψ,ψ〉 =∑ij
cic∗jeλ(ai)〈ri, rj〉 =
∑i
eλ(ai)|ci|2.
It follows that when measuring the physical quantity correspond-
ing to the operator A, we obtain only the eigenvalues ai with non-zero
probability. The probability of obtaining the value a in the state ψ
is equal to∑
i |〈ψ, ri〉|2, where the sum is taken over those i where
Ari = ari.
To each state vector ψ, we can assign the sequence (c1, . . . , ci, . . .)
of coefficients that arise from the decomposition of the vector ψ with
respect to the basis ri, consisting of the eigenvectors of operator A.
This construction specifies an isomorphism of the space R and the
space l2. We call this isomorphism an A-representation and the
sequence (c1, . . . , ci, . . .), we call the A-representation of the vector ψ.
The A-representation of the vector ψ allows us to easily calculate the
probabilities of values of the observable a in the state ψ.
The notion of an A-representation can also be defined when the
operator A has continuous spectrum. Moreover, it can be generalized
to the case when we are dealing with a family of commutative self-
adjoint operators. Namely, if A1, . . . , Ak are self-adjoint, pairwise
commuting operators, then an (A1, . . . , Ak)-representation is an
isomorphism of Hilbert spaces R and L2(M) in which the operators
A1, . . . , Ak are transformed to the multiplication operators by the
functions a1(m), . . . , ak(m) (hereM is a measure space, withm∈M).
In the (A1, . . . , Ak)-representation of the vector ψ ∈ R, one can easily
March 26, 2020 11:53 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch01 page 5
Principles of Quantum Theory 5
calculate the probability distribution of the observables A1, . . . , Akin the state ψ. It is also possible to prove that the (A1, . . . , Ak)-
presentation exists for any system of commuting self-adjoint opera-
tors A1, . . . , Ak. (If the operators A1, . . . , Ak have discrete spectrum,
then there exists an orthonormal basis of the space R consisting of
the common eigenfunctions of the operators A1, . . . , Ak).
The (A1, . . . , Ak)-presentation is closely tied to the generalized
eigenfunctions of the operators A1, . . . , Ak. That is, let rm, where m
belongs to the space with measure M , be a δ-normalized, generalized
basis for the space R (for details, see A.8). If the basis is an
eigenbasis for the operators A1, . . . , Ak (i.e. Airm = ai(m)rm), then
the isomorphism of the spaces R and L2(M), given by the formula
a(m) = 〈x, rm〉, is a (A1, . . . , Ak)-presentation.
In what follows, we will identify physical quantities with their
corresponding self-adjoint operators.
1.4 Heisenberg operators
Up to this point, we have considered our state vectors to be
dependent on time, while the operators for the physical quantities
were constant (Schrodinger’s picture). However, it is possible to view
this from a different but equivalent angle, where the operators depend
on time, but the vectors stay constant (Heisenberg’s picture).
Let us define for every operator A a time-dependent operator At(a Heisenberg operator) by the formula At = U∗(t, 0)AU(t, 0).
We note the following relations:
(AB)t = U∗(t, 0)ABU(t, 0)
= U∗(t, 0)AU(t, )U∗(t, 0)BU(t, 0)
= AtBt,
(A∗)t = U∗(t, 0)A∗U(t, 0)
= (U∗(t, 0)AU(t, 0))∗
= (At)∗,
f(At) = f(A)t,
〈Atψ,ψ〉 = 〈Aψt, ψt〉.
March 26, 2020 11:53 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch01 page 6
6 Mathematical Foundations of Quantum Field Theory
Combining the last two relations, we can verify that calculating
the probability distributions for the operator A and the state
ψt = U(t, 0)ψ, we obtain the same answer as when we calculate the
probability distributions for the operator At and the state ψ. Hence,
the Heisenberg and Schrodinger pictures are equivalent.
From (1.1) and the Hermitian conjugate equation i∂U∗(t,0)∂t =
−U∗(t, 0)H(t), we obtain the following equation for the operator At(the Heisenberg equation of motion):
dAtdt
= i[Ht, At],
where Ht = U∗(t, 0)H(t)U(t, 0). When the Hamiltonian H does
not depend on time, Ht = H,At = exp(itH)A exp(−itH), and the
Heisenberg equation takes the form
dAtdt
= i[H,At].
Let us calculate the Heisenberg equation of motion in a concrete
example. Let R be the space of square-integrable functions of one
variable x and H = p2
2m + V(x, t), where p = 1iddx and x is the
operator of multiplication by x. Using the fact that time evolution
commutes with summation and multiplication of operators, as well
as the operation of taking a function of an operator, we obtain
Ht =p2t2m
+ V(xt, t).
(In the case when H does not depend on time, then H = Ht.)
From the relation [p,V(x)] = 1iV′(x), it follows that [pt,
p2t2m +
V(xt, t)] = 1iV′(xt, t) and therefore dp
dt = −V ′(xt, t). (We are using the
fact that evolution preserves commutation relations.) Analogously,
we have that dxtdt = 1
m pt. Similar reasoning can be applied whenever it
is necessary to write the Heisenberg equations for a specific operator.
1.5 Integrals of motion and stationary states
A physical quantity (an observable) A is called an integral of motion
if for any state vector ψ, the probability distribution of the quantity
A in the state ψt = U(t, 0)ψ does not depend on the time t.
March 26, 2020 11:53 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch01 page 7
Principles of Quantum Theory 7
In the Heisenberg picture, the quantity A is an integral of motion
when the Heisenberg operator At does not depend on t. We will
assume for the rest of this section that the Hamiltonian H does not
depend on time. It is clear from the Heisenberg equations that the
quantity A will be an integral of motion if and only if the operator
A commutes with the Hamiltonian H. In particular, the physical
quantity that corresponds to a time-independent operator H is an
integral of motion; as we have noted, an observable of this form has
the physical meaning of energy.
The state vectors corresponding to the eigenvectors of the oper-
ator H (states with definite energy E) are called stationary states.
Indeed, if Hψ = Eψ, then ψt = exp(−iEt)ψ satisfies the Schrodinger
equation. This means that ψt differs from ψ only by a numerical
factor and consequently describes the same state; hence, the solution
for the equation Hψ = Eψ (the stationary Schrodinger equation)
is “stationary”. Conversely, when the vector ψt = U(t, 0)ψ for any
t differs only by a factor C(t) from the vector ψ, it follows from
the Schrodinger equation that iC ′(t)ψ = C(t)Hψ, hence C(t) =
exp(−iEt) and Hψ = Eψ. Therefore, any state that doesn’t change
with time is stationary in the sense of the above definition.
A stationary state corresponding to the smallest energy value is
called the ground state. It is easy to check that ground states can
also be characterized as states with the smallest average energy; in
other words, if Φ0 is a ground state and Ψ is any other state, then
〈HΨ,Ψ〉 ≥ 〈HΦ0,Φ0〉 (we assume that Φ0 and Ψ are normalized
vectors). Indeed, let us consider for simplicity the case when the
Hamiltonian H only has discrete spectrum. Decomposing the vector
Ψ into the eigenvectors Φn of the operator H, we have
〈HΨ,Ψ〉 =⟨H
∑cnΦn,
∑cnΦn
⟩=
∑En|cn|2
≥ E0
∑|cn|2
≥ E0 = 〈HΦ0,Φ0〉
(here, En are the eigenvalues of the operator H corresponding to the
eigenvectors Φn and E0 = minEn is the energy of the ground state).
b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 9
Chapter 2
Quantum Mechanics of Single-Particleand Non-Identical Particle Systems
2.1 Quantum mechanics of a single scalar particle
To describe a quantum mechanical system, we specify the Hilbert
space R of quantum states and the Hamiltonian operator H defining
time evolution. Similarly, we need to specify the operators for the
major physical quantities. In the following, we describe how to write
simple quantum systems.
The state for a single scalar particle in three-dimensional space
is described by a wave function — a square-integrable function ψ(r)
(i.e. the Hilbert space R in this case is L2(E3)). The Hamiltonian
operator for a non-relativistic particle in a potential field with energy
V(r) is written in the form H = − 12m∆ + V(r), where m is the mass
of the particle, ∆ is the Laplace operator, and V(r) is the operator
of multiplication by the function V(r). The coordinate operator x
for the coordinate x is defined as the multiplication operator by x
(i.e. ψ(r) = xψ(r)). Operators y and z are defined analogously. Since
the operators x, y, z commute, we can obtain the joint probability
distribution for the coordinates x, y, z in the state ψ(r). From the
previously formulated postulates, it follows that the density of this
probability distribution is |ψ(r)|2 (as always, we consider the state
vectors ψ(r) to be normalized). The operators for projections of the
momentum p are px = 1i∂∂x , py = 1
i∂∂y , pz = 1
i∂∂z , respectively.
The components of angular momentum are operators defined by the
formulas: Mx = ypz − zpy; My = zpx − xpz; Mz = xpy − ypx.
9
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 10
10 Mathematical Foundations of Quantum Field Theory
For brevity, we will often use the vector operators: r = (x, y, z),
p = (px, py, pz), M = r× p = (Mx, My, Mz).
For the Heisenberg operators rt and pt, the equations of motion
are
drtdt
=ptm
;dptdt
= −dVdr
(rt).
Up to this point, the states for one particle were written as ψ(r)
with r ∈ E3. If α is a unitary operator, transforming the space
L2(E3) to the space L2(M), where M is some measure space (an
isomorphism of the space L2(E3) and the space L2(M)), then we
can equivalently represent the state vectors ψ ∈ L2(E3) as vectors
ψ = αψ ∈ L(M). For every operator A acting on the space L2(E3),
there will be a corresponding operator A = αAα−1 acting on L2(M),
and from 〈f(A)ψ, ψ〉 = 〈f(A)ψ,ψ〉, it follows that the probabilities
obtained from the operator A and the state vector ψ will be the same
as those obtained from A and ψ.
Changing the representation of a state vector is useful in many
cases. For example, let α be the Fourier transform
ψ(p) = (2π)−3/2
∫exp(−ipr)ψ(r)dr
(in this case M = E3). The Fourier transform of the operator px =1i∂∂x is the multiplication operator ˜pxψ(p) = pxψ(p) (analogously for
the other variables). It is clear that the joint probability density of
px, py, pz is equal to |ψ(p)|2.
Let us describe the geometric origin of momentum and angular
momentum operators. Note that to every translation T in the space
E3, there corresponds a unitary operator WT on the space L2(E3),
that is, WT transforms wave functions when the underlying space
is transformed by T . WT can be written in the form WTψ(r) =
ψ(T−1r). It is obvious that WT1WT2 = WT1T2 and therefore the
operators WT form a unitary representation of the translation group.
The component operator for the momentum along an axis can
be written as the operator of infinitesimal translation along the
axis, that is, the operator px, for example, can be written as
i lima→0WTa−1
a , where Ta is translation by a along the axis x
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 11
Single-Particle and Non-Identical Particle Systems 11
(Ta(x, y, z) = (x+a, y, z)). Accordingly, the component of the angular
momentum can be written as the operator of infinitesimal small
rotation around the corresponding axis, that is, Mz can be written
as i limψ→01ψ (WSψ − 1) where Sψ denotes rotation by the angle
ψ around the z axis (i.e. Sφ(x, y, z) = (x cosφ − y sinφ, x sinφ +
y cosφ, z)).
This geometric interpretation allows us to understand when
components of momentum and angular momentum are integrals of
motion. For example, if the Hamiltonian is invariant with respect to
rotation around the z axis (i.e. if H commutes with WSφ), then the
Hamiltonian will commute with the operator Mz and therefore Mz
will be an integral of motion.
2.2 Quantum mechanics of particles with spin
The state of a particle with spin is written as a column vector ψ of
k square-integrable functions defined on three-dimensional space
ψ =
ψ1(r)
ψ2(r)
...
ψk(r)
,or, in other words, a function ψ(r) with k-column outputs. Instead
of notation ψi(r), we will use ψ(r, i), where i is a parameter, that
takes the values 1, 2, . . . , k. In other words, the space of states R can
be taken to be the space of square-integrable functions ψ(ξ), where
ξ = (r, i) ∈ E3 ×B, r ∈ E3, i ∈ B = 1, . . . , k.Integration over ξ has the same effect as integrating over r and
summing over i, for example, taking two functions φ ∈ R and ψ ∈ R,
we have
〈φ, ψ〉 =
∫φ(ξ)ψ(ξ)dξ =
k∑i=1
∫φ(r, i)ψ(r, i)dr.
The operators of position, momentum, and the Hamiltonian, if
the particle is in a potential field, are the same as in the case of the
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 12
12 Mathematical Foundations of Quantum Field Theory
spinless particle. The discrete variable acts like a parameter under
the action of the listed operators on the function ψ(ξ) = ψ(r, i).
However, the operators of angular momentum are different. As we
have seen, these operators are related to the action of rotations on the
wave functions and, in general, the rotations act not only on space
variables but also on discrete variables. Therefore, these operators
are represented as sums of the orbital part and the part coming from
discrete variables (the spin part). More precisely, the components of
operators for the angular momentum Mx, My, Mz can be written
in the form Mx = lx + sx; My = ly + sy; Mz = lz + sz, where
l = (lx, ly, lz) is the operator of the orbital angular momentum and
s = (sx, sy, sz) is the operator for the spin angular momentum. The
operator for the orbital angular momentum obeys the same formulas
as the operator of angular momentum for spinless particles (i.e.
l = r × p); the operator for the spin angular momentum acts only
on the discrete variable (we say that the operator A acts only on the
discrete variable if there exist matrices aij , where 1 ≤ i, j ≤ k, such
that Aψ(r, i) =∑k
j=1 aijψ(r, j)).
Remark. We can verify these relationships by following the con-
siderations in Section 2.1. To every rotation T in three-dimensional
space, there corresponds a unitary transformation WT of the space
R, this transformation changes the argument r of the wave function
(as in scalar case), but we should take into account discrete variables.
For example, if k = 3, we can consider ψ(r) as a function of r, taking
values in three-dimensional vectors. Then the operator WT acts on
the function ψ(r) in the natural way, that is, the function ψ′ = WTψ
is given by the equation ψ′i(r) =∑3
j=1 Tijψj(T−1r), where Tij is the
matrix for the transformation T . In general, the operator WT should
be written in the form
(WTψ)(r, i) =
k∑j=1
DTijψ(T−1r, j).
This form of the operator WT immediately leads to the formulas
previously stated for the components of angular momentum, which
are identified with operators of infinitesimally small rotation around
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 13
Single-Particle and Non-Identical Particle Systems 13
an axis. Operators WT must satisfy the relation WT1T2 = WT1WT2 ;
this means that they specify a unitary representation of the rotation
group. It is clear that matrices DT define k-dimensional unitary
representation of the rotation group. Knowing the finite-dimensional
unitary representation of the rotation group, it’s not difficult to
establish the form of spin projection operators. The correspondence
between rotation T with the operator WT and the matrices DT is in
general two-valued (the operator WT and the matrix DT are defined
up to a sign). This does not affect our calculations since state vectors
in quantum mechanics are defined up to multiplication by a number.
It is well known that for every value k ≥ 0, there exists one (up to
equivalence) k-dimensional unitary irreducible representation of the
rotation group. For k odd, the representation is single-valued; for k
even, it is two-valued. The functions
ψ(r) =
ψ1(r)
ψ2(r)
...
ψk(r)
,taking values in the state space, describe a particle with spin s = k−1
2 .
This terminology comes from the remark that the projection of the
spin takes values between −s and s (more precisely, every operator
sx, sy, sz, constructed as described in this paragraph, has eigenvalues
−s,−s+ 1, . . . , s− 1, s).
Especially important is the case of particles with spin 1/2, due to
many elementary particles, such as the electron, proton, and neutron,
having spin 1/2. Components of the spin can be written in this case
as matrices
sx =1
2
[0 1
1 0
]; sy =
1
2
[0 −ii 0
]; sz =
1
2
[1 0
0 −1
].
The above apparatus is also useful when particles have various other
quantum numbers, not just spin (e.g. isospin). The only difference
is in that when there are no quantum numbers besides spin, the
matrices DT specify an irreducible representation; otherwise, they
define a reducible representation.
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 14
14 Mathematical Foundations of Quantum Field Theory
2.3 Quantum description of a system with
non-identical particles
Two electrons (or two protons) have the same mass, the same charge;
their other properties are also identical. Therefore, we say that
all electrons are identical particles. If two particles belonging to a
system of particles are identical, then we must essentially change the
quantum mechanical description of the system. In this section, we
will consider only a system of particles where any two particles are
not identical.
If we consider a system of n three-dimensional particles with spin
zero, then the Hilbert space R of the states of such a system can
be considered as a space L2(E3n) of square-integrable functions of n
spatial variables.
Let us consider the case of a system with n particles with spin.
We assume that the jth particle can be described by a function ψ(ξ),
defined on the set Bj (in coordinate representation Bj is the set
of pairs (r, i), where r is a point in three-dimensional space and i
runs over the discrete set 1, 2, . . . , kj). Then the space of states of
the whole system is the space R = L2(B1 × · · · × Bn) of square-
integrable functions ψ(ξ1, . . . , ξn) of variables ξ1, . . . , ξn that run
over the sets B1, . . . , Bn. In the coordinate representation, the wave
function ψ(x1, y1, z1, i1, . . . , xn, yn, zn, in) of the system of particles
depends on the coordinates (xj , yj , zj) and spin variables ij of every
particle.
Every operator corresponding to a physical quantity related to the
jth particle can be considered in a natural way as an operator on the
space R. For example, in the coordinate representation, the operators
xj , yj , zj of the coordinates of the jth particle act in R as operators
of multiplication by xj , yj , zj ; the operators pjx, pjy, pjz of the
momentum components of the jth particle act in R as operators of
coordinate differentiation of the jth particle: pjx = 1i∂∂xj
, pjy = 1i∂∂yj
,
pjz = 1i∂∂zj
(which we can write more compactly as pj = 1i∂∂rj
).
In general, if the operator A (acting on the space L(Bj)) trans-
forms the function ψ(ξ) into the function ψ′(ξ) =∫A(ξ, η)ψ(η)dη,
then the corresponding operator in the space R transforms the
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 15
Single-Particle and Non-Identical Particle Systems 15
function ψ(ξ1, . . . , ξn) into the function
ψ′(ξ1, . . . , ξn) =
∫A(ξj , η)ψ(ξ1, . . . , ξj−1, η, ξj+1, . . . , ξn)dη.
The operator of total momentum p is given by the formula
p =n∑j=1
pj =n∑j=1
1
i
∂
∂rj. (2.1)
The Hamiltonian operator of a system of n particles, in the
simplest case, can be written in the form
H =
n∑j=1
p2j
2mj+ V(r1, . . . , rn) =
n∑j=1
(− ∆j
2mj) + V(r1, . . . , rn) (2.2)
(here, the function V typically takes the form V(r1, . . . , rn) =∑nj=1 Vj(rj) +
∑j<l Vjl(|rj − rl|)).
If V(r1, . . . , rn) = V1(r1) + · · · + Vn(rn) (the case of non-
interacting particles in an external field), then the problem of the
motion of n particles can be reduced to n single-particle problems.
Let us calculate, as an example, the stationary states of n non-
interacting particles. Let us denote by ψ(j)1 (ξj), . . . , ψ
(j)r (ξj), . . . the
eigenfunctions of the operator Hj =p2j
2mj+ Vj(rj) in the space
L2(Bj) (the stationary states of the jth particle). The corresponding
eigenvalues will be denoted by E(j)r . It is easy to check that
the function φ(1)r1 (ξ1)φ
(2)r2 (ξ2) . . . φ
(n)rn (ξn) is a stationary state of the
Hamiltonian H = H1+· · ·+Hn in the space R with the corresponding
eigenvalue E(1)r1 + · · · + E
(n)rn . If, for every j, the functions φ
(j)r (ξj)
form a complete orthonormal system in the space L2(Bj), then the
products φ(1)r1 (ξ1)φ
(2)r2 (ξ2) . . . φ
(n)rn (ξn) form a complete orthonormal
system in the space R.
Until now, we have worked with coordinate representations, which
means that we considered wave functions as functions of coordinates
and spin variables. Let us use another representation for every
particle; let us choose an isomorphism αj of the space L2(Bj) and
the space L2(Mj), where Mj is a measure space. Then it is easy to
construct an isomorphism of the space R = L2(B1×· · ·×Bn) and the
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 16
16 Mathematical Foundations of Quantum Field Theory
space L2(M1×· · ·×Mn) of square-integrable functions of n variables
(m1, . . . ,mn) that run over the sets M1, . . . ,Mn. In other words,
the state of a system of n particles can be represented by functions
ψ(m1, . . . ,mn), where mj ∈Mj . Denoting the space of states of the
j-th particle by Rj , we can say that the space of states R of an n
particle system is isomorphic to the tensor product R1 ⊗ · · · ⊗ Rn(this follows immediately from the definition of the tensor product
given in A.4).
In particular, it is very often convenient to use the momentum
representation. In the momentum representation, the system of n
particles is described as a function of momentum variables p1, . . . ,pnand spin variables i1, . . . , in, given by the formula
ψ(p1, i1, . . . ,pn, in) = (2π)−3n/2
∫exp
(−i
n∑k=1
pkrk
)×ψ(r1, i1, . . . , rn, in)dnr,
where ψ(r1, i1, . . . , rn, in) is the wave function in the coordinate
representation.
The momentum operator of the ith particle pi in the momentum
representation can be considered as the operator of multiplication by
pi; the operator of total momentum, which is equal to p =∑
pi, can
be considered as an operator of multiplication by p1 + · · ·+ pn.
2.4 A particle in a box with periodic
boundary conditions
Let us consider, as an example, a scalar particle moving freely in
the cube Ω, specified by the inequalities 0 ≤ x ≤ L, 0 ≤ y ≤L, 0 ≤ z ≤ L. To make this problem precise, we will impose periodic
boundary conditions on the wave functions of stationary states. The
space of states R in this case can be identified with the space of all
square-integrable functions on the cube Ω and the Hamiltonian is
the operator H = (− 12m)∆ with periodic boundary conditions (more
precise definition of this Hamiltonian will be given later).
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 17
Single-Particle and Non-Identical Particle Systems 17
Let us fix a vector a in three-dimensional space and consider the
operation of parallel translation by the vector a. The corresponding
unitary shift operator Wa on the space R transforms state vectors as
translation (Waψ)(r) = ψ(r− a), where ψ(r) is a periodic extension
of ψ(r), given on the cube Ω, to the whole three-dimensional
space. The momentum operators px, py, pz can be defined as
infinitesimal shift operators along the axes x, y, z (for example,
px = i lima→0W(a,0,0)−1
a ).
It is easy to check that pxψ = 1i∂ψ∂x . The operator px is defined
on the functions ψ ∈ L2(Ω), obeying the condition ∂ψ∂x ∈ L
2(Ω) and
satisfying the periodic boundary conditions ψ(0, y, z) = ψ(L, y, z).
In terms of the operator p, the Hamiltonian can be written in the
form H = 12m p2.
The functions φk = L−3/2 exp(−ikr), where k runs over the
lattice TΩ consisting of vectors 2πL n (where n is a vector of integer
coordinates), form a complete orthonormal system of common
eigenfunctions of the operators H, px, py, pz. Every function ψ ∈ Rcan be represented in the form ψ(r) =
∑ckφk (where the sum is
taken over the lattice TΩ); the function ck on the lattice TΩ is called
the wave function in the momentum representation.
2.5 One-dimensional harmonic oscillator
Let us consider a one-dimensional non-relativistic quantum particle
with mass m and potential energy V (q). We assume that V (q) has
one global minimum at the point q0 and V ′′(q0)) = k > 0. Then we
can study the ground state and low-lying energy levels replacing V (q)
with V (q0)+ k2 (q−q0)2. This means that we should study a quantum
harmonic oscillator. Taking into account higher order terms in the
Taylor expansion of V (q), we obtain the Hamiltonian of anharmonic
oscillator. Without loss of generality, we can assume that m = 1,
k = 1, V (q0) = 0. Then in the variable x = q − q0, the Hamiltonian
of the harmonic oscillator takes the form
H =p2
2+x2
2= −1
2
d
dx2+
1
2x2.
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 18
18 Mathematical Foundations of Quantum Field Theory
First, let us find the Heisenberg operators pt and xt from the
Heisenberg equations
dptdt
= −xt;dxtdt
= pt.
This system of operator equations is linear and therefore it can be
solved precisely as the corresponding system of numerical equations.
One of the possible ways to solve it is based on the introduction of
auxiliary operators
a =1√2
(x+ ip); a+ =1√2
(x− ip).
The equations for the Heisenberg operators at and a+t have a very
simple form
datdt
= −iat;da+
t
dt= ia+
t ,
hence a+t = a+ exp(it); at = a exp(−it). Using these relations, we
can immediately obtain the formulae xt = 1√2(at + a+
t ) and pt =1√2(a+t − at).
The operators a+ and a are very convenient for solving problems
related to the harmonic oscillator.
Note, first of all, that the Hamiltonian can be expressed in terms
of these operators by the formula H = a+a + 1/2. Second, note
that the commutation relations with the Hamiltonian have the form
[H, a+] = a+, [H, a] = −a, and their commutator is [a, a+] = 1.
Let us find the stationary states of the Hamiltonian H. We use
the following statement: If Hφ = Eφ, then H(aφ) = (E−1)aφ (hence
if φ is a stationary state and aφ 6= 0, then aφ is also a stationary
state). This statement follows from the relations H(aφ) = a(H −1)φ = (E − 1)aφ.
Similarly, H(a+φ) = a+(H + 1)φ = (E + 1)a+φ.
Note that the ground state φ0 should satisfy the relation aφ0 = 0
(otherwise, aφ0 is a stationary state with lower energy). Solving
the equation aφ0 = 1√2(x + d
dx)φ0 = 0, we obtain that φ0 =
π−1/4 exp(−12x
2) (the constant π−1/4 comes from the normalization
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 19
Single-Particle and Non-Identical Particle Systems 19
condition) and that the corresponding eigenvalue is E0 = 1/2.
Starting with φ0, we can get an infinite number of stationary states
φn = cn(a+)nφ0 = cn2−n/2(x− ddx)n(π−1/4 exp(−x2
2 )) (here cn is the
normalization constant). Noting that the operator a+ increases the
energy by 1, we have Hφn = (n + 12)φn. It is easy to see that the
functions φn have the form φn(x) = Hn(x) exp(−12x
2), where Hn(x)
are polynomials of n-th degree; one can check that Hn(x) coincides
with the n-th Hermite polynomial up to a constant factor. Let us
calculate the normalization constant cn assuming that all of them
are taken to be real and positive. It is obvious that φn = γna+φn−1,
where γn = cn/cn−1. Taking the scalar square of this equality, we
obtain
1 = 〈φn, φn〉
= γ2n
⟨a+φn−1, a
+φn−1
⟩= γ2
n
⟨aa+φn−1, φn−1
⟩= γ2
n
⟨(H +
1
2
)φn−1, φn−1
⟩= nγ2
n,
hence, γn = 1/√n and cn = (n!)−1/2.
The orthonormal system of stationary states φn = 1√n!a+nφ0,
with energy levels En = n+ 1/2, exhausts all stationary states. One
can check this by proving the completeness of functions φn in the
space L2(E1). However, one can give a more direct proof. Let φ
denote a stationary state of H. Let us denote by n then, the minimal
number satisfying anφ0 = 0 (such numbers necessarily exist because
the eigenvalues of the Hamiltonian H are bounded from below).
Then, an−1φ = λφ0, where λ 6= 0. Applying the operator (a+)n−1
to this equation, after some easy calculations, we obtain that φ is
proportional to φn−1.
The more general Hamiltonian H = p2
2m + mω2x2
2 can be reduced
to the Hamiltonian we have considered by means of changing the
system of units. One can, however, use the same consideration in
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 20
20 Mathematical Foundations of Quantum Field Theory
this case, taking
a =1√2
(√mωx+
ip√mω
); a+ =
1√2
(√mωx− ip√
mω
).
These operators are also related by the formula [a, a+] = 1, however,
the expression of the Hamiltonian in terms of these operators has
the form H = ω(a+a + 1/2), hence [H, a] = −ωa, [H, a+] = ωa+.
The dependence of operators at and a+t on time has the form at =
a exp(−iωt), a+t = a+ exp(iωt). The stationary states can be written
in terms of the ground state φ0 by the formula φn = (n!)−1/2(a+)nφ0,
with the energies equal to En = (n+ 1/2)ω.
Adding to the quadratic potential energy some higher order
terms with respect to x, we obtain the Hamiltonian of anharmonic
oscillators. It is convenient to express the Hamiltonian of anharmonic
oscillators in terms of the operators a+ and a. For example, if
H = p2
2m + mω2x2
2 + αx3 + βx4, then its expression in terms of a+
and a looks as follows:
H =ω
2+ ωa+a+ γ(a+ + a)3 + δ(a+ + a)4,
where γ = α(2mω)−3/2, δ = β(2mω)−2.
2.6 Multidimensional harmonic oscillator
Let us consider a quantum mechanical system with space of states
R = L2(En) and the Hamiltonian
H = −n∑j=1
αj∂2
∂x2j
+∑i,j
ki,j xixj
(here, αj > 0 and the quadratic form∑kijxixj is positively definite).
In this way, we can write down, for example, the Hamiltonian for a
system of oscillators coupled by means of elastic forces. By means
of a linear change of variables ξi =∑
j aijxj (the coordinates ξi are
called normal coordinates), the Hamiltonian can be written in the
March 26, 2020 12:16 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch02 page 21
Single-Particle and Non-Identical Particle Systems 21
following form:
H = −1
2
n∑i=1
∂2
∂ξ2i
+1
2
n∑i=1
ω2i ξ
2i .
(This statement follows immediately from the theorem that a
symmetric matrix can be diagonalized by means of an orthogonal
transformation, or from the equivalent theorem that for any two
quadratic forms, one of which is positively definite, one can transform
both into a sum of squares by means of a linear transformation.)
The Hamiltonian, written in normal coordinates, represents a
system of n non-interacting one-dimensional oscillators. Using the
results of Section 2.3, we can calculate its eigenvectors and eigen-
values. Note that the operators a+i = 1√
2(√ωiξi − 1√
ωi∂∂ξi
); ai =1√2(√ωiξi + 1√
ωi∂∂ξi
) allow us to write the Hamiltonian in the form
H =∑n
i=1 ωi(a+i ai + 1/2). The operators a+
i , ai(1 ≤ i ≤ n) satisfy
the relations
[a+i , a
+i ] = [ai, aj ] = 0; [ai, a
+j ] = δij
(these relations, called the canonical commutation relations, will
be explored further in a later section). The ground state Φ of
the Hamiltonian H satisfies the relations aiΦ = 0 for all i =
1, . . . , n. We can obtain the rest of the stationary states from Φ by
means of the operators a+i , i.e. the stationary states have the form
a+v11 a+v2
2 . . . a+vnn Φ, where v1, . . . , vn are non-negative integers; the
corresponding energies are (v1 + 12)ω1 + · · ·+ (vn + 1
2)ωn.
The Hamiltonian H ′ for a multidimensional anharmonic oscillator
has the form H ′ = H + V , where H is the Hamiltonian of harmonic
oscillator and V =∑
k1,...,knck1,...,kn x
k11 . . . xknn . Similar to H, the
Hamiltonian H ′ is also conveniently represented in terms of the
operators a+i , ai. We obtain
H ′ = const +
n∑i=1
ωia+i ai
+∑
k1,...,km,l1,...,ln
Γk1,...,km|l1,...,ln a+k11 . . . a+kn
n al11 . . . alnn ,
where the coefficients Γk|l can be expressed in terms of ck1,...,kn .
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22 Mathematical Foundations of Quantum Field Theory
In general, one cannot calculate explicitly the stationary states
of the Hamiltonian H ′ and other physical quantities associated with
this Hamiltonian. However, in the case that the anharmonic terms V
can be considered as a small perturbation, there exist techniques that
allow us to calculate the expansions of physical quantities associated
with H ′ as a power series with respect to the small parameter. These
techniques (Feynman diagrams) are described in Section 6.4.
March 26, 2020 13:43 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch03 page 23
Chapter 3
Quantum Mechanics of a Systemof Identical Particles
3.1 A system of n identical particles
In Section 2.3, we mentioned that two electrons are identical
particles and that a system of two electrons cannot be described
by means of the prescriptions of Section 2.3. If the system of
two electrons were specified by a wave function ψ(ξ1, ξ2), where
ξi = (ri, si) stands for the coordinate variable ri and the spin
variable si, then we could ask about the physical meaning of the
number∑
s1,s2|ψ(r1, s1, r2, s2)|2 = p(r1, r2). It would be natural to
consider this number as the probability, or more precisely as the
probability density, of the first electron occupying the point r1 and
the second electron occupying the point r2. However, the first and
second electrons are identical, therefore we can only talk about the
probability of finding one of these electrons at the point r1 and the
other electron at the point r2. So we should make some changes
and the right way to make these changes is as follows. For the wave
function of two electrons, we should consider only square-integrable
functions ψ(ξ1, ξ2) = ψ(r1, s1, r2, s2) that are anti-symmetric with
respect to the variables ξ1, ξ2 (i.e. ψ(ξ2, ξ1) = −ψ(ξ1, ξ2)). Then, it
is obvious that p(r1, r2) = p(r2, r1) and that this quantity makes
sense as the probability density measuring the probability of finding
one of the electrons at the point r1 and the other at r2. For the
system of two protons, two neutrons, or two µ-mesons, the situation
is precisely the same. However, for the system of two π0-mesons, we
23
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24 Mathematical Foundations of Quantum Field Theory
should consider as wave functions only functions ψ(r1, r2) that are
symmetric with respect to the variables r1, r2.
Let us now give precise definitions.
All particles are divided into two classes — bosons and fermions.
The space Fn of the states of the systems of n identical bosons
(fermions) is a Hilbert space of square-integrable functions of n
variables ψ(ξ1, . . . , ξn) that are symmetric (antisymmetric) with
respect to the variables (ξ1, . . . , ξn) (here, ξi ∈ X, where X is a
measure space). To emphasize that we are dealing with bosons, we
use the notation F sn for the space Fn and in the case of fermions,
we use the notation F an . The space L2(Xn) = L2(X × · · · × X)
will be denoted by Bn. The spaces F sn and F an are subspaces of this
space.
Experiments show that particles with half-integer spin (electrons,
protons, and so on) are fermions and particles with integer spin
(π-mesons, photons, and so on) are bosons. In relativistic quantum
theories, this statement can be derived from theoretical considera-
tions.
It is easy to check that the space Fn is completely determined
by the Hilbert space F1 of states of one particle and the number
n (i.e. it doesn’t depend on the representation of the space F1 of
one particle in the form L2(X)). Indeed, as we have mentioned
already in Section 2.3, the isomorphism α of the spaces L2(X) and
L2(Y ) can be naturally extended to an isomorphism αn of the spaces
L2(Xn) = L2(X×· · ·×X) and L2(Y n) = L2(Y ×· · ·×Y ). It is easy to
check that the isomorphism αn transforms symmetric functions into
symmetric functions and antisymmetric functions into antisymmetric
functions.
If B is a Hilbert space of states of one particle, then the space
of states of n identical particles Fn can be described as an nth
symmetric tensor power of the space B in the case of bosons and
an nth antisymmetric tensor power of the space B in the case of
fermions.
Let us define the operators of symmetrization Ps and
antisymmetrization Pa in the space Bn = L2(Xn) by means of the
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Quantum Mechanics of a System of Identical Particles 25
formulas
Psψ(ξ) =1
n!
∑π
ψ(π(ξ)),
Paψ(ξ) =1
n!
∑π
(−1)γπψ(π(ξ))
(here, ξ = (ξ1, . . . , ξn) ∈ Xn, π is a permutation (i1, . . . , in) of the
indices 1, . . . , n, with π(ξ) = (ξi1 , . . . , ξin) ∈ Xn, γπ is the parity of
the permutation π, and∑
π is a sum over all permutations π). In
particular, for n = 2,
Psψ(ξ1, ξ2) =1
2(ψ(ξ1, ξ2) + ψ(ξ2, ξ1)),
Paψ(ξ1, ξ2) =1
2(ψ(ξ1, ξ2)− ψ(ξ2, ξ1)).
It is clear that PsBn = F sn; PaBn = F an ; Psψ = ψ if ψ ∈ F sn, and
Paψ = ψ if ψ ∈ F an .
It is easy to check that Ps and Pa are orthogonal projections of
the space Bn onto the subspaces F sn and F an , respectively.
Let us denote by H an operator on Bn that commutes with
operators Ps and Pa. Then, F sn and F an are H-invariant subspaces.
If ψ ∈ Bn is an eigenfunction of the operator H, then Psψ and
Paψ (if they are non-zero) are also eigenfunctions of the operator
H that belong to subspaces F sn and F an , respectively, with the same
eigenvalue.
Let us consider a system of n non-interacting identical parti-
cles, i.e. a system having a Hamiltonian transforming the function
ψ(ξ1, . . . , ξn) into the function
ψ(ξ1, . . . , ξn) =n∑i=1
∫A(ξi, ηi)ψ(ξ1, . . . , ξi−1, ηi, ξi+1, . . . , ξn)dηi,
(3.1)
in particular, a single-particle Hamiltonian H1 transforms the
function ψ(ξ) into the function ψ(ξ) =∫A(ξ, η)ψ(η)dη (i.e.
March 26, 2020 13:43 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch03 page 26
26 Mathematical Foundations of Quantum Field Theory
the generalized function A(ξ, η) is a kernel of the operator H1).
Formula (3.1) specifies an operator Hn on the space Bn, commuting
with the operators Ps and Pa, hence this operator transforms the
subspaces F sn and F an into themselves.1 The operators on the spaces
F sn and F an , defined by relation (3.1), will be denoted by Hsn and Ha
n,
respectively. These operators are the Hamiltonians of systems of n
non-interacting bosons or fermions.
Let us assume, for definiteness, that the single-particle Hamilto-
nian H1 has discrete spectrum, and let us denote by φn the complete
orthonormal system of its eigenvectors, with corresponding eigen-
values En. Then it is obvious that the functions φk1(ξ1) . . . φkn(ξn)
constitute a complete orthonormal system of eigenvectors of the
operator Hn in the space Bn (see Section 2.3). It follows that
Psφk1(ξ1) . . . φkn(ξn) are eigenfunctions of the Hamiltonian Hsn and
belong to the space F sn. Similarly, Paφk1(ξ1) . . . φkn(ξn) are eigenvec-
tors of the Hamiltonian Han and belong to F an . The corresponding
eigenvalues, in both cases, are equal to Ek1 + · · ·+Ekn . In the case of
fermions, the functions Paφk1(ξ1) . . . φkn(ξn) should only be consid-
ered in the case of distinct indices k1, . . . , kn because in the case when
a pair of indices coincides, the antisymmetrization gives zero.2 It is
easy to check that the functions Psφk1(ξ1) . . . φkn(ξn) form a complete
system of functions in F sn and the functions Paφk1(ξ1) . . . φkn(ξn)
form a complete system in F an . These functions are not normalized,
however, one can check that two functions belonging to one of these
systems are either orthogonal or proportional. In the coordinate
representation, the space of states of one particle is realized as the
space L2(E3 × B), where B is a finite set, therefore the space of
states of n identical particles can be considered as a space of square-
integrable symmetric (antisymmetric) functions ψ(r1, i1, . . . , rn, in)
1More precisely, the definition of the operator Hn can be given as follows: Hn isa self-adjoint operator on Bn that transforms functions φ1(ξ1) . . . φn(ξn), wherethe functions φi belong to the domain of the operator H1, into the function∑ni=1 φ1(ξ1) . . . φi−1(ξi−1)(H1φi)(ξi)φi+1(ξi+1) . . . φn(ξn). Such an operator is
unique.2In other words, in the system of non-interacting identical fermions, two fermions
cannot be found in the same state. This statement is called the Pauli exclusionprinciple.
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Quantum Mechanics of a System of Identical Particles 27
depending on the coordinate variables r1, . . . , rn ∈ E3 and discrete
variables i1, . . . , in ∈ B (the functions ψ, in the case of bosons, are
invariant with respect to simultaneous permutation of coordinates rland rm and discrete variables il and im, while in the case of fermions,
this function changes sign under such permutation of variables).
The operator of total momentum Pn and the Hamiltonian Hn
of a system of n non-relativistic identical particles can be written
precisely in the same way as specified in Section 2.3 ((2.1) and (2.2));
the only difference is that the masses of identical particles are
equal and the potential energy Vn(r1, . . . , rn) should be a symmetric
function of the coordinates. We will always assume that
Vn(r1, . . . , rn) =
n∑j=1
V(rj) +∑
1≤j≤l≤nW(|rj − rl|)
(the first sum corresponds to the potential energy of particles in an
external field and the second sum corresponds to the interaction). It
is easy to see that the operators Pn and Hn transform symmetric
functions and antisymmetric functions into antisymmetric functions,
i.e. they can be considered as operators on the spaces F sn and F an .
The operators Pj = 1i∂∂rj
do not have this property; this means that
for the identical particles, we cannot speak of the momentum of the
jth particle.
In the case of non-interacting particles, the operator of potential
energy has the form Vn(r1, . . . , rn) = V(r1) + · · · + V(rn); it was
shown earlier that the stationary states of n particles, in this case,
can be expressed in terms of stationary states of a single particle.
3.2 Fock space
In Section 3.1, we fixed a measure space X. We have considered
Hilbert spaces F sn and F an of square-integrable symmetric and
antisymmetric functions ψ(ξ1, . . . , ξn), depending on n variables
ξ1, . . . , ξn ∈ X. It is often convenient to consider the direct sums
F s = F s0 + F s1 + · · ·+ F sn + · · · ,
F a = F a0 + F a1 + · · ·+ F an + · · · .
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28 Mathematical Foundations of Quantum Field Theory
Here, F s0 and F a0 are spaces of constant functions (i.e. one-
dimensional spaces).
The spaces F s and F a are called Fock spaces. The elements of
Fock spaces represent the states of a system of identical bosons
or fermions in the case when the number of particles is not fixed.
Speaking of bosons and fermions simultaneously, we will use the
notation F for one of the spaces F s or F a. Vectors in the space
F s or F a can be considered as sequences (f0, f1, . . . , fn, . . . ), where
fn ∈ F sn (or correspondingly fn ∈ F an ), satisfying the condition∑∞n=0 ‖fn‖2 < ∞. In other words, the elements of the space F can
be considered as column vectors
f =
f0
f1(ξ1)...
fn(ξ1, . . . , ξn)...
of symmetric (antisymmetric) functions fn(ξ1, . . . , ξn) as entries; they
should satisfy the condition3
∞∑n=0
∫|fn(ξ1, . . . , ξn)|2dξ1 . . . dξn <∞.
Linear combination and scalar product of two such column vectors
are defined in the natural way, in particular, the scalar product of f
and g is
〈f, g〉 =
∞∑n=0
∫fn(ξ1, . . . , ξn)gn(ξ1, . . . , ξn)dξ1 . . . dξn.
The spaces Fn are naturally embedded in the Fock space F (to the
vector f ∈ Fn, there corresponds a sequence (f0, . . . , fn, . . . ) ∈ F ,
where fn = f, fk = 0 for k 6= n).
Let us introduce the operator of the number of particles
N = N0 +N1 + · · ·+Nn + · · · ,
3Such column vectors are called Fock states.
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Quantum Mechanics of a System of Identical Particles 29
where Nn is an operator on the space Fn, multiplying every vector
by the number of particles n. The spaces Fn are eigenspaces of the
operator N .
The operator of total momentum P and the Hamiltonian of the
system of non-relativistic particles H in the Fock space are defined
as direct sums of operators Pn and Hn introduced in Section 3.1
P = P0 + P1 + · · ·+ Pn + · · · ,
H = H0 +H1 + · · ·+Hn + · · ·
(the operators P and H act on Fock space, constructed starting with
the measure space E3 ×B).
In non-relativistic quantum mechanics, the operators of all observ-
ables leave the spaces Fn ⊂ F invariant (i.e. they commute with
the operator of the number of particles N). However, even in the
non-relativistic case, it is very convenient to express the operators
of physical quantities in terms of auxiliary operators that do not
commute with N . Namely, we will introduce the operators a(f) and
a+(f) that can be considered as the operators of annihilation and
creation of a particle with the wave function f ∈ L2(X).
First of all, let us define the operator an(f), where f ∈ L2(X),
acting on the space Fn into the space Fn−1. Namely, we will assume
that the operator an(f) transforms a function φ(ξ1, . . . , ξn) ∈ Fn into
the function φ(ξ1, . . . , ξn−1) ∈ Fn−1, defined by the formula
φ(ξ1, . . . , ξn−1) =√n
∫φ(ξ1, . . . , ξn−1, ξ)f(ξ)dξ
(the operator a0(f) transforms the space F0 into zero).
The operator a+n (f), defined on the space Fn−1 and taking values
in the space Fn, can be described as the operator adjoint to an(f). It
is easy to calculate that the operator a+n (f) transforms the function
φ(ξ1, . . . , ξn−1) into the function√nP s(φ(ξ1, . . . , ξn−1)f(ξn)) in the
bosonic case and into the function√nP a(φ(ξ1, . . . , ξn−1)f(ξn)) in the
fermionic case.
Let us consider a subset D of the space F , consisting of finite
sequences (φ0, . . . , φn, . . . ) (in other words, D is the smallest linear
subspace that contains all subspaces Fn). The operators a(f) and
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30 Mathematical Foundations of Quantum Field Theory
a+(f), where f ∈ L2(X), will be defined on the space D, namely,
a(f) transforms the sequence (φ0, . . . , φn, . . . ) ∈ D into the sequence
(a1(f)φ1, a2(f)φ2, . . . , an+1(f)φn+1, . . . )
and the operator a+(f), transforms the same sequence into
(0, a∗1(f)φ0, . . . , a∗n(f)φn−1, . . . ).
It is obvious that these operators transform the subspace D into
itself and that they are Hermitian conjugate on D.
It is easy to check (see Section 6.1) that in the case of fermions, the
operators a(f), a+(f) are bounded (‖a(f)‖ = ‖f‖) and therefore can
be extended by continuity to the whole Fock space F ; however, in the
bosonic case, the operators a(f), a+(f) are not bounded (‖an(f)‖ =√n‖f‖).By means of straightforward calculations, it is easy to find the
commutation relations for the operators a(f), a+(f). In the bosonic
case, the operators satisfy the conditions
[a(f), a(g)] = [a+(f), a+(g)] = 0; [a(f), a+(g)] = 〈f, g〉 (3.2)
(these relations are called the canonical commutation relations or
CCR). In the fermionic case, we have similar conditions for the
anticommutators
[a(f), a(g)]+ = [a+(f), a+(g)]+ = 0; [a(f), a+(g)]+ = 〈f, g〉 (3.3)
(these relations are called the canonical anticommutation relations
or CAR).
The operators a(f) depend linearly on f ∈ L2(X). This means
that they can be considered as operator generalized functions on X.
In other words, we can introduce the symbols a(x), a+(x), that are
related to the operators a(f), a+(f) by the following formulas:
a(f) =
∫f(x)a(x)dx,
a+(f) =
∫f(x)a+(x)dx.
March 26, 2020 13:43 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch03 page 31
Quantum Mechanics of a System of Identical Particles 31
The symbols a(x), a+(x) are not operators. The integral∫f(x)
a(x)dx is considered simply as another notation for the opera-
tors a(f).
Note, however, that in principle, the symbol a(x) can be consid-
ered as an operator on some subset of the space F and the symbol
a+(x) can be considered as an operator acting from the space F into
some larger space (see Appendix A.7).
In what follows, we will consider the expression of the form
A =
∫f(x1, . . . , xm|y1, . . . , yn)a+(x1) . . . a
+(xm)a(y1) . . .
× a(yn)dmxdny, (3.4)
where f is a generalized function. Clearly, such expressions define an
operator on D if the function f takes the form
f1(x1) . . . fm(xm)g1(y1) . . . gn(yn), (3.5)
where fi, gi ∈ L2(X) (then one should assume that A = a+(f1) . . .
a+(fm)a(g1) . . . a(gn)). This definition can be extended to the case
when the function f is a finite sum of products of the form (3.5).
One can prove easily that under this condition the operator A
transforms a sequence φ = (φ0, . . . , φk, . . . ) ∈ D, into the sequence
ψ = (ψ0, . . . , ψk, . . . ) ∈ D, where
ψk(ξ1, . . . , ξk) =
√k!
(k −m)!· (k −m+ n)!
(k −m)!
×P∫φn−m+k(ξ1, . . . , ξk−m, x1, . . . , xn)
× f(ξk−m+1, . . . , ξk|x1, . . . , xn)dnx (3.6)
(P denotes the operator of symmetrization in the bosonic case and
antisymmetrization in the fermionic case).
We can use the formula (3.6) to define the operator A for an
arbitrary function f . The domain of the operator A, in this case,
consists of all sequences φ ∈ D, for which the functions ψk(ξ1, . . . , ξk),
obtained by the formula (3.6), are square-integrable. Considering
March 26, 2020 13:43 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch03 page 32
32 Mathematical Foundations of Quantum Field Theory
expressions of the form
A =
∞∑m,n
∫Am,n(x1, . . . , xm|y1, . . . , yn)a+(x1)
. . . a+(xm)a(y1) . . . a(yn)dmxdny, (3.7)
we define every summand using the consideration above and the
sum of the series we understand in terms of strong convergence.
The representation of the operator A in the form (3.7) is called the
representation in normal form (in the expression (3.7), the creation
operators stand to the left of the operators of annihilation).
Every bounded operator can be represented in the form (3.7) if
the convergence of the series is understood as weak convergence.
The vector θ, specified by the sequence (1, 0, 0, . . . ), plays a special
role in the Fock space F . This vector corresponds to the state that
contains no particles (θ ∈ F0) and is called the vacuum vector ; it
satisfies the condition a(f)θ = 0 for all functions f ∈ L2(X). It is easy
to check that the vector θ ∈ F is a cyclic vector with respect to the
family of operators a+(f) (in other words, linear combinations of the
vectors of the form a+(f1) . . . a+(fs)θ are dense in F ). One can write
down an explicit expression of every vector φ = (φ0, . . . , φn, . . . ) ∈ Fin terms of the generalized operator functions a+(x) and the vector
θ, namely, by applying the formula (3.6), we see that
f =∞∑n=0
1√n!
∫fn(ξ1, . . . , ξn)a+(ξ1) . . . a
+(ξn)θdnξ. (3.8)
Let us consider the simplest operators on the Fock space. Let us
start with operators of the form∫A(x, y)a+(x)a(y)dxdy. (3.9)
The operator (3.9) transforms the sequence (φ0, . . . , φk, . . . ) ∈ D into
the sequence (ψ0, . . . , ψk, . . . ), where
ψk(ξ1, . . . , ξk) = kP
∫φk(ξ1, . . . , ξk−1, x)A(ξk, x)dx. (3.10)
March 26, 2020 13:43 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch03 page 33
Quantum Mechanics of a System of Identical Particles 33
It is easy to check that in the case when the function A(x, y) is
a kernel of a self-adjoint operator on L2(X), the operator (3.9) is
essentially self-adjoint on F . Let us denote by A the self-adjoint
extension of the operator∫A(x, y)a+(x)a(y)dxdy. The operator A
commutes with the operator N , hence, it transforms the space Fninto itself. The operator induced by the operator A on the space Fnwill be denoted by An. Operators of the form An were considered
earlier, namely, the Hamiltonian of a system of n non-interacting
identical particles can be written in this form. In Section 3.1, we
have found eigenvectors and eigenvalues of the operator An in the
assumption that the operator A1 is a self-adjoint operator with
discrete spectrum in L2(X). Using this result, we can obtain the
description of eigenvectors and eigenvalues of the self-adjoint oper-
ator A. Namely, the vectors∏∞i=1(a
+(φi))niθ constitute a complete
system of eigenvectors of the operator A and the corresponding eigen-
values are equal to∑∞
i=1 niEi (here, φ1, . . . , φi, . . . is the complete
system of eigenfunctions of the operator A1 and E1, . . . , En, . . . are
the corresponding eigenvalues; the numbers ni constitute a finite
sequence. In the fermionic case, the numbers ni are equal to 0 or 1 and
in the bosonic case n = 0, 1, 2, . . . ). The proof can be reduced to the
remark that the vector a+(f1) . . . a+(fn)θ is equal to the vector that
corresponds in Fock space to the function Pf1(ξ1) . . . fn(ξn)θ ∈ Fn,
up to a constant factor (recall that Fn is naturally embedded in F ).
The operators
B =
∫B(x1, x2|y1, y2)a+(x1)a
+(x2)a(y1)a(y2)dx1dx2dy1dy2
(3.11)
also commute with the operator of the number of particles (as well as
the operators of the form (3.4), obeying m = n). This means that the
operator B transforms the subspace Fn into itself and it follows from
formula (3.6) that after restricting the operator B to the subspace
Fn, we obtain an operator Bn transforming the vector φ into function
ψk(ξ1, . . . , ξk) = k(k − 1)P
∫φk(ξ1, . . . , ξk−2, x1, x2)
×B(ξk−1, ξk|x1, x2)dx1dx2. (3.12)
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34 Mathematical Foundations of Quantum Field Theory
Now we can write down the operators N,P, and H in the normal
form.
The operator N can be represented as
N =
∫a+(ξ)a(ξ)dξ (3.13)
(or, more precisely, the operator N can be expressed by the for-
mula (3.9), where A(ξ, η) = δ(ξ, η)). This follows immediately from
the relation (3.10), which implies that the operator N transforms the
sequence (φ0, . . . , φk, . . . ) ∈ D into the sequence (ψ0, . . . , ψk, . . . ),
where
ψk(ξ1, . . . , ξk) = kP
∫φk(ξ1, . . . , ξk−1, x)δ(ξk, x)dx
= kPφk(ξ1, . . . , ξk) = kφk(ξ1, . . . , ξk).
The momentum operator P and the Hamiltonian H of a system
of non-relativistic identical particles act on Fock space, constructed
starting from the measure space E3 × B. The operators acting on
this space can be expressed in terms of the operator generalized
functions a+(x, s), a(x, s) in the coordinate representation and in
terms of the operator generalized functions a+(k, s), a(k, s) in the
momentum representation (here, x,k ∈ E3, s ∈ B). In the following,
we will sometimes use the notation a+s (k) = a+(k, s), as(k) = a(k, s).
Again, using the relation (3.10), it is easy to derive that in the
coordinate representation, the operator P is
P =∑s
1
i
∫a+(x, s)
∂
∂xa(x, s)dx, (3.14)
or, more precisely,
P =∑s,s′
∫A(x, s,y, s′)a+(x, s)a(x, s′)dxdy,
where A(x, s,y, s′) = δs′s
1i∂∂xδ(x− y). In the momentum representa-
tion, the operator P takes the form
P =∑s
∫ka+(k, s)a(k, s)dk. (3.15)
March 26, 2020 13:43 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch03 page 35
Quantum Mechanics of a System of Identical Particles 35
It follows from the relations (3.10) and (3.12) that the Hamiltonian H
of the system of non-relativistic identical particles can be represented
as a sum of operators of the form (3.9) and (3.11). Namely, in the
coordinate representation,
H =∑s
∫a+(x, s)
(− ∆
2m
)a(x, s)dx
+∑s
V(x)a+(x, s)a(x, s)dx +∑s
1
2
∫W(|x1 − x2|)a+(x1, s)
× a+(x2, s)a(x2, s)a(x1, s)dx1dx2, (3.16)
and in the momentum representation,
H =∑s
∫k2
2ma+(k, s)a(k, s)dk
+∑s
V(k1 − k2)a+(k1, s)a(k2, s)dk1dk2
+∑s
1
2
∫W(k1 − k4)δ(k1 + k2 − k3 − k4)
× a+(k1, s)a+(k2, s)a(k3, s)a(k4, s)d
4k, (3.17)
where V(k), W(k) are the Fourier transforms of the functions
V(x),W(x).
It is easy to construct examples of expressions of the form (3.7)
that do not define an operator in Fock space. For example, the
expression
A =
∫α(x1, . . . , xm)a+(x1) . . . a
+(xm)dx1 . . . dxm (3.18)
can define an operator on Fock space only in the case when the
function α is square integrable. Namely, by the general definition,
the operator defined by the expression (3.8) should transform the
sequence (φ0, . . . , φn, . . . ) ∈ D into the sequence (ψ0, . . . , ψn, . . . ),
March 26, 2020 13:43 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch03 page 36
36 Mathematical Foundations of Quantum Field Theory
where
ψn(ξ1, . . . , ξn) =
√n!
(n−m)!Pφn−m(ξ1, . . . , ξn−m)α(ξn−m+1, . . . , ξn).
If the function α is square integrable, then the functions ψn are also
square integrable, hence the expression (3.18) specifies an operator
A on the whole space D and this operator transforms D into
itself. However, if the function α is not square integrable, then
the function ψn may be square integrable only under the condition
φn−m ≡ 0. This means that the domain of the operator specified by
the expression (3.18) contains only the zero vector; in other words,
the expression (3.18) does not specify any operator because the
domain of the operator should be dense in Fock space.
Let us consider the Fock space F (L2(E3)), constructed starting
with the measure space E3, and operators defined by expressions of
the form
A =
m+n≤s∑m,n
∫Am,n(k1, . . . ,km|l1, . . . , ln)
× δ(k1 + · · ·+ km − l1 − · · · − ln)
× a+(k1) . . . a+(km)a(l1) . . . a(ln)dmkdnl (3.19)
(operators of this form commute with the momentum operator). Let
us suppose that the functions Am,n belong to the space S(E3(m+n))
of smooth, rapidly decreasing functions (faster than any power
function). Let us single out the subspace S∞ ⊂ F that consists
of sequences (φ0, . . . , φk, . . . ) ∈ D, obeying φk ∈ S(E3k). In what
follows, it will be convenient to consider operators on the space
F (L2(E3)) only on the set S∞. By means of the relation (3.6), it is
easy to check that in the case when Am,0 ≡ 0, the operator specified
by the expression (3.19) is well defined on all elements of the set
S∞. This operator transforms every sequence in S∞ into a sequence
belonging to the same set. If one of the functions Am,0 is non-zero,
then the expression (3.19) cannot define an operator on Fock space
because the function Am,0(k1, . . . ,km)δ(k1 + · · ·+km) is not square
integrable.
March 26, 2020 13:43 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch03 page 37
Quantum Mechanics of a System of Identical Particles 37
Let us consider the fermionic case. In this case, we can introduce
the notion of even vectors as vectors belonging to a direct sum
of subspaces F2n and odd vectors as vectors belonging to a direct
sum of subspaces F2n+1. An operator on Fock space is called
parity-preserving if it transforms an even vector to an even vector
and an odd vector to an odd vector. Introducing an involution τ ,
transforming a sequence (φ0, φ1, . . . , φ2n, φ2n+1, . . . ) into a sequence
(φ0,−φ1, . . . , φ2n,−φ2n+1, . . . ), we can say that an even vector is
invariant with respect to this involution, τx = x, and an odd
vector satisfies the condition τx = −x. A parity-preserving operator
commutes with this involution. An operator represented in the
form (3.7) (normal form) is parity preserving if the functions Am,ndo not vanish only in the case when the numbers m and n have the
same parity.
Operators corresponding to physical quantities should be parity
preserving. In particular, the Hamiltonian is always a parity-
preserving operator.
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March 26, 2020 14:5 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch04 page 39
Chapter 4
Operators of Time EvolutionS(t, t0) and Sα(t, t0)
4.1 Non-stationary perturbation theory
Let us consider a Hamiltonian H(t) having the form H(t) = H0 +
gV (t). Let us assume that we can calculate the operator of evolution
U0(t, t0) = exp(−iH0(t − t0)) corresponding to the Hamiltonian
H0. We will solve the problem of finding the evolution operator
U(t, t0) from the Hamiltonian H(t) as a series with respect to the
parameter g. It is convenient to introduce the operator S(t, t0) =
exp(iH0t)U(t, t0) exp(−iH0t0), expressed in terms of the operator
U(t, t0), and calculate it. We will find out later that the operator
S(t, t0) is very important on its own. In the case when we should
emphasize that the operators U(t, t0) and S(t, t0) depend on the
parameter g, we will use the notation U(t, t0|g) and S(t, t0|g).
The operator U(t, t0) satisfies equation (1.1). Using this equation,
we can easily obtain the following equation for the operator S(t, t0):
i∂S(t, t0)
∂t= gV (t)S(t, t0), (4.1)
where V (t) = exp(iH0t)V (t) exp(−iH0t); the initial condition is
specified in the form S(t0, t0) = 1.
In some physical situations, the operator H0 can be considered as
a free Hamiltonian (i.e. it describes non-interacting particles), and
the operator H − H0 = gV (t) is the interaction Hamiltonian. This
terminology is also commonly used in quantum field theory. However,
39
March 26, 2020 14:5 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch04 page 40
40 Mathematical Foundations of Quantum Field Theory
in quantum field theory, as we will see later, there exists no natural
partition of the Hamiltonian into the free and interaction parts. We
will not use these terms in order to avoid misleading associations.
We will express S(t, t0) as a power series in the variable g as
follows:
S(t, t0) =∞∑n=0
gnSn(t, t0).
Substituting this series into the equation for the operator S(t, t0), we
obtain the recurrence relation
i∂Sn(t, t0)
∂t= V (t)Sn−1(t, t0).
Using the initial condition S(t0, t0) = 1, we see that S0(t0, t0) = 1
and Sn(t0, t0) = 0 for n ≥ 1, hence
Sn(t, t0) =1
i
∫ t
t0
V (τ)Sn−1(τ, t0)dτ. (4.2)
The differential equation for S with an initial condition is equivalent
to the integral equation
S(t, t0) = 1 + g
∫ t
t0
V (τ)S(τ, t0)dτ,
one can solve the integral equation using the method of iterations.
From (4.2) or from the integral equation, we can conclude that
S1(t, t0) =1
i
∫ t
t0
V (τ)dτ,
S2(t, t0) =
(1
i
)2 ∫ t
t0
dτ1
∫ τ1
t0
dτ2V (τ1)V (τ2),
...
Sn(t, t0) =
(1
i
)n ∫ t
t0
dτ1
∫ τ1
t0
dτ2· · ·∫ τn−1
t0
dτnV (τ1) . . . V (τn).
March 26, 2020 14:5 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch04 page 41
Operators of Time Evolution S(t, t0) and Sα(t, t0) 41
We can also write
Sn(t, t0) =
(1
i
)n ∫Γn
dτ1 . . . dτnV (τ1) . . . V (τn),
where the domain of integration Γn is defined by t ≥ τ1 ≥ · · · ≥τn≥ t0.
We introduce the following notation:
T (V (τ1) . . . V (τn)) = V (τi1) . . . V (τin),
where i1, . . . , in is a permutation of the indices 1, . . . , n, satisfying
the relation τi1 ≥ · · · ≥ τin . In other words, T (V (τ1) . . . V (τn))
(chronological or a T-product of the operators V (τ1) . . . V (τn)) is
defined as the product of the operators V (τ1), . . . , V (τn) in order
of decreasing time τi. If some of the times τi are identical, then the
permutation for which τi1 ≥ · · · ≥ τin is not unique, however, it is
easy to see that the T -product does not depend on this choice of
permutation. Using the T -product, we can write Sn(t, t0) in the form
Sn(t, t0) =1
n!
(1
i
)n ∫ t
t0
· · ·∫ t
t0
T (V (τ1) . . . V (τn))dτ1 . . . dτn.
Indeed, the domain of integration in this integral can be partitioned
into n! regions GP , with each region GP corresponding to a permu-
tation P = (j1, . . . , jn) of the indices 1, . . . , n. Here, GP is the region
singled out by the inequalities t ≥ τj1 ≥ · · · ≥ τjn ≥ t0. It is clear
that (1
i
)n ∫GP
T (V (τ1) . . . V (τn))dτ1 . . . dτn
=
(1
i
)n ∫GP
V (τj1) . . . V (τjn)dτ1 . . . dτn,
and that this integral, after a change of variables, is reduced to
the integral Sn(t, t0). This means, that the integral 1n!(
1i )n∫ tt0. . .∫ t
t0T (V (τ1) . . . V (τn))dτ1 . . . dτn splits into n! identical integrals,
equal to Sn(t, t0); this gives the needed formula. The full operator
March 26, 2020 14:5 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch04 page 42
42 Mathematical Foundations of Quantum Field Theory
S(t, t0) can be written as the series
S(t, t0) =∞∑n=0
1
n!
(1
i
)ngn∫ t
t0
· · ·∫ t
t0
T (V (τ1) . . . V (τn))dτ1 . . . dτn.
(4.3)
This series can be compactly written in the form
S(t, t0) = T exp
(1
ig
∫ t
t0
V (τ)dτ
), (4.4)
which is called T-exponential. If the operator V (t) is bounded for
every t and is continuous (in the sense of strong limits) in t, then the
series (4.3) is convergent in norm (hence, in this case, the existence
of solutions of equations (1.1) and (4.1) follows). Indeed, we have
that supt0≤τ≤t ‖V (τ)‖ = supt0≤τ≤t ‖V (τ)‖ < ∞ and ‖Sn(t, t0)‖ ≤1n!(supt0≤τ≤t ‖V (τ)‖)n|t− t0|n. In the case when it is not possible to
prove the convergence of the series (4.3), the corresponding results
are conditional (if a series expansion in g is possible, then the series
takes the form (4.3)). Regardless of the possibility of the series
expansion in g, one can prove the following relation:
∂nS(t, t0|g)
∂gn|g=0 =
(1
i
)n ∫ n
t0
· · ·∫ t
t0
T (V (τ1) . . . V (τn))dτ1 . . . dτn.
If the HamiltonianH can be represented in the formH0+V , where
H0 and V do not depend on time, then it is convenient to consider the
Hamiltonian Hα(t) = H0 + exp(−α|t|)V . We introduce the operator
Sα(t, t0) = exp(iH0t)Uα(t, t0) exp(−iH0t0), where Uα(t, t0) is the
operator of evolution, constructed with Hα(t).
In a similar way, we can define the operator Sα(t, t0) as the
solution of the differential equation
i∂Sα(t, t0)
∂t= exp(−α|t|)V (t)Sα(t, t0)
with the initial condition Sα(t0, t0) = 1 (here, V (t) = exp(iH0t)
V exp(−iH0t)). Following the discussion above, we can write Sα(t, t0)
March 26, 2020 14:5 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch04 page 43
Operators of Time Evolution S(t, t0) and Sα(t, t0) 43
in the form of the T -exponent:
Sα(t, t0) = T exp
(1
i
∫ t
t0
exp(−α|τ |)V (τ)dτ
).
The operator Sα(∞,−∞) = slim t→∞t0→−∞
Sα(t, t0), denoted by Sα,
plays a special role and is called the adiabatic S-matrix. Two other
important operators Sα(0,±∞) = slimt→±∞ Sα(0, t) will be denoted
Sα+, and Sα− and are called the adiabatic Møller operators.
In the case when H0 can be considered a free Hamiltonian
and V is the interaction, considering the Hamiltonian Hα(t) with
α → 0 describes adiabatic (infinitesimally slow) turning on and
off the interaction; this explains the above-introduced terminology.
In place of the Hamiltonian H = H0 + V , it is often useful
to consider the family of Hamiltonians Hg = H0 + gV , where
g is a constant, which is typically called the coupling constant.
The operators Sα(t, t0), Sα(∞,−∞), Sα(0,±∞), constructed with
the Hamiltonian Hg, will be denoted by Sα(t, t0|g), Sα(∞,−∞|g) =
Sα(g), and Sα(0,±∞|g) = Sα±(g).
4.2 Stationary states of Hamiltonians depending
on a parameter
Let us consider the family of Hamiltonians H(g), where 0 ≤ g ≤ g0.
We will assume that for every g, there exists a normalized eigenvector
φg of the operator H(g), with eigenvalue E(g), which is differentiable
with respect to the parameter g. Without loss of generality, we can
assume that ⟨φg,
dφgdg
⟩= 0.
Indeed, if this condition is not satisfied, one can always find a phase
factor exp(iα(g)), such that the vector φg = exp(−iα(g))φg satisfies
the condition: namely, we can choose α(g) = 1i
∫ g0
⟨φλ,
dφλdλ
⟩dλ (the
function α(g) is real, since from the relation 〈φλ, φλ〉 = 1, one can
March 26, 2020 14:5 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch04 page 44
44 Mathematical Foundations of Quantum Field Theory
obtain that⟨φλ,
dφλdλ
⟩+
⟨dφλdλ
, φλ
⟩=
⟨φλ,
dφλdλ
⟩+
⟨φλ,
dφλdλ
⟩= 0).
By differentiating in g the relation
H(g)φg = E(g)φg,
we obtain
(H(g)− E(g))dφgdg
= −dH(g)
dgφg +
dE(g)
dgφg. (4.5)
Taking the scalar product of this relation with φg, we obtain
dE(g)
dg=
⟨dH(g)
dgφg, φg
⟩. (4.6)
Let us consider the commonly occurring case of H(g) = H0 + gV ;
then, it is clear that
dE(g)
dg= 〈V φg, φg〉 , (4.7)
(H0 + gV − E(g))dφgdg
= (〈V φg, φg〉 − V )φg. (4.8)
We will sometimes assume that the vector φg is analytic with respect
to g in the neighborhood around g = 0, and we will search for an
expansion of the vector φg and the eigenvalue E(g) in a series in
g (stationary perturbation theory). Formulas (4.7) and (4.8), when
g = 0, provide the linear terms in g (first-order terms), namely,
E(g) = E(0) + g 〈V φ0, φ0〉+ · · · ,
φg = φ0 + gψ + · · · ,
where ψ satisfies the relation
(H0 − E(0))ψ = (〈V φ0, φ0〉 − V )φ0. (4.9)
Given that the vector φg satisfies the equation⟨φg,
dφgdg
⟩= 0, we
obtain the following condition on the vector ψ:
〈ψ, φ0〉 = 0. (4.10)
April 6, 2020 15:51 Mathematical Foundations of Quantum Field Theory 9in x 6in 11222-ch04 page 45
Operators of Time Evolution S(t, t0) and Sα(t, t0) 45
In the case when E(0) is a simple eigenvalue of the Hamiltonian H0,
then equations (4.9) and (4.10) give an unambiguous definition of the
vector ψ. The derivation of the relations (4.7) and (4.8) is rigorous
if all the operators H0 + gV have the same domain.
We did not consider the more subtle question of the existence of
eigenvectors φg that depend on the parameter g smoothly. With mild
assumptions, the answer to this question is given by the following
theorem [Kato, 2013].
Let H0 denote a self-adjoint operator and V a Hermitian operator
with domain that contains the domain of H0. Let us suppose that
E0 is a simple isolated eigenvalue of the operator H0 and φ0 is
the corresponding eigenvector. Then for sufficiently small g: (1) the
operator H(g) = H0 + gV is a self-adjoint operator with the
same domain as the operator H0; (2) there exists an eigenvector
φg of the operator H(g), which is analytic in g and is equal to
φ0 for g = 0; (3) the corresponding eigenvalue E(g) also depends
analytically on g and is simple; (4) the relations (4.7) and (4.8) are
valid.
We do not wish to give a direct calculation of the higher terms
with respect to g in the series for φg and E(g). Instead, we will prove
the formula that allows us to get the decomposition of eigenvectors
and eigenvalues in a series in g from non-stationary perturbation
theory. Let us suppose that for 0 ≤ g ≤ g0, the self-adjoint operators
H(g) = H0 + gV have the same domain and E(g) is an isolated
eigenvalue of the operator H(g) that depends continuously on the
parameter g in the interval 0 ≤ g ≤ g0, and φg is the eigenvector
corresponding to this eigenvalue, obeying⟨φg,
dφgdg
⟩= 0. Then
φg = lima→0
exp
(iC(g)
α
)Sα(0,−∞|g)φ0,
φg = lima→0
exp
(iC(g)
α
)Sα(0,+∞|g)φ0,
(4.11)
where C(g) =∫ g
0E(λ)−E(0)
λ dλ and Sα(0,∓∞|g) are adiabatic Møller
matrices, constructed with the pair of operators H(g), H0. This
statement is proven in Section 4.3.
March 26, 2020 14:5 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch04 page 46
46 Mathematical Foundations of Quantum Field Theory
4.3 Adiabatic variation of stationary state
Let us suppose that the Hamiltonian H(t) varies slowly (adiabati-
cally) with time.
Let us consider a solution of the Schrodinger equation (1.2)
with initial condition ψt0 = φ0, where φ0 is a stationary state of
the Hamiltonian H(t0) corresponding to a non-degenerate energy
level. We will show that for every t this solution is very close
to the stationary state of the Hamiltonian H(t). (When we are
talking about the stationary state of Hamiltonian H(t), we always
assume that we consider an eigenvector of H(t) for a fixed t.) For
definiteness, we restrict ourselves to the case when we consider a
family of Hamiltonians Hg that depend on the parameter g, and the
Hamiltonian H(t) is defined by the formula H(t) = Hαt, where α is
a small positive number. Then equation (1.2) can be reduced to the
equation
iαdψ(g)
dg= Hgψ(g), (4.12)
by means of the change of variables g = αt. In what follows, we will
consider equation (4.12).
Let us assume that Hg is a family of self-adjoint operators with
the same domain D, smoothly depending on the parameter g in the
interval g0 ≤ g ≤ g1 (i.e. for any x ∈ D, the vector Hgx smoothly
depends on g). We will assume that for every g in the interval
g0 ≤ g ≤ g1, there exists a stationary state φg of the Hamiltonian
Hg, smoothly depending on the parameter g. The energy level E(g),
corresponding to the state φg, will be supposed isolated and non-
degenerate. The state φg is assumed to be normalized and obeying
〈φg, dφgdg 〉 = 0. Finally, we will assume thatdnφgdgn |g=g0 = 0 for
n = 1, 2, . . . .
Lemma 4.1. Assuming the conditions above, the solution of equa-
tion (4.12) that coincides with φg0 with g = g0 can be written in the
form
ψ(g) = exp
[− iαC(g)
](φg + αsg + α2r(g, α)),
April 6, 2020 15:51 Mathematical Foundations of Quantum Field Theory 9in x 6in 11222-ch04 page 47
Operators of Time Evolution S(t, t0) and Sα(t, t0) 47
where C(g) =∫ g
0 E(λ)dλ, sg is defined by the relation
idφgdg
= (Hg − E(g))sg,
⟨dsgdg
, φg
⟩= 0, sg0 = 0,
and the norm of the vector r(g, α) is bounded above by a constant that
does not depend on g or α (here, g lies in the interval g0 ≤ g ≤ g1).
To prove this lemma, let us first use the change of variables ψ(g) =
exp( iαC(g))σ(g) to transform equation (4.12) to the form
iαdσ(g)
dg= (Hg − E(g))σ(g). (4.13)
Let us assume that the solution of equation (4.13) has the form
σ(g) =
∞∑n=0
αnσn(g). (4.14)
By comparing terms with equal powers of α, we obtain the relations
(Hg − E(g))σ0(g) = 0, (4.15)
idσ0(g)
dg= (Hg − E(g))σ1(g),
...
idσn−1(g)
dg= (Hg − E(g))σn(g), (4.16)
σn(g0) = 0 for n ≥ 1. (4.17)
Furthermore, taking the scalar product of these relations with φg, we
obtain ⟨dσ0(g)
dg, φg
⟩= 0, (4.18)⟨
dσ1(g)
dg, φg
⟩= 0, (4.19)
...⟨dσn(g)
dg, φg
⟩= 0. (4.20)
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48 Mathematical Foundations of Quantum Field Theory
Equation (4.15) and relation (4.18) are satisfied for σ0(g) = φg. It
then follows that σ1(g) can be found from relations (4.16) and (4.19);
we also obtain that σ1(g) = sg. By the recursive application of
relations (4.17) and (4.20), we can find σn(g). Note that the σn(g)
are specified by these relations uniquely (this follows from the fact
that the eigenvalue E(g) is non-degenerate and isolated) and are
smooth in g. From equationdnφgdgn |g=g0 = 0, it follows that all the
above relations can be satisfied.
Let us show now that
σ(g) =
N∑n=0
αnσn(g) + αN+1rN (g, α), (4.21)
where |rN (g, α)| ≤ K. Indeed, by inserting (4.21) into (4.14), we
obtain the following equations for rN (g, α):
iα∂rN (g, α)
∂g= (Hg − E(g))rN (g, α)− idσN (g)
dg(4.22)
with initial condition rN (g0, α) = 0. Let us denote by V (g) the
unitary operator defined by the relation
iαdV (g)
dg= (Hg − E(g))V (g),
V (g0) = 1.
Now, we can make a change of variables rN (g, α) = V (g)ρN (g, α) in
the equation (4.22). We obtain
iα∂ρN (g, α)
∂g= −iV −1(g)
dσN (g)
dg,
hence
‖rN (g, α)‖ = ‖ρN (g, α)‖ ≤∫ g
g0
∥∥∥∥dρN (g′, α)
dg′
∥∥∥∥ dg′≤ 1
α
∫ g
g0
∥∥∥∥dσN (g′)
dg′
∥∥∥∥ dg′ ≤ const
α. (4.23)
April 6, 2020 15:51 Mathematical Foundations of Quantum Field Theory 9in x 6in 11222-ch04 page 49
Operators of Time Evolution S(t, t0) and Sα(t, t0) 49
We can now see that
rN (g, α) = σN (g) + αrN+1(g, α).
Applying inequality (4.23) to rN+1, we can derive the following
inequality for rN :
‖rN (g, α)‖ ≤ ‖σN (g)‖+ αconst
α≤ const.
Since r(g, α) = r1(g, α), the above estimate with N = 1 confirms
Lemma 4.1.
Remark 4.1. In the above discussion, the condition of smooth
dependence of Hg and φg on g can be relaxed to the requirement
of three-fold differentiability; the conditiondnφgdgn |g=g0 = 0 can be
relaxed to hold only for n = 1.
Remark 4.2. If we do not require thatdφgdg |g=g0 = 0, then it is
clear, from the above proof, that the proof of Lemma 4.1 applies
to the solution of equation (4.12), satisfying the initial condition
ψg0 = φg0 + αsg0 .
We will now show how to obtain the relation (4.11) using
Lemma 4.1. Let us assume that the family of Hamiltonians Hg satisfy
the conditions of Lemma 4.1, with the exception of the requirement
thatdnφgdgn |g=g0 = 0 for n ≥ 1. Let us consider the operator of evolution
Uα(t, T ) constructed with the Hamiltonian Hα(t) = Hh(αt), where
h(τ) is a smooth function defined on τ ≤ 0 and taking values in the
interval [g0, g1]. Let us further assume that h(τ) = g0 for τ ≤ a.
Then the family of Hamiltonians Hλ = Hh(λ) with λ in [a, 0] and the
vectors φλ = φh(λ) satisfy the conditions of Lemma 4.1, including the
condition dnφλdλn |λ=a = 0 (the latter condition follows from the relation
dnh(λ)dλn |λ=a = 0). Applying the statement of this lemma, we see
that ∥∥∥∥Uα (0,a
α
)φα − exp
(− 1
αC
)φ0
∥∥∥∥→ 0,
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50 Mathematical Foundations of Quantum Field Theory
where C =∫ 0a E(h(λ))dλ. We obtain
φh(0) = φ0 = lima→0
exp
(i
αC
)Uα
(0,a
α
)φh(a). (4.24)
Introducing the operator
Sα(t, T ) = exp(iHg0t)Uα(t, T ) exp(−iHg0T ),
we can rewrite the equation (4.14) in the form
φh(0) = lima→0
exp
i
α
∫ 0
a[E(h(λ))− E(h(a))]dλ
Sα
(0,a
α
)φh(a).
(4.25)
Since Sα( aα , T ) = 1 for T < aα and h(λ) = h(a) for λ < a, using the
relation (4.25), we can show that
φh(0) = lima→0
exp
i
α
∫ 0
−∞[E(h(λ))− E(h(−∞))]dλ
×Sα (0,−∞)φh(−∞). (4.26)
To obtain (4.11) from (4.26), we must choose for the family of
Hamiltonians Hg the family H0 + gV with 0 ≤ g ≤ g1, and for
h(τ) choose the function g exp(−α|τ |). The function exp(−α|τ |) does
not vanish for τ 0 as required of the function h(τ), therefore,
strictly speaking, we cannot use equation (4.26). However, a slight
modification of the above proof, based on Remark 4.2, allows us to
verify (4.11).
Remark 4.3. If the family of Hamiltonians Hg also depends on
another parameter Ω, then it is not difficult to outline the conditions
under which the limit in (4.26) is uniform in Ω (for this, it is
necessary to give uniform in Ω estimates in the proof of the lemma).
In particular, if HΩg = HΩ
0 +gV Ω (0 ≤ g ≤ g1,Ω ∈ O), then the limit
in (4.26) will be uniform in g and in Ω if the norm of the operators
V Ω is bounded by a constant not depending on Ω and it is possible
to find a δ in such a way that the interval (EΩ(g)− δ, EΩ(g) + δ) for
any Ω and g contains no eigenvalues of the operator HΩg except for
EΩ(g) (see Tyupkin and Shvarts, 1972).
March 26, 2020 14:41 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch05 page 51
Chapter 5
The Theory of Potential Scattering
5.1 Formal scattering theory
Let H and H0 be two self-adjoint operators on the space R. The
Møller matrices S+ and S− of the operator pair (H,H0) are defined
as strong limits
S− = slimt→−∞
exp(itH) exp(−itH0) = slimt→−∞
S(0, t), (5.1)
S+ = slimt→+∞
exp(itH) exp(−itH0) = slimt→+∞
S(0, t). (5.2)
The operators S− and S+ are isometries, as strong limits of
unitary operators, however, they are not necessarily unitary. If they
are unitary, one can check that
S∗− = slimt→−∞
exp(itH0) exp(−itH),
S∗− = slimt→+∞
exp(itH0) exp(−itH)
(see Appendix A.5).
The S-matrix of the pair of operators (H,H0) is defined by the
formula
S = S∗+S−.
If the operators S+ and S− are unitary, then the S-matrix is also
unitary and can be written in the form
S = slimt→∞t0→−∞
S(t, t0), (5.3)
51
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52 Mathematical Foundations of Quantum Field Theory
where
S(t, t0) = exp(iH0t) exp(−iH(t− t0)) exp(−iH0t0).
However, the S-matrix can sometimes be unitary when the Møller
matrices are not unitary. Namely, for the S-matrix to be unitary, it is
sufficient (and necessary) to assume that the ranges of the operators
S− and S+ coincide: S−R = S+R.
In what follows, we will see that, under certain conditions, one
can use the S-matrix to describe the process of scattering; in these
conditions, it is usually possible to prove that the S-matrix is unitary.
Let us assume that the operators H and H0 have the same
domain; we will denote by the symbol V the difference H − H0.
Let us prove a simple sufficient condition for the existence of Møller
matrices (Cook’s condition).
If the integral∫∞
0 ‖V exp(−iH0t)x‖dt converges for all vectors x
in a dense set T ⊂ R, then the Møller matrices S+ and S− exist.
Proof. Consider the vector Φx(t) = exp(iHt) exp(−iH0t)x. Note
that∥∥∥∥dΦx(t)
dt
∥∥∥∥ = ‖i exp(iHt)V exp(−iH0t)x‖ = ‖V exp(−iH0t)x‖.
Hence, we have that
‖Φx(t1)− Φx(t2)‖ =
∥∥∥∥∫ t2
t1
dΦx(t)
dtdt
∥∥∥∥ ≤ ∫ t2
t1
∥∥∥∥dΦx(t)
dt
∥∥∥∥ dt=
∫ t2
t1
‖V exp(−iH0t)x‖dt
and therefore, it follows from our assumptions that for x ∈ T , we
have
limt1,t2→±∞
‖Φx(t1)− Φx(t2)‖ = 0.
This means that the limits of Φx(t) for t→ ±∞ exist for x ∈ T and
therefore, the limits exist for all x ∈ R (see Appendix A.5).
Let us consider the relation of Møller matrices S± and the
S-matrices S to the adiabatic Møller matrices Sα± and the adiabatic
March 26, 2020 14:41 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch05 page 53
The Theory of Potential Scattering 53
S-matrices Sα defined in Section 4.1. Let us assume that for any t
and t0,
slimα→0
Sα(t, t0) = S(t, t0).
It is easy to check that this condition is satisfied if the operator V
is bounded (in this case, we even have convergence in norm); the
condition is also satisfied in a much larger class of situations.
We will now show that the Cook condition implies
slimα→0
Sα+ = S+, (5.4)
slimα→0
Sα− = S−. (5.5)
If the operator S+ is unitary, then the relations (5.4) and (5.5) imply
slimα→0
Sα = S. (5.6)
To prove these statements, let us consider the vectors
Φαx(t) = Sα(0, t)x = S∗α(t, 0)x,
where x ∈ T . It is clear that∥∥∥∥dΦαx(t)
dt
∥∥∥∥ = ‖iUα(0, t)V exp(−α|t|) exp(−iH0t)x‖
= exp(−α|t|)‖V exp(−iH0t)x‖,
from which it follows that
‖Φαx(t1)− Φα
x(t2)‖ ≤∫ t2
t1
‖V exp(−iH0t)x‖ exp(−α|t|)dt
≤∫ t2
t1
‖V exp(−iH0t)x‖dt.
It follows from this inequality that the limits Φαx(±∞) exist, hence
the operators Sα± also exist. Furthermore, it follows from this
March 26, 2020 14:41 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch05 page 54
54 Mathematical Foundations of Quantum Field Theory
inequality that the limit
limt→±∞
Φαx(t) = Φα
x(±∞)
is uniform in α; this allows us to take the limit α→ 0 under the limit
sign. Taking this limit, we obtain, for x ∈ T , the relation
limα→0
Sα±x = limα→0
limt→±∞
Φαx(t) = lim
t→±∞limα→0
Sα(0, t)x
= limt→±∞
S(0, t)x = S±x,
implying that the strong limit of Sα± for α→ 0 equals S±.
If V is a bounded operator, then
‖Φαx(t1)− Φα
x(t2)‖ ≤ ‖V ‖ · ‖x‖ ·∣∣∣∣∫ t2
t1
exp(−α|t|)dt∣∣∣∣.
This inequality implies that in the relations (5.4)–(5.6), one can
talk about norm convergence (instead of strong convergence), hence
the operators Sα+, Sα−, Sα are unitary.
Taking the limit of t to ±∞ in the identity,
exp(iHτ)S(0, t) = S(0, t+ τ) exp(iH0τ),
we obtain the important relation
exp(iHτ)S± = S± exp(iH0τ),
which implies
HS± = S±H0, H0S = SH0. (5.7)
If S± are unitary operators, then this relation establishes the unitary
equivalence between H and H0. It follows that if, for example, H0
does not have a discrete spectrum but H does, then the operators
S+ and S− are not unitary (it is easy to check that the ranges of the
operators S+ and S− are orthogonal to eigenvectors in the discrete
spectrum).
Let φλ be a complete system of δ-normalized, generalized eigen-
functions of the operator H0 and Eλ the corresponding energy levels
(for simplicity of notation, let us assume that the operator H0 has no
discrete spectrum). Then the functions ψ±λ = S±φλ are generalized
March 26, 2020 14:41 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch05 page 55
The Theory of Potential Scattering 55
eigenfunctions of the operator H with the same eigenvalues Eλ. This
follows from the relation
Hψ±λ = HS±φλ = S±H0φλ = EλS±φλ = Eλψ±λ .
The matrix elements of S-matrices in the basis φλ can be easily
expressed in terms of the functions ψ±λ , namely,
〈Sφλ, φµ〉 =⟨S∗+S−φλ, φµ
⟩= 〈S−φλ, S+φµ〉 =
⟨ψ−λ , ψ
+µ
⟩.
Therefore, it is convenient to obtain the equations for the functions
ψ±λ . With this goal in mind, consider the operator
Σ±ε = ±ε∫ ±∞
0exp(−ε|t|)S(0, t)dt.
It is easy to check that in the case when the operators S+ and S−exist, we have
limε→+0
Σ±ε = S±.
For the proof, it is sufficient to use the relation
limε→+0
±ε∫ ±∞
0exp(−ε|t|)f(t)dt = f(±∞), (5.8)
which holds if the vector function f(t) is bounded and has the limit
f(±∞) = limt→±∞ f(t).1 It is clear that ψ±λ = limε→+0 ψ±ελ , where
ψ±ελ = Σ±εφλ = ±ε∫ ±∞
0exp(−ε|t|) exp(iHt) exp(−iEλt)φλdt
=±iε
H − Eλ ± iεφλ.
1If ‖f(t)‖ ≤ A and for t ≥ T , we have that ‖f(t)− f(+∞)‖ ≤ δ, then∥∥∥∥ε ∫ ∞
0
exp(−εt)f(t)dt− f(+∞)
∥∥∥∥ = ε
∥∥∥∥∫ ∞
0
exp(−εt)(f(t)− f(+∞))dt
∥∥∥∥≤ 2εTA+ δ.
March 26, 2020 14:41 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch05 page 56
56 Mathematical Foundations of Quantum Field Theory
We have obtained that ψ±ελ satisfies the equation
(H − Eλ ± iε)ψ±ελ = ±iεφλ,
which can be rewritten in the form
φ±ελ = φλ + (Eλ −H0 ∓ iε)−1V ψ±ελ . (5.9)
If the limit limε→0(Eλ −H0 ± iε)−1 = (Eλ −H0 ± i0)−1 exists in an
appropriate sense, then one can take the limit ε→ 0 in equation (5.9);
we obtain the following equation for ψ±λ :
ψ±λ = φλ + (Eλ −H0 ∓ i0)−1V ψ±λ .
This equation is called the Lippman–Schwinger equation.2 One can
also express the matrix elements of the S-matrix in terms of the
functions ψ+λ only (or of the functions ψ−λ only), namely,
〈Sφλ, φµ〉 =⟨ψ−λ , ψ
+µ
⟩= δ(λ− µ)− 2πiδ(Eλ − Eµ)
⟨φλ, V ψ
+µ
⟩= δ(λ− µ)− 2πiδ(Eλ − Eµ)
⟨V ψ−λ , φµ
⟩. (5.10)
Indeed, using the Lippman–Schwinger equation, we can show that⟨ψ−λ , ψ
+µ
⟩= 〈φλ, φµ〉+
⟨(Eλ −H0 + i0)−1V ψ−λ , φµ
⟩+⟨φλ, (Eµ −H0 − i0)−1V ψ+
µ
⟩+⟨V ψ−λ , (Eλ −H0 − i0)−1(Eµ −H0 − i0)−1V ψ+
µ
⟩= δ(λ− µ) +
⟨V ψ−λ , (Eλ − iH0 − i0)−1φµ
⟩+⟨(Eµ −H0 + i0)−1φλ, V ψ
+µ
⟩− 1
Eλ − Eµ(⟨ψ−λ − φλ, V ψ
+µ
⟩−⟨V ψ−λ , ψ
+µ − φµ
⟩)
= δ(λ− µ) +
(1
Eλ − Eµ + i0− 1
Eλ − Eµ
)⟨V ψ−λ , φµ
⟩+
(1
Eµ − Eλ + i0+
1
Eλ − Eµ
)⟨φλ, V ψ
+µ
⟩. (5.11)
2We will not go further into the delicate question of specifying the precisemeaning of the Lippman–Schwinger equation. The calculations at the end of thissection are also informal.
March 26, 2020 14:41 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch05 page 57
The Theory of Potential Scattering 57
In the above, we used the formula
(Eλ −H0 − i0)−1(Eµ −H0 − i0)−1 = − 1
Eλ − Eµ× [(Eλ −H0 − i0)−1 − (Eµ −H0 − i0)−1]. (5.12)
Both parts of this formula can be seen as generalized operator
functions of λ and µ. The numerical generalized function 1Eλ−Eµ can
be understood as one of the functions (Eλ −Eµ + i0)−1, (Eλ −Eµ −i0)−1 or P 1
Eλ−Eµ (all these functions coincide up to a summand of
the form cδ(Eλ − Eµ)); formula (5.12) is correct in all cases, since
δ(Eλ − Eµ)[(Eλ −H0 − i0)−1 − (Eµ −H0 − i0)−1] = 0).
Replacing 1Eλ−Eµ with 1
Eλ−Eµ+i0 or 1Eλ−Eµ−i0 in (5.11), we obtain
the formula we wanted to prove.
5.2 Single-particle scattering
Let us consider the scattering of non-relativistic particles without
spin in the potential field V(x) that decays to zero at infinity. The
state space R in this case is the space L2(E3) and the Hamiltonian
has the form H = p2/2m+V(x). When the particle is very far from
the scattering center, we can describe the motion of the particle with
the Hamiltonian H0 = p2/2m, assuming that the potential energy is
equal to zero. Accordingly, the scattering of the particle in the field
V(x) can be described by the scattering matrix S corresponding to
the pair of operators (H,H0).
Let us check that in the case when the potential V(x) is square-
integrable, the Cook condition is satisfied, hence the Møller matrices
exist. Note that for a dense set of functions f ∈ R, we have the
inequality |ft(x)| ≤ C|t|−3/2, where ft = exp(−iH0t)f (for the
proof of this inequality, see Section 10.3, Lemma 10.5). This implies
that the norm of the function ψt = V exp(−iH0t)f does not exceed
C|t|−3/2√∫|V(x)|2dx, since ψt(x) = V(x)ft(x). This shows that the
Cook condition is satisfied. One can also prove this condition with
weaker assumptions.
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58 Mathematical Foundations of Quantum Field Theory
If the potential V(x) is simultaneously square and absolutely
integrable (V ∈ L2 ∩ L1), then one can show that the S-matrix is
unitary. We will not give the proof of this non-trivial fact. We will
only note that the unitarity of the S-matrix in different assumptions
was considered in numerous papers. The proof of the theorem in the
above formulation is given by Kuroda (1959).
Let us introduce the notation
S(p,q) = 〈Sφq, φp〉,
where φp = (2π)−3/2 exp(ipx) is a generalized eigenfunction of the
momentum operator (S(p,q) can be considered as a kernel of the
operator S in the momentum representation).
Since the operator S commutes with the operator H0, the function
S(p,q) takes the form
S(p,q) = S1(p,q)δ(p2 − q2) =S1(p,q)
2pδ(p− q).
The expression
dσ = π2|S1(k,p)|2dω, (5.13)
where dω = sin θdθdφ is the element of solid angle and k is a
vector of the same length as p and directed at the angle dω has the
physical meaning of differential cross-section, it gives the number of
outgoing particles in the solid angle dω under the condition that the
incoming particles have momentum p (we assume that the beam of
incoming particles has unit flux).3
By the formula (5.10),
S(k,p) = 〈Sφp, φk〉 = δ(k− p)− 2πiδ
(p2
2m− k2
2m
)⟨V ψ−p , φk
⟩,
and therefore the formula for the differential cross-section can be
written in the form
dσ = 16m2π4|⟨V ψ−p , φk
⟩|2dω.
To understand the physical meaning of the quantity (5.13), let
us assume that the incoming particles are described by the wave
3We consider only the scattering on non-zero angles.
March 26, 2020 14:41 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch05 page 59
The Theory of Potential Scattering 59
function exp(−iH0t)φ with t → −∞ (in other words, we assume
that the particles are described by the function φ at time t = 0 if the
interaction with the potential field is neglected). From the definition
of the S-matrix, it follows that for t → +∞, the wave function has
the form exp(−iH0t)Sφ. In the momentum representation, we have
(Sφ)(p) = ψ(p) =
∫S(p,q)φ(q)dq,
(exp(−iH0t)Sφ)(p) = exp
(−i p
2
2mt
)ψ(p),
hence the probability of having the momentum of outgoing particles
directed in the solid angle Ω is equal to∫Ω|ψ(p)|2dp =
∫Ωdp
∫dqdq′S(p,q)S(p,q′)φ(q)φ(q′).
In actual scattering experiments, we never know the wave function
of the incoming particle. In classical mechanics, this means that we
know the initial momentum p0 of the incoming particle, however, we
do not know its impact parameter (recall that the impact parameter
is the distance of the particle trajectory from the scattering center
in the case when we neglect the interaction and assume that the
trajectory is a straight line). Therefore, in classical mechanics, one
considers a particle beam with particles having the same initial
momentum p0 but different impact parameters; we assume that the
beam has a unit flux (i.e. the number of particles going through a unit
area orthogonal to p0 in unit time is equal to one). The differential
cross-section in the solid angle dΩ is the number of outgoing particles
that are directed into the solid angle dΩ.
In quantum mechanics, the notion of the impact parameter cannot
be defined. However, we can consider a family of wave functions
ρa(p) = α(p) exp(ipa),
where α(p) is a normalized wave function that doesn’t vanish only
in a small neighborhood of p0 and the vector a is orthogonal
to p0. Then one can say that the vector a is the analog of the
impact parameter (recall that multiplication by exp(ipa) in the
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60 Mathematical Foundations of Quantum Field Theory
momentum representation is equivalent to a shift by the vector a
in the coordinate representation). The scattering cross-section in the
angle Ω of the incoming particle with initial momentum p0 can be
defined as
σΩ =
∫a⊥p0
da
∫Ωdp|ψa(p)|2,
where ψa = Sρa. Using
ψa(p) =
∫S(p,q)ρa(q)dq =
∫S1(p,q)
δ(p− q)
2pα(q) exp(iqa)dq,
we can say that
σΩ =
∫a⊥p0
da
∫Ωdp
∫dqdq′S1(p,q)S1(p,q′)
× δ(p− q)
2p· δ(p− q′)
2pα(q)α(q′) exp(i(q− q′)a),
Integrating over a, we obtain
σΩ = (2π)2
∫Ωdp
∫dqdq′S1(p,q)S1(p,q′)
δ(p− q)
4p2
×α(q)α(q′)δ(q− q′)δ(qT − q′T );
here qT , q′T are projections of the vectors q and q′ onto the plane
orthogonal to the vector p0; we used the fact that∫a⊥p0
exp(i(q− q′)a)da = (2π)2δ(qT − q′T )].
It is easy to check that
δ(q − q′)δ(qT − qT ′) = 2qδ(q2 − q′2)δ(qT − qT ′)
= 2qδ(q2n − q′2n )δ(qT − q′T )
=q
qnδ(qn − q′n)δ(qT − q′T )
+q
qnδ(qn + q′n)δ(qT − q′T )
=q
qn[δ(q− q′) + δ(Iq− q′)],
where qn, q′n are the projections of the vectors q, q′ on the vector p0
and I is the symmetry with respect to the plane orthogonal to p0.
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The Theory of Potential Scattering 61
Using the δ-functions entering the integrand, we can do the integral
over dq′ and using spherical coordinates in the integral over dp, we
can instead integrate over dp. We obtain
σΩ = π2
∫ω
sin θdθdφ
∫dq|S1(q, θ, φ|q)|2 q
qn|α(q)|2
+π2
∫ω
sin θdθdφ
∫dqS1(q, θ, φ|q)S1(q, θ, φ|Iq)
×(− q
qn
)α(q)α(Iq)
(when p runs over Ω, then the spherical coordinates θ, φ run over
ω; if p is a vector with spherical coordinates (p, θ, φ), then we use
the notation S1(p,q) = S1(p, θ, φ|q)). If the function S1(p,q) is
continuous when q changes in a neighborhood of the point p0 and p
changes in the angle Ω, we obtain that
σΩ ≈ π2
∫ω|S1(p0, θ, φ|p0)|2 sin θdθdφ.
This agrees with the expression for the differential cross-section that
was written earlier. We have used that for a normalized function α(q)
with support in a small neighborhood of the point p0 and continuous
function f(q), we have that∫f(q)|α(q)|2dq ≈ f(p0).
If p0 6= 0, then ∫f(q)α(q)α(Iq)dq ≈ 0.4
4It is not difficult to convert the above considerations into a rigorous proof.To be precise, one needs to consider a sequence of normalized functions αn withsupport tending towards the point p0 6= 0 and use the fact that for such a sequenceand a continuous function f(q), we have∫
f(q)|αn(q)|2dq→ f(p0),∫f(q)αn(q)αn(Iq)dq→ 0.
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62 Mathematical Foundations of Quantum Field Theory
Let us write down the Lippman–Schwinger equations in this case.
For the full system of generalized eigenfunctions of the Hamiltonian
H0, we will take the unctions φp(x) = (2π)−3/2 exp(ipx). In the
momentum representation, the Hamiltonian H0 can be considered as
multiplication by p2
2m , the operator (Eq−H0± iε)−1 can be identified
with multiplication by ( q2
2m −p2
2m ± iε)−1, and the operator (Eq −H0 ± i0)−1 can be identified with multiplication by the generalized
function ( q2
2m −p2
2m ± i0)−1. Hence, the momentum representation of
the Lippman–Schwinger equation takes the form
ψ±q (p) = δ(p− q) +
∫V(p− p′)ψ±q (p′)dp′
q2
2m −p2
2m ∓ i0, (5.14)
where
V(p) = (2π)−3/2
∫V(x) exp(−ipx)dx.
Taking into account that∫exp(ipx)dpq2
2m −p2
2m ± i0= −4mπ2 exp(±iqx)
x,
we can write the Lippman–Schwinger equation in the coordinate
representation
ψ±q (x) = φq(x)− m
2π
∫dx′
exp(∓iq|x− x′|)|x− x′|
V(x′)ψ±q (x′). (5.15)
Let us calculate the asymptotic behavior of the function ψ±q (x) for
x → ∞. We have assumed that the potential V(x) tends to zero
sufficiently fast at infinity, hence we can assume that in the integral
in the formula (5.15), we have x x′, and hence in the exponential,
we can replace the |x−x′| in the numerator with x− xxx′ and |x−x′|
in the denominator with x. We obtain that for large x,
ψ±(x) ≈ φq(x) + f±q
(∓xx
) exp(∓iqx)
x,
where
f±q (e) = −m2π
∫dx′ exp(−iqex′)V(x′)ψ±q (x′).
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The Theory of Potential Scattering 63
Hence, it follows that
f±q (e) = −m√
2π⟨V ψ±q , φk
⟩, (5.16)
where k = qe. Formula (5.16) together with the relations (5.10)
and (5.13) allows us to express the matrix entries of the S-matrix
and the differential cross-section in terms of the function f−q (e) that
is specified by the asymptotic behavior at infinity of the function
ψ−q (x). For example, the scattering cross-section in the solid angle
dω is equal to
dσ = (2π)3|f−p (e)|2dω; (5.17)
here p is the momentum of the incoming particles and e is a unit
vector directed in the solid angle dω. The function ψ+q (x) satisfies
the equation (− 1
2m∆ + V(x)
)ψ±q (x) =
q2
2mψ±q (x) (5.18)
and the boundary condition at infinity
ψ±q (x) ≈ φq(x) + f±q
(∓xx
) exp(∓iqx)
x. (5.19)
It is easy to check that these two conditions characterize the
function ψ±q (x). In other words, the function ψ−q (x) can be described
as the solution of the stationary Schrodinger equation that can be
represented at infinity as a sum of a plane wave and an outgoing
spherical wave. The function ψ+q (x) can be described similarly, but
the outgoing spherical wave should be replaced by an incoming
spherical wave.
Hence, we come to the stationary formulation of the scattering
problem for non-relativistic particles. In this formulation, one can
find the scattering cross-section by solving the stationary Schrodinger
equation with the boundary condition (5.19). It is useful to note that
in the case that V(x) = V(−x),
ψ+q (x) = ψ−q (−x)
(this follows from the fact that the complex conjugate of the incoming
spherical wave is the outgoing spherical wave and equation (5.18) is
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64 Mathematical Foundations of Quantum Field Theory
invariant with respect to complex conjugation). Therefore, f+q (e) =
f−q (e), and hence the scattering cross-section can be written in the
form
dσ = (2π)3|f+p (e)|2dω.
5.3 Multi-particle scattering
The multi-particle scattering problem can be considered on the Fock
space F , corresponding to the measure space E3 × B, where B is a
finite set. The momentum operator P on this space is defined by the
formula (3.15). Let us consider a translation-invariant HamiltonianH
on the space F (i.e. a self-adjoint operator H, commuting with the
momentum operator P).
The definition of the scattering matrix given in Section 5.1 is
not always applicable to the problem of scattering in a system of
n particles. This is because composite particles can be generated
in the process of collision. For example, if we start with protons
and neutrons, then it is possible that after collision we will have a
deuteron (a bound state of a proton and neutron) or an α-particle
(a bound state of two protons and two neutrons). Therefore, we
should first give a general definition of a particle that includes the
objects we will call composite particles.
The vector Φ(k) that describes a single-particle state with
momentum k should be an eigenvector of the operators P and H
(in the single-particle state, having the momentum k, the energy
has definite value ω(k)). However, the operator P has only one
normalized eigenvector (the Fock vacuum), therefore, the vector
function Φ(k) should be a generalized function. This consideration
allows us to give the following definition of a particle.
We define a particle corresponding to the Hamiltonian H, as a
generalized vector function Φ(k), obeying
HΦ(k) = ω(k)Φ(k), (5.20)
PΦ(k) = kΦ(k), (5.21)⟨Φ(k),Φ(k′)
⟩= δ(k− k′). (5.22)
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The Theory of Potential Scattering 65
The function ω(k) is called the energy of a single-particle state or
the dispersion law.
For example, the particles corresponding to the Hamiltonian
H0 =
n∑s=1
∫εs(k)a+
s (k)as(k)dk (5.23)
are generalized vector functions
Φs(k) = a+s (k)θ. (5.24)
The same generalized vector functions can also be considered as
particles corresponding to the Hamiltonian
H = H0 +W, (5.25)
where
W =∑
m≥2,n≥2
∫Wm,n(k1, . . . ,km|l1, . . . , ln)a+(k1)
. . . a+(km)a(l1) . . . a(ln)dmkdnl. (5.26)
These particles are called elementary particles. For translation-
invariant Hamiltonians of the form H0 + V , where V is an arbitrary
self-adjoint operator, a particle is called an elementary particle if
it can be obtained from the particle a+s (k)θ, corresponding to the
Hamiltonian H0 by means of perturbation theory (in other words,
if the Hamiltonian H0 + gV has a particle Ψg(k) that depends
continuously on the parameter g in the interval [0, 1], and we also
have Ψ0(k) = a+s (k)θ, then the particle Ψ1(k) is called an elementary
particle of the Hamiltonian H0 + V ).
The particles Φ(k),Φ′(k) are orthogonal if 〈Φ(k),Φ′(k)〉 = 0; the
system of particles Φ1(k), . . . ,ΦN (k) is called complete if there is
no particle orthogonal to every particle Φi(k).
For example, for the Hamiltonian, H0 =∫ε(k)a+(k)a(k)dk,
where ε(k) is a strongly convex function, the system of particles
consisting of a single particle Φ(k) = a+(k)θ (elementary particle),
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66 Mathematical Foundations of Quantum Field Theory
is complete. To prove this, let us consider a particle
Ψ(k) =∑n
∫ψn(k,k1, . . . ,kn)a+(k1) . . . a+(kn)θ
corresponding to the Hamiltonian H0. From the condition PΨ(k) =
kΨ(k), it follows that
(k1 + · · ·+ kn)ψn(k,k1, . . . ,kn) = kψn(k,k1, . . . ,kn),
and hence
ψn(k,k1, . . . ,kn) = δ(k− k1 − · · · − kn)φn(k1, . . . ,kn).
The condition HΨ(k) = ω(k)Ψ(k) gives the relation
(ε(k1) + · · ·+ ε(kn))ψn(k,k1, . . . ,kn) = ω(k)ψn(k,k1, . . . ,kn),
from which it is clear that
(ω(k1 + · · ·+ kn)− ε(k1)− · · · − ε(kn))φn(k1, . . . ,kn) = 0.
(5.27)
Equation (5.27) allows us to conclude that φn ≡ 0 for n > 1 (to
prove this, we should note that a strongly convex function cannot be
constant on a set of positive measure and that the function ε(p) +
ε(k − p) is a strongly convex function of the variable p). Hence,
Ψ(k) = φ1(k)a+(k)θ; this concludes the proof.
The complete system of particles does not always consist only
of elementary particles. It is possible that there exist particles that
are orthogonal to every elementary particle (such particles are called
composite particles).
We will consider, for example, the Hamiltonian
H =
∫k2
2ma+(k)a(k)dk +
1
2
∫W(k1 − k3)a+(k1)
× a+(k2)a(k3)a(k4)δ(k1 + k2 − k3 − k4)dk1dk2dk3dk4
(5.28)
that describes a system of identical spinless bosons (see Section 3.2).
This Hamiltonian commutes with N (the operator of number of
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The Theory of Potential Scattering 67
particles), therefore, it is sufficient to find only particles that belong
to the n-particle state Fn. Recall that Fn can be represented as the
space of symmetric functions φ(k1, . . . ,kn), where ki ∈ E3. Let us
introduce new variables
p =k1 + · · ·+ kn
n,
pi = ki − kn (i = 1, . . . , n− 1).
The space F ′n will be defined as the space of functions ψ(p1, . . . ,pn−1)
obeying the condition that the function f(p)ψ(p1, . . . ,pn−1), where
f(p) ∈ L2(E3), belongs to the space Fn. Functions from the space F ′ncan be considered as wave functions of relative motion of n particles.
The consideration of motion of n particles in terms of functions from
the space F ′n corresponds to the separation of the motion of the center
of inertia. Mathematically, the possibility to separate the motion of
the center of inertia for the Hamiltonian Hn means that there exists
a Hamiltonian H ′n in the space F ′n, that for every function,
f(p)ψ(p1, . . . ,pn−1) ∈ Fn,
where f ∈ L2(E3), ψ ∈ Fn, we have
Hn(f(p)ψ(p1, . . . ,pn−1)) =
(p2
2nmf(p)
)ψ(p1, . . . ,pn−1)
+ f(p)(H ′nψ)(p1, . . . ,pn−1).
Let us assume that ψ ∈ F ′n is a normalized eigenvector of
the operator H ′n with eigenvalue E. It is easy to check that the
generalized vector function
Φ(k) = δ(k− p)ψ(p1, . . . ,pn−1)
is a particle in the sense of the above definition. It is also easy to
check that
HΦ(k) =
(k2
2nm+ E
)Φ(k).
For n > 1, the particle Φ(k) is a composite particle (a bound state
of a system of n particles).
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68 Mathematical Foundations of Quantum Field Theory
Let us now ask what we should consider to be a scattering matrix
for a Hamiltonian of the form (5.25) (for more general translation-
invariant Hamiltonians, the definition of the scattering matrix will
be given in Chapter 9). It is natural to try to define the scattering
matrix of the Hamiltonian (5.25) as an S-matrix corresponding to the
pair of operators (H,H0). However, this works only in the case when
elementary particles Φs(k) = a+s (k)θ constitute a complete system of
particles (in other words, there exist no composite particles). From
the physical viewpoint, it is clear that the S-matrix of the operators
(H,H0) cannot describe the scattering of composite particles. If
composite particles are present, then the S-matrix corresponding to
the pair of operators (H,H0) will not be a unitary operator.
To formulate the definition of scattering matrix, we will restrict
ourselves to the case when all particles are bosons.
Note, first of all, that every particle Φ(k) can be written in the
form
Φ(k) =
∞∑n=1
∑i1,...,in
∫δ(k− k1 − · · · − kn)
×φn(k1, i1, . . . ,kn, in)a+i1
(k1) . . . a+in
(kn)θdk1 . . . dkn.
Let us assign to the particle Φ(k) an operator generalized function
defined by the formula
A(k) =∞∑n=1
∑i1,...,in
∫δ(k− k1 − · · · − kn)
×φn(k1, i1, . . . ,kn, in)ai1(k1) . . . ain(kn)θdk1 . . . dkn.
The operator generalized function A(k) satisfies the conditions:
(1) A+(k)θ = Φ(k) (the operator A+(k) creates the particle Φ(k)
when applied to the vacuum) and, (2) A(k) is a superposition of
operators ai1(k1) . . . ain(kn), i.e. it can be represented in the form
A(k) =
∞∑n=1
∑i1,...,in
∫λn(k,k1, i1, . . . ,kn, in)
× ai1(k1) . . . ain(kn)dk1 . . . dkn.
Moreover, these conditions specify A(k) uniquely.
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The Theory of Potential Scattering 69
Let us now define the in- and out-operators Ain(k, τ ) and
Aout(k, τ ), corresponding to the particle Φ(k), as the limits
Ain(k, τ ) = limt→−∞
exp(iω(k)(t− τ))A(k, t), (5.29)
Aout(k, τ ) = limt→+∞
exp(iω(k)(t− τ))A(k, t), (5.30)
where A(k, t) = exp(iHt)A(k) exp(−iHt) and ω(k) is the dispersion
law for the particle Φ(k). The limit is understood in the sense of the
limit of generalized functions (in other words, we assume that for
every function f(k) ∈ S(E3), we have∫f(k)A in
out(k, τ)dk = lim
t→∓∞
∫f(k) exp(iω(k)(t− τ))A(k, t)dk
in the sense of strong operator limit on the linear subspace D, where
D is defined as the smallest linear subspace containing all spaces Fn).
When talking about the operators Ain and Aout at the same time,
we will use the notation Aex.
Under certain conditions on the Hamiltonian H (in particular,
for the Hamiltonians of the form (5.28), under the assumption that
interaction potential is square integrable), we can prove that
(1) the limits (5.29) and (5.30) exist;
(2) if the particles Φ1(k), . . . ,Φm(k) are orthogonal, then the corre-
sponding operators Aex(k, i, τ), A+ex(k, i, τ), where i = 1, . . . ,m,
for fixed τ obey the canonical commutation relations (CCR)
[A+ex(k, i, τ), A+
ex(k′, i′, τ)] = [Aex(k, i, τ), Aex(k′, i′, τ)] = 0;
[Aex(k, i, τ), A+ex(k′, i′, τ)] = δiiδ(k− k′);
(3) Aex(k, i, τ) = exp(iHτ)Aex(k, i) exp(−iHτ) = exp(−ωi(k)τ)
Aex(k, i), where Aex(k, i) = Aex(k, i, 0);
(4) Aex(k, i)θ = 0, A+ex(k, i)θ = Φi(k).
The proof of these statements can be found, for example, in Hepp
and Epstein (1971) (the last two statements are trivial).
The generalized vector functions
Ψin(k1, i1, . . . ,kn, in) = A+in(k1, i1) . . . A+
in(kn, in)θ,
Ψout(k1, i1, . . . ,kn, in) = A+out(k1, i1) . . . A+
out(kn, in)θ
are called in- and out-states.
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70 Mathematical Foundations of Quantum Field Theory
Sometimes it is useful to note that the vectors Ψin and Ψout can
also be represented in the form
Ψex(k1, i1, . . . ,kn, in) = limt→±∞
exp(−i(ωi1(k1) + · · ·+ ωin(kn))t)
×A+(k1, i1, t) . . . A+(kn, in, t)θ. (5.31)
This follows immediately from the definition of the operators A+in
and A+out. Applying the formula (5.8), we can get from (5.31) the
following representation of in- and out-states:
Ψ inout
(k1, i1, . . . ,kn, in) = ± limα→0
iα(H − (ωi1(k1)
+ · · ·+ ωin(kn))± iα)−1A+(k1, i1) . . . A+(kn, in)θ.
Let us fix a complete orthonormal system of particles Φ1(k),
. . . ,ΦN (k) and the operators Aex(k, 1), . . . , Aex(k, N) corresponding
to the particles of this system.
We define the matrix elements of the S-matrix (or scattering
amplitudes) as functions
Sm,n(k1, i1, . . . ,km, im|l1, j1, . . . , ln, jn)
=⟨A+
in(l1, j1) . . . A+in(ln, jn)θ,A+
out(k1, i1) . . . A+out(km, im)θ
⟩.
Knowing these matrix elements, we can express the probability that
by collision of n particles with quantum numbers l1, j1, . . . , ln, jn, we
get particles with quantum numbers k1, i1, . . . ,km, im (more general
situations will be discussed in Chapter 10).
Let us consider the relation of the definition of the scattering
matrix in terms of the in- and out-operators, with the definition of the
S-matrix given in Section 5.1. Let us suppose that the Møller matrices
S∓, corresponding to the pair of operators (H,H0) are unitary (this
assumption is always satisfied when there are no composite particles).
The operators A(k, s) corresponding to elementary particles Φs(k) =
a+s (k)θ are clearly equal to as(k). Let us show that
Ain(k, s) = ain(k, s) = S−as(k)S∗−, (5.32)
Aout(k, s) = aout(k, s) = S+as(k)S∗+. (5.33)
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The Theory of Potential Scattering 71
To check these equalities, we will use the relations (5.1) and (5.2). It
follows from these relations that
S∓as(k)S+∓ = slim
t→∓∞exp(iHt) exp(−iH0t)as(k)
× exp(iH0t) exp(−iHt)
= slimt→∓∞
exp(iεs(k)t)as(k, t) = a inout
(k, s).
The relations (5.32) and (5.33) imply that⟨a+
in(l1, σ1) . . . a+in(ln, σn)θ, a+
out(k1, s1) . . . a+out(km, sm)θ
⟩=⟨S−a
+σ1(l1) . . . a+
σn(ln)θ, S+a+s1(k1) . . . a+
sm(km)θ⟩
=⟨Sa+
σ1(l1) . . . a+σn(ln)θ, a+
s1(k1) . . . a+sm(km)θ
⟩(we have used S∗−θ = S∗+θ = θ). In other words, the matrix elements
of the S-matrix defined in the present section under the assumption
we have made coincide with the matrix elements of the operator S
in the generalized basis a+s1(k1) . . . a+
sm(km)θ.
In the case when there exist bound states, we can also construct
an operator S having the functions Sm,n as matrix elements. This
operator S acts on the space Fas that is called the space of asymptotic
states. Fas can be defined as a Fock space constructed from the
measure space E3×N , where N is a set of types of particles. (Let us
recall that we have fixed a complete orthonormal system of particles.)
The operators of creation and annihilation in Fas will be denoted
b+(k, i) and b(k, i) (here, i ∈ N). The vectors from the space Fas can
be considered as initial and final states of the scattering process (for
example, the vector b+i1(f1) . . . b+in(fn)θ, where bi(f) =∫f(k)b(k, i)dk
corresponds to a state with the particles of the types i1, . . . , in with
the wave functions f1, . . . , fn). Let us define the isometries S− and
S+ acting from the space Fas into the space F by means of the
relations
Ain(k, i)S− = S−b(k, i), S−θ = θ, (5.34)
Aout(k, i)S+ = S+b(k, i), S+θ = θ. (5.35)
(Such operators do exist and they are defined by the relations (5.34)
and (5.35) in a unique way; the proof of this fact is given in
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72 Mathematical Foundations of Quantum Field Theory
Section 6.1.) The scattering matrix is defined as the operator S =
S∗+S− acting on the space Fas. It is obvious that
Sm,n(k1, i1, . . . ,km, im|l1, j1, . . . , ln, jn)
=⟨A+
in(l1, j1) . . . A+in(ln, jn)θ,A+
out(k1, i1) . . . A+out(km, im)θ
⟩=⟨S−b
+(l1, j1) . . . b+(ln, jn)θ, S+b+(k1, i1) . . . b+(km, im)θ
⟩=⟨Sb+(l1, j1) . . . b+(ln, jn)θ, b+(k1, i1) . . . b+(km, im)θ
⟩.
In other words, the matrix elements of the operator S in the general-
ized basis b+(k1, i1) . . . b+(km, im)θ coincide with the functions Sm,n(the scattering amplitudes).
Hamiltonians of the form (5.28) commute with the operator of
the number of particles, therefore, one can study separately the
n-particle scattering for n = 2, 3, . . . . For n = 2, this problem can
be reduced to the problem of potential scattering considered in the
preceding section. The proof of unitarity of the scattering matrix
for the three-particle problem was obtained by Faddeev (1963).
Faddeev’s method is based on the construction of equations for
the in- and out-states. Faddeev equations are also very useful for
calculations. They were also generalized in different ways for the
case of n-particle scattering problem for n > 3. The unitarity of the
scattering matrix for any number of non-relativistic particles was
proven in Sigal and Soffer (1987), see Hunziker and Sigal (2000) for
review.
In conclusion, let us present the definition of the Møller matrices
S± and the scattering matrix S in the form that is more convenient
in general situations (Chapter 10). Let us say that the operator B is
a good operator if it can be represented as∑m
∑s1,...,sm
∫f(k1, s1, . . . ,km, sm)
× a+s1(k1) . . . a+
sm(km)dk1 . . . dkm
and obeys the condition
Bθ =
∫φ(k)Φi(k)dk,
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The Theory of Potential Scattering 73
where Φi(k) is one of the particles in the fixed complete system
Φ1(k), . . . ,Φn(k). The isometric operator S−(S+) acting on the
asymptotic space Fas into the space F is called the Møller matrix if for
all good operators B1, . . . , Bm and smooth functions with compact
support f1(p), . . . , fm(p), we have
limt→∓∞
B1(f1, t) . . . Bm(fm, t)θ
= S∓b+(φ1, f1, i1) . . . b+(φm, fm, im)θ.
Here, the operators Bα(fα, t) are defined by the formula
Bα(fα, t) =
∫fα(x, t)Bα(x, t)dx,
where
Bα(x, t) = exp(iHt− iPx)Bα exp(−iHt+ iPx),
fα(x, t) =
∫exp(−iωiα(p)t+ ipx)fα(p)
dp
(2π)3,
and the numbers iα and the functions φα(p), ωiα(p) are defined by
the relations
Bαθ =
∫φα(p)Φiα(p)dp,
HΦiα(p) = ωiα(p)Φiα(p).
The scattering matrix S, as always, is defined in terms of the Møller
matrices S = S∗+S−.
To check that this definition of Møller matrices is equivalent to the
above definition, we note that a good operator Bα can be represented
in the form
Bα =
∫φα(p)A+
iα(p)dp,
where Ai(p) is an operator generalized function corresponding to the
particle Φi(p). Using the obvious relation
exp(−iPx)A+i (k) exp(iPx) = exp(−ikx)A+
i (k),
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74 Mathematical Foundations of Quantum Field Theory
we see that
Bα(fα, t) =
∫fα(x, t) exp(−ikx)φα(k)A+
iα(k, t)dk
=
∫exp(−iωiα(k)t)fα(k)φα(k)A+
iα(k, t)dk,
hence
slimt→∓∞
Bα(fα, t) =
∫fα(k)φα(k)A+
inout
(k, iα)dk. (5.36)
To finish the proof of equivalence of the two definitions of the Møller
matrices, we note that it follows from (5.36), (5.34) and (5.35) that
limt→∓∞
B1(f1, t) . . . Bn(fn, t)θ =
∫f1(k1)φ1(k1)A+
inout
(k1, i1)
. . . fn(kn)φn(kn)A+in
out
(kn, in)dnkθ = S∓b+(f1, φ1, i1)
. . . b+(fnφn, in)θ.
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Chapter 6
Operators on Fock Space
6.1 The representations of canonical and
anticommutation relations: Fock representation
Let us assume that we have assigned to every vector f of a pre-Hilbert
space B two conjugate operators a(f) and a+(f) acting on a dense
subspace of the Hilbert space H, in such a way that a(f) linearly
depends on f (i.e. a(λf + µg) = λa(f) + µa(g)). If the operators
satisfy the relations
[a(f), a(g)] = [a+(f), a+(g)] = 0, [a(f), a+(g)] = 〈f, g〉 (6.1)
(where f, g ∈ B and λ, µ are complex numbers), then we say that
these operators specify a representation of canonical commutation
relations, or CCR, on the space H. If the operators satisfy the
analogous relations
[a(f), a(g)]+ = [a+(f), a+(g)]+ = 0, [a(f), a+(g)]+ = 〈f, g〉, (6.2)
then we say that these operators specify a representation of canonical
anticommutation relations or CAR.1
We will also introduce the notation a(f,−1) = a(f), a(f,+1) =
a+(f). Using this notation, we can write down CCR and CAR in the
following form:
[a(f, ε), a(g, ε′)]∓ = Aεε′ 〈f, g〉, (6.3)
1In the definition of the representation of CCR and CAR, one should assumethat the operators a(f), a+(f) are defined on the same linear subspace D that isdense in the space H. They should transform D into itself. The operators a(f)and a+(f) are conjugate (i.e. 〈a(f)x, y〉 =
⟨x, a+(f)y
⟩for all x, y ∈ D).
75
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76 Mathematical Foundations of Quantum Field Theory
where Aεε′ is a matrix with the elements A11 = A−1−1 = 0, A−11 =
1, A1−1 = ∓1 (the upper sign is used in the case of CCR and the
lower sign is used in the case of CAR).
In the case of CAR, the operators a(f) and a+(f) are
bounded (the equation a(f)a+(f) + a+(f)a(f) = 〈f, f〉 implies that
〈a(f)x, a(f)x〉 + 〈a+(f)x, a+(f)x〉 = 〈f, f〉 〈x, x〉, hence ‖a(f)‖ ≤‖f‖, ‖a+(f)‖ ≤ ‖f‖). Therefore, one can assume that the operators
a(f), a+(f) are defined on the whole space H. In the case of CCR,
the operators a(f), a+(f) are unbounded.
Speaking simultaneously of CCR and CAR, we will use the term
“canonical relations” (CR). When it is necessary to stress that CR
are constructed by means of a pre-Hilbert space B, we will use the
notation CR(B). If in the space B we have fixed an orthonormal
basis φn, then the operators an = a(φn) and a+n = a+(φn) satisfy the
relations
[am, an] = [a+m, a+n ] = 0, [am, a
+n ] = δm,n (6.4)
in the case of CCR and the relations
[am, an]+ = [a+m, a+n ]+ = 0, [am, a
+n ]+ = δm,n (6.5)
in the case of CAR (these relations are also called CR).
The most important example of operators satisfying CR includes
the operators of creation a+(f) and annihilation a(f), defined in
Section 3.2. (They specify a representation of CCR in the bosonic
case and the representation of CAR in the fermionic case.) The
operators a+(f), a(f) are called creation and annihilation operators
in other cases as well; note, however, that these names don’t always
agree with their physical meaning.
One more important representation of CCR is given by the
following operators:
a(f) =
N∑n=1
fnan,
a+(f) =
N∑n=1
fna+n ,
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Operators on Fock Space 77
where a+n , an are operators satisfying the relations (6.4) that were
constructed in Section 2.6 where we studied the system of coupled
oscillators (here f = (f1, . . . , fN ) runs over a complex N -dimensional
space).
The representation of CR is called the Fock representation if one
can find a cyclic vector θ in the space H that satisfies the condition
a(f)θ = 0. More precisely, we assume that the vector θ is a cyclic
vector with respect to the operators a+(f), a(f) (this means that
applying these operators to θ and taking linear combinations we get
a subset of H that is dense in H). The vector θ is called the vacuum
vector2 (see Section 3.2).
Let us prove that two Fock representations are equivalent. This
means that there exists a unitary operator α mapping the space
H1 onto the space H2 and satisfying the conditions αa1(f) =
a2(f)α, αθ1 = θ2 (here ai(f) is a Fock representation of CR in the
spaceHi with the vacuum vector θi). In the case of CCR, the equation
αa1(f) = a2(f)α should be satisfied on a dense subset of the space
H1 (not necessarily on the whole domain of a1(f)).
Let us construct an operator α. For vectors of the form
a1(f1, ε1) . . . a1(fn, εn)θ1, (6.6)
we define the operator α, assuming that
α(a1(f1, ε1) . . . a1(fn, εn)θ1) = a2(f1, ε1) . . . a2(fn, εn)θ2.
If x and y are two vectors of the form (6.6), then 〈x, y〉 = 〈αx, αy〉.This follows from the fact that the scalar product of vectors of
the form (6.6) can be calculated using only CR and the relation
ai(f)θi = 0. Let us denote by S1 the linear subspace consisting
of linear combinations of vectors of the form (6.6). The operator
α, by linearity, can be extended to S1 and satisfies the condition
〈x, y〉 = 〈αx, αy〉 for all x, y ∈ S1. Using the assumption that the
vector θi is cyclic with respect to the operators ai(f, ε), we obtain
2Using CR, it is easy to check that every vector of the form a(f1, ε1) . . . a(fn, εn)θcan be represented as a linear combination of vectors of the forma+(φ1) . . . a+(φm)θ. Hence, θ is also a cyclic vector with respect to the familyof operators a+(f).
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78 Mathematical Foundations of Quantum Field Theory
that the set S1 is dense in H2 and the set S2 = αS1 is dense in H2.
The operator α can be extended to the closure S1 = H1 continuously;
the density of the set S2 in H2 implies the relation αH1 = H2. It is
clear that on the set S1 we have αa1(f, ε) = a2(f, ε)α. This completes
the proof of the equivalence of two Fock representations.3
Hence, it is sufficient to describe just one Fock representation; this
was done in Section 3.2 (and will be done again in slightly different
terms later).
The space of Fock representation CR(B) will be called Fock space
and denoted as F (B).
Let us assume that the pre-Hilbert space B is realized as a dense
subset of the space L2(Er) (the space of square-integrable functions
on the Euclidean space Er). Then, together with the operators
a+(f), a(f), it is convenient to introduce the operator generalized
functions a+(x), a(x) on Er, where
a(f) =
∫f(x)a(x)dx, a+(f) =
∫f(x)a+(x)dx.
The functions a+(x), a(x) satisfy the following relations:
[a(x), a(y)]∓ = [a+(x), a+(y)]∓ = 0,
[a(x), a+(y)]∓ = δ(x, y); a(x)θ = 0.(6.7)
This definition of the generalized operator functions a+(x), a(x) can
also be used in the case when B is a dense subset of the space
L2(X), where X = Er × B consists of pairs (e, s) with e ∈ Er a
point in Eucliean space and s ∈ B from a finite set B (then the
integration over X is understood as integration over Eucliean space
and summation over a finite set). We can also use this definition
in the more general case when X is an arbitrary measure space;
however, we do not need this level of generality. Later in this section,
3If the first representation of CR is a Fock representation, but the secondrepresentation has a vector θ2 ∈ R2 satisfying the condition a2(f)θ2 = 0 (notnecessarily cyclic), then precisely in the same way, we can construct the operatorα satisfying the relation αa1(f) = a2(f)α, αθ1 = θ2. However, in general, α will bean isometry that is not necessarily unitary (this statement is used in Section 5.3to construct the operators S− and S+).
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Operators on Fock Space 79
we assume that X is any measure space; however, for simplicity, the
reader can assume that X = Er or X = Er ×B.
Operator generalized functions a+(x), a(x), in the case of the Fock
representations of CR, were considered in Section 3.2. In particular,
we have shown in this section that an expression of the form (3.4) can
be understood as an operator acting on Fock space. For an arbitrary
representation of CR, the expression (3.4) can be easily defined in the
case when the function entering this expression has the form (3.5).
If f is an arbitrary function, then it can be represented as a limit of
functions fk of the form (3.5) and the operator A corresponding to
the function f should be defined, in this case, as a limit of operators
Ak corresponding to the functions fk. Of course, this definition of the
operator A is not completely precise because we have not defined the
notion of limit for the functions fk and the operators Ak. We will not
be concerned with these definitions; however, we will note that in the
case when B = S(E3) and the functional 〈a(f, ε)u, v〉 (for arbitrary
u, v ∈ D) continuously depends on f in the topology of the space
S(E3), then using the operator analog of the kernel theorem (see
Appendix A.5), one can define the operator (3.4) for the functions
f(x1, . . . ,xm|y1, . . . ,yn) from the space S(E3(m+n)).
In the case when B = L2(X), every element Φ of Fock space can
be written in a unique way in the form
Φ =∑n
∫fn(x1, . . . , xn)a+(x1) . . . a
+(xn)θdx1 . . . dxn, (6.8)
where the functions fn(x1, . . . , xn) are symmetric in the case of CCR
and antisymmetric in the case of CAR in the arguments x1, . . . , xn ∈X. The norm of the vector Φ is equal to∑
n
n!
∫|fn(x1, . . . , xn)|2dx1 . . . dxn.
Conversely, to every sequence f0, f1(x), . . . , fn(x1, . . . , xn), . . . of
symmetric (antisymmetric) functions obeying∑n
n!
∫|fn(x1, . . . , xn)|2dx1 . . . dxn <∞,
we can assign a vector in Fock space. This description of the Fock
space can be obtained from the formulation described in Section 3.2,
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80 Mathematical Foundations of Quantum Field Theory
namely, it is easy to check that the Fock state (Fock column)
Φ =
φ0
φ1(x1)...
φn(x1, . . . , xn)...
can be represented in the form
Φ =∑n
1√n!
∫φn(x1, . . . , xn)a+(x1) . . . a
+(xn)θdx1 . . . dxn.
This statement can also be formulated in slightly different
terms. Namely, one can say that vector generalized functions
a+(x1) . . . a+(xn)θ, where n = 0, 1, 2, . . . , constitute a generalized
basis of the space F (L2(X)) in the following sense: every vector
from F (L2(X)) can be decomposed as a linear combination of these
vector generalized functions (i.e. represented in the form (6.8)) and if
the coefficient functions fn are symmetric (antisymmetric), then this
decomposition is unique. (Note that we used the term generalized
basis not precisely in the form that is described in A.7.) Generalized
functions
Akl(x1, . . . , xk|y1, . . . , yl) =⟨Aa+(y1) . . . a
+(yl)θ, a+(x1) . . . a
+(xk)θ⟩
are called matrix elements or matrix entries of the operator A in the
generalized basis a+(x1) . . . a+(xn)θ.
In conclusion, we will describe an example of a representation
of CCR that is not a Fock representation. Let us consider, in the
space F (L2(En)), the operator generalized functions b(x) = a(x) +
φ(x), b+(x) = a+(x) + φ(x), where φ ∈ S ′ is a numerical generalized
function. The operators b(x), b+(x) specify a representation of CCR
(we assign the operators
b(f) =
∫f(x)b(x)dx = a(f) +
∫f(x)φ(x)dx,
b+(f) =
∫f(x)b+(x)dx = a+(f) +
∫f(x)φ(x)dx
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Operators on Fock Space 81
to every function f ∈ S). It is easy to see that this representation
of CCR can be a Fock representation only in the case when the
function φ is square integrable. To check this, consider the vector
ξ ∈ F (L2(En)) obeying the condition
b(x)ξ = a(x)ξ + φ(x)ξ = 0. (6.9)
Then ⟨ξ, a+(x)θ
⟩= 〈a(x)ξ, θ〉 = −φ(x) 〈ξ, θ〉.
For every vector ξ ∈ F (L2(En)), the function 〈ξ, a+(x)θ〉 is square
integrable because this function enters the Fock column correspond-
ing to the vector ξ.4 This means that in the case when φ /∈ L2(En),
equation (6.9) does not have a solution in the Fock space.
A broader class of representations of CCR is described by the
following formulas:
b(f) = a(Φf) + a+(Ψf) +
∫f(x)φ(x)dx,
b+(f) = a(Ψf) + a+(Φf) +
∫f(x)φ(x)dx
(6.10)
(for concreteness, we assume that f ∈ S(En), φ ∈ S ′(En), and Φ,Ψ
are operators transforming the space S(En) into the space L2(En)).
It is easy to check that the relations (6.10) specify a representation
of CCR in the case when
Φ∗Φ−Ψ∗Ψ = 1,
Φ∗Ψ−Ψ∗Φ = 0,
where the operators Φ,Ψ are defined by the formulas Φf = Φf,Ψf =
Ψf . The relation (6.10) can be written in the form
b(x) =
∫Φ(y, x)a(y)dy +
∫Ψ(y, x)a+(y)dy + f(x),
b+(x) =
∫Ψ(y, x)a(y)dy +
∫Φ(y, x)a+(y)dy + f(x),
(6.11)
4We have used the fact that 〈ξ, θ〉 6= 0. If 〈ξ, θ〉 = 0, then 〈ξ, a+(x1)a+(x2) . . .a+(xn)θ〉 = −φ(x1)
⟨ξ, a+(x2), . . . , a+(xn)θ
⟩. By induction on n, we derive⟨
ξ, a+(x1) . . . a+(xn)θ⟩
= 0, hence ξ = 0.
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82 Mathematical Foundations of Quantum Field Theory
where Φ(x, y) and Ψ(x, y) are the kernels of the operators Φ
and Ψ.
One can prove that the representation of CCR specified in (6.10)
is equivalent to the Fock representation if and only if the functions
Ψ(x, y) and f(x) are square integrable (see Berezin (2012), where this
statement and the analogous statement for CAR were formulated and
proved).
The transition from the operators a+(x), a(x) to the operators
b+(x), b(x), described by the formulas (6.11), is called a canonical
transformation in the case when the new operators also satisfy
CR. If the operators b+(f), b(f) generate a Fock representation
of CCR, then one can find a unitary operator U satisfying the
condition
Ua(f)U−1 = b(f),
Ua+(f)U−1 = b+(f).
(This follows from the uniqueness of Fock representation that we
have proven.) Otherwise, such a unitary operator doesn’t exist.
6.2 The simplest operators on Fock space
Let us fix an orthonormal basis φk in the space B, where the index k
runs over the set M . The operators a+(φk), a(φk) in the Fock space
F (B) are denoted a+k , ak. We will use the notation a(k, ε) = aεk, where
ε = ±1, assuming that a1k = a(k, 1) = a+k , a−1k = a(k,−1) = ak.
Let us consider the operators Nk = a+k ak in the Fock space F (B).
It is easy to check that all the operators Nk commute. Let us find
their common eigenvectors.
We consider first the case of CCR. Then the commutation
relations of the operators Nk with the operators a+l , al have the form
[Nk, a+l ] = δk,la
+l , [Nk, al] = −δk,lal.
If φ is a common eigenvector of the operators Nk, in other words
Nkφ = nkφ, then Nka+l φ = a+l Nkφ = nka
+l φ for l 6= k,Nka
+k φ =
a+k (Nk + 1)φ = (nk + 1)a+k φ. Similarly, Nkalφ = nkalφ for l 6=k,Nkakφ = (nk − 1)akφ. We conclude that vectors of the form
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Operators on Fock Space 83
a+n1k1
. . . a+nrkr
θ, where ni are non-negative integers and ki 6= kj ,
form a complete orthogonal5 (not orthonormal) system of common
eigenvectors for the operators Nk. The eigenvalue of the operator Nk
in the state a+n1k1
. . . a+nrkr
θ is equal to ni, if k = ki, and equal to zero,
if k does not coincide with any ki.
Let us consider the case of CAR. The operators Nk satisfy the
relation N2k = Nk in this case, which implies that the eigenvalues of
Nk are equal to 0 or 1. Using the equations Nka+l = a+l Nk, Nkal =
alNk for l 6= k, and a+k Nk = 0, Nkak = 0 and Nka+k = a+k (1 −
Nk), akNk = (1−Nk)ak, we can analyze the action of the operators
a+k , ak on the common eigenvector φ of the operators Nk. Namely,
if Nkφ = nkφ and nl = 1, then a+l φ = 0; if nl = 0, then a+l φ is
an eigenvector of the operators Nk; and Nka+l φ = nka
+l φ for l 6=
k,Nka+k φ = a+k φ. Similarly, for nl = 0 we have alφ = 0, and for
nl = 1 we obtain that alφ is an eigenvector of the operators Nk,
and we have Nkalφ = nkalφ for l 6= k,Nkakφ = 0. We see that the
vectors a+k1 . . . a+knθ, where all ki are different, constitute a complete
orthogonal6 (but not orthonormal) system of common eigenvectors
of the operators Nk, namely we have Nkia+k1. . . a+krθ = a+k1 . . . a
+krθ
and Nka+k1. . . a+krθ = 0, if k is not equal to any ki. When we consider
simultaneously the case CCR and CAR, we will denote the vectors
of the complete system that we have constructed as a+n1k1
. . . a+nsks
θ;
however, we should remember that in the case of CAR the numbers
ni cannot be greater than one (i.e. ni = 0, 1).
The operator Nk has the physical meaning of the operator of the
number of particles in the state φk, and the operators a+k and akare called the creation and annihilation operators of a particle in
the state φk. The properties of the operators Nk, a+k , ak agree with
this terminology. The operator N =∑
kNk =∑
k a+k ak is called the
operator of the number of particles and the state x ∈ F (B) satisfying
Nx = nx is called an n-particle state.
5More precisely, two vectors of this kind are either proportional or orthogonal.The completeness of this system of eigenvectors follows from the cyclicity of thevector θ with respect to the family of operators a+(f).6More precisely, two vectors of this kind are either orthogonal or proportional.
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84 Mathematical Foundations of Quantum Field Theory
It is easy to see that these definitions agree with the definitions of
Section 3.2. In particular, the set of n-particle states coincides with
the subspace Fn. This implies that the operator of the number of
particles N doesn’t depend on the choice of orthonormal basis φk.
Let us consider the Hamiltonian
H0 =∑
ωka+k ak =
∑ωkNk, (6.12)
where ωk are real numbers.7 Note that the Heisenberg operators
ak(t) = exp(−iH0t)ak exp(−iH0t) and a+k (t) = exp(iH0t)a+k
exp(−iH0t) can be calculated; namely, it is clear that the operators
ak(t) = ak exp(−iωkt) and a+k (t) = a+k exp(iωkt) satisfy the following
Heisenberg equations:
dak(t)
dt= i[H0, ak(t)] = −iωkak(t),
da+k (t)
dt= i[H0, a
+k (t)] = iωka
+k (t).
The operator H0 commutes with the operators Nk, therefore
the system of vectors a+n1k1
. . . a+nrkr
θ that we have constructed is
a complete system of eigenvectors of the operator H0 and the
eigenvalues are equal to∑r
i=1 ωkini.
Let us find the ground state of the Hamiltonian H0. In the case
of CCR, if at least one ωk < 0, then the spectrum of energies is
unbounded from below and therefore the ground state does not exist.
If all ωk ≥ 0, then all energy levels are non-negative and the vacuum
vector θ is the ground state. In the case of CAR, the lowest energy
level is equal to E0 =∑
k∈L ωk, where the sum is taken over the set L
consisting of k, for which ωk < 0. To obtain the ground state Φ from
the vacuum vector θ, we should apply all operators a+k with k ∈ Lto the vacuum vector (in other words, Φ = (
∏k∈L a
+k )θ). If the set
L is infinite, then the ground state doesn’t exist. Let’s introduce the
7The operator H0 is an essentially self-adjoint operator for any choice of realnumbers ωk (this follows from the fact that the operator H0 has a completesystem of eigenvectors). In particular, the operator of the number of particles Nis essentially self-adjoint. As usual, we identify the essentially self-adjoint operatorwith its self-adjoint extension.
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Operators on Fock Space 85
operators bk in the following way: bk = a+k for k ∈ L and bk = ak for
k 6∈ L. The operators b+k , bk satisfy the relations (6.5). To check that
these operators generate a representation of CAR, we should define
the operators b+(f), b(f) for all f ∈ B. It is easy to do this, setting
b(f) = a+(Pf) + a((1 − P )f), where Pf =∑
k∈L 〈f, φk〉φk. It is
convenient to express the Hamiltonian H0 in terms of the operators
b+k , bk. If the set L is finite, then
H0 =∑|ωk|b+k bk +
∑k∈L
ωk,
and then the ground state of the Hamiltonian H0 is the vacuum
vector for the operators bk, i.e. it satisfies the conditions bkΦ = 0. If
the set L is infinite, then the vector satisfying the condition bkΦ =
0 in the space F (B) doesn’t exist. In other words, the operators
b+(f), b(f) specify a representation of CAR that is not equivalent to
the Fock representation.
The operators acting in Fock space F (B), where B = L2(X), can
be expressed in terms of operator generalized functions a+(x), a(x),
satisfying the relations (6.7). The statements proven in this section
can also be proven for the operators in the space F (B) that are
expressed in terms of a+(x), a(x). In particular, one can introduce
the commuting operators (or more precisely operator generalized
functions) N(x) = a+(x)a(x). The operator of the number of
particles N can be expressed in terms of the operators N(x) by the
formula
N =
∫N(x)dx
(more details about the operator of the number of particles can be
found in Section 3.2). A complete system of generalized eigenvectors
(generalized eigenbasis) of the operator
H0 =
∫ω(x)a+(x)a(x)dx =
∫ω(x)N(x)dx (6.13)
consists of vectors (vector generalized functions) a+(x1) . . . a+(xn)θ.
Operators of the form (6.12), (6.13) are among the simplest
operators in Fock space. Slightly more general operators (3.9) are
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86 Mathematical Foundations of Quantum Field Theory
considered in Section 3.2. The analysis of Hamiltonians of the form
H =
∫A(x, y)a+(x)a(y)dxdy +
∫B(x, y)a+(x)a+(y)dxdy
+
∫B(x, y)a(y)a(x)dxdy
+
∫C(x)a+(x)dx+
∫C(x)a(x)dx (6.14)
(quadratic Hamiltonians) is more complicated. Under certain condi-
tions, they can be reduced to the form (6.13) by means of a linear
canonical transformation of the form (6.11). A complete analysis of
the Hamiltonian (6.14) can be found in the book by Berezin (2012).
Much harder is the situation for the Hamiltonians of degree higher
than two, with respect to the operators of creation and annihilation.
Very rarely one can find precisely the eigenvalues or the eigenvectors
of these operators and hence it is necessary to apply other methods.
In Chapter 11, we will explain a method of calculating the evolution
operator and the eigenvalues of the Hamiltonian H = H0 + gV ,
where H0 =∑ωka
+k ak, in the form of power series with respect to
the parameter g (perturbation theory).
In this section, we will consider the conditions under which the
formal expression
H =∑m
∑ki,εi
Γmk1,ε1,...,km,εmaε1k1. . . aεmkm (6.15)
defines an operator on the Fock space F (B) (here, as always, a1k =
a+k = a+(φk), a−1k = ak = a(φk), where φk is an orthonormal basis
for B, ki are taken from the set M , and εi = ±1). In the case of CAR,
one can prove the following statement. If the numerical series∑m
∑ki,εi
Γmk1,ε1,...,km,εm (6.16)
is absolutely convergent, then the operator series (6.15) is also
absolutely convergent (in the sense of norm convergence) and specifies
a bounded operator in F (B). This follows from the remark that
in the case of CAR we have that ‖aεk‖ ≤ ‖φk‖ = 1 and hence,
‖aε1k1 . . . aεmkm‖ ≤ 1.
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Operators on Fock Space 87
If, in addition, the expression (6.15) is formally Hermitian, i.e.
obeys
Γmk1,ε1,...,km,εm = Γmk1,−ε1,...,km,−εm , (6.17)
then the operator H defined by the formula (6.15) is self-adjoint
(since bounded Hermitian operators are self-adjoint).
In the case of CCR, we can use the following estimates for an
n-particle state x:
‖aεkx‖ ≤√n‖x‖,
‖aε1k1 . . . aεmkmx‖ ≤
√n√n+ 1 . . .
√n+m− 1‖x‖ (6.18)
(the first of which can be obtained from the relation
‖akx‖2 =⟨a+k akx, x
⟩≤
⟨∑k
a+k akx, x
⟩= 〈Nx, x〉 = n‖x‖2,
and the second follows from the first).
Using the estimate (6.18), one can check that the expression (6.15)
specifies an operator with everywhere dense domain in F (B), if this
expression is polynomial (Γmk1,ε1,...,km,εm = 0 for m > s) and the
numerical series (6.16) is absolutely convergent. This follows from
the remark that the series
s∑m=1
∑ki,εi
Γmk1,ε1,...,km,εmaε1k1. . . aεmkmx
converges for x ∈ Fn, and hence the domain of the operator H
contains D — the union of all linear spaces Fn. If the operator H is
formally Hermitian, i.e. the condition (6.17) is satisfied, then H will
be Hermitian on D. However, it is not clear whether this operator
is essentially self-adjoint on D; results in this direction can be found
in the paper by Hepp (1966). The operators of interest for physics
are usually bounded from below and hence they define a self-adjoint
operator by means of Friedrich’s extension (see Appendix A.5). The
analysis of the operator H defined by the expression (6.15) is much
easier when Γmk1,ε1,...,km,εm 6= 0 only for ε1 + · · · + εm = 0 (i.e.
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88 Mathematical Foundations of Quantum Field Theory
if the operator H commutes with the operator of the number of
particles N). Under this condition, one can consider the operator H
separately on each subspace Fn. One can extend the class of operators
of the form (6.15) that defines self-adjoint operators, by noting that
adding to a self-adjoint operator (for example, to H0) a bounded
self-adjoint operator, we again obtain a self-adjoint operator.
We will not analyze the conditions under which expressions of the
form
H =∑m
∑ε1,...,εm
∫Γm(x1, ε1, . . . , xn, εn)a(x1, ε1)
. . . a(xn, εn)dx1 . . . dxn (6.19)
define an operator on the space F (B) = F (L2(X)). Note that only
for the case X = Er some results in this direction are proven in
Section 3.2.
In the case of CAR an operator corresponding to a physical quantity
(in particular, a Hamiltonian) should always contain an even number
of creation and annihilation operators (i.e. Γm = 0 for odd m).
6.3 The normal form of an operator: Wick’s theorem
Let us say that an operator C in the Fock space F (B) is presented
in normal form if the operators of creation are to the left of the
annihilation operators:
C =∑m,n
Γm,n(k1, . . . , km|l1, . . . , ln)a+k1 . . . a+kmal1 . . . aln (6.20)
(here ak = a(φk), where φk is an orthonormal basis in B. As usual, we
sum over repeated indices ki, lj). The sum in the expression above can
be infinite, in which case the convergence of the series is understood
in the sense of strong operator convergence. The set of indices for
the operators ak will be denoted by M . Since M is countable, we
can identify M with the set of natural numbers; however, this is not
always convenient.
The product of an arbitrary number of operators a+k , al, taken in
any order, as well as any finite sum of such operators, can be written
in normal form by means of CR. It can be shown that any bounded
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Operators on Fock Space 89
operator can be written in normal form if the convergence in (6.20)
is understood in the sense of weak convergence.
The representation of an operator in normal form is especially
convenient by the calculation of the vacuum average (vacuum
expectation value) 〈Cθ, θ〉 of the operator C. It is clear that 〈Cθ, θ〉 =
Γ0,0 (the constant term in the representation of the operator C in
normal form).
Let us assume that C1, . . . , Ck are operators written in normal
form. Let us consider the following question: How can we write the
product C1 . . . Ck of these operators in normal form?
Before we answer this question, let us consider a simpler (though,
in some sense, more general) situation. Let A1, . . . , Am, B1, . . . , Bnbe operators with the property that the commutators [Aα, Bβ]
are numbers (i.e. they can be written as product of the iden-
tity operator and a number). The number BβAα = [Aα, Bβ] is
called the contraction of the operators Aα and Bβ. The prod-
uct B1 . . . BnA1 . . . Am with one contraction BβAα will be under-
stood as this product with the operators Bβ and Aα deleted
and with the factor BβAα inserted instead of these operators;
to denote the product with the contraction BβAα we will write
B1 . . . Bβ . . . BnA1 . . . Aα . . . Am. The product B1 . . . BnA1 . . . Am
with k contractions Bβ1Aα1 , . . . , BβkAαkwill be understood as a
product where the operators Bβ1 , . . . , Bβk , Aα1 , . . . , Aαkare deleted
and the contractions Bβ1Aα1 , . . . , BβkAαkare included in their
place. Hence, in order to define a product with k contractions, we
should select k operators Bβ1 , . . . , Bβk (the order of the selection is
unimportant), and to every one of these operators, we should assign
one of the operators A1, . . . , An, assuming that different indices
β1, . . . , βk correspond to different indices α1, . . . , αk.
Let us now formulate the following statement, which we will call
Wick’s theorem, noting that it presents a simplified form of this
theorem:
The product A1 . . . AmB1 . . . Bn is equal to the product B1 . . . BnA1
. . . Am with additional summand terms obtained by all possible
contractions.
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90 Mathematical Foundations of Quantum Field Theory
For example,A1A2B1B2 = B1B2A1A2 +B1B2A1A2 +B1B2A1A2 +B1B2A1A2
+B1B2A1A2 +B1B2A1A2 +B1B2A1A2
+B1B2A1A2 .
This statement is almost obvious: when we transfer an operator
Bβ to the left, we use the relation AαBβ = BβAα + BβAα, and
therefore every time we change the order of Bβ and Aα, we will add
two summands: one with the same number of contractions as in the
original summand and another where the number of contractions is
greater by one.
In order to give a formal proof, we can start by demonstrating
the following statement (either directly or by induction over m):
A1 . . . AmB = BA1 . . . Am +BA1 . . . Am +BA1A2 . . . Am
+ · · ·+BA1 . . . Am . (6.21)
In other words, we check Wick’s theorem for n = 1. Then, we can
use mathematical induction with respect to the number of operators
Bβ; at every induction step, we use the formula (6.21).
Let us now consider the reduction to normal form of the product
of two operators K and L, both represented in normal form. Without
loss of generality, we can assume that K and L have definite degree
with respect to creation and annihilation operators:
K = K(k1, . . . , km|l1, . . . , ln)a+k1 . . . a+kmal1 . . . aln ,
L = L(p1, . . . , pr|q1, . . . , qs)a+p1 . . . a+praq1 . . . aqs
(as always, we have in mind a summation over the repeated indices
from the set M).
Let us begin with the case of CCR. In this case, applying Wick’s
theorem to the product al1 . . . alna+p1 . . . a
+pr , one can show that KL
is equal to
K(k1, . . . , km|l1, . . . , ln)L(p1, . . . , pr|q1, . . . , qs)
×a+k1 . . . a+kma+p1 . . . a
+pral1 . . . alnaq1 . . . aqs ,
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Operators on Fock Space 91
with the addition of summands obtained from this expression by
means of arbitrary contractions between the operators a+pi and alj(recall that a+p al = [al, a
+p ] = δpl).
For example, if K = K(k|l)a+k al and L = L(p1, p2|q1, q2)a+p1a+p2aq1aq2 , then
KL = K(k|l)L(p1, p2|q1, q2)a+k a+p1a
+p2alaq1aq2
+K(k|l)L(p1, p2|q1, q2)a+k a+p1a
+p2al aq1aq2
+K(k|l)L(p1, p2|q1, q2)a+k a+p1 a
+p2al aq1aq2
= K(k|l)L(p1, p2|q1, q2)a+k a+p1a
+p2alaq1aq2
+K(k|l)L(l, p2|q1, q2)a+k a+p2aq1aq2
+K(k|l)L(p1, l|q1, q2)a+k a+p1aq1aq2 .
The functions K(k1, . . . , km|l1, . . . , ln) and L(p1, . . . , pr|q1, . . . , qs)can be assumed to be symmetric with respect to every group of
indices. In what follows, we will assume that this symmetry condition
is always satisfied. Using this assumption, we can considerably
reduce the number of summands in the expression for KL, since
the summands corresponding to the same number of contractions
are identical. In our example, using the symmetry of coefficients, one
notices that the second and third summands are equal.
The reduction to normal form can be represented graphically by
means of diagrams suggested by Feynman. The operator
K = K(k1, . . . , km|l1, . . . , ln)a+k1 . . . a+kmal1 . . . aln
will be denoted by a diagram consisting of one internal vertex with
m dotted incoming lines and n dotted outgoing lines (we consider
the lines to be directed topological intervals). We will say that such a
diagram is a star (see Fig. 6.1). The beginning of an incoming line will
be called an in-vertex and the end of an outgoing line will be called an
out-vertex. We will assign the indices k1, . . . , km ∈M to in-vertices;
to out-vertices, we will assign the indices l1, . . . , ln ∈ M ; and to the
internal vertex, we will assign the function K(k1, . . . , km|l1, . . . , ln).
The operator L is depicted by a similar diagram.
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92 Mathematical Foundations of Quantum Field Theory
Fig. 6.1 Star: Feyman diagram of one internal vertex.
The normal form of the operator KL can be expressed now as
a collection of all different diagrams, consisting of diagrams of the
operators K and L, where some of the out-vertices of the diagram for
the operator K are connected with solid lines with the in-vertices of
the diagram for the operator L. These solid lines will be called edges
of the diagram and we assign the contractions a+p al = [al, a+p ] = δlp
to the edges. (Here, l is the index of the end of the edge and p is the
index of the beginning of the edge.) The direction of the edge will
always be fixed as the direction from the out-vertex of the diagram of
the operator K to the in-vertex of the diagram of the operator L. For
every diagram, we construct a product in which the internal vertices
of the diagram contribute the factors K(k1, . . . , km|l1, . . . , ln) and
L(p1, . . . , pr|q1, . . . , qs), where k1, . . . , km, p1, . . . , pr are the indices
of the ends of the lines entering the vertex and l1, . . . , ln, q1, . . . , qsare the indices of the ends of the lines exiting the in-vertex. The
factor a+p al = δlp is assigned to every connecting edge. Finally, the
lines with a free end contribute the operator a+k or ak, if the line is
incoming or outgoing, respectively. We assume a summation over the
set M for each index. The operators a+k , al are assumed to be written
in normal order.
It is easy to check that the operator constructed by the diagram
described above coincides with one of the summands in the product
of the operators K and L written in normal form. Therefore, the
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Operators on Fock Space 93
operator KL is equal to the sum of operators corresponding to all the
possible diagrams (more shortly, to the sum of diagrams).
By induction, it is easy to obtain diagrams for the reduction to
normal form of a product of r operators K1, . . . ,Kr, where
Ki = Ki(k1, . . . , kmi |l1, . . . , lni)a+k1. . . a+kmi
al1 . . . alni.
The normal form of the operator K1 . . .Kr is depicted by a sum of
diagrams consisting of the diagrams of the operators K1, . . . ,Kr and
some number of lines connecting the outgoing vertices of the diagram
for the operator Ki with the incoming vertices of the diagram of the
operator Kj (here i, j = 1, . . . , r; i < j). The prescription that assigns
to a diagram an operator remains the same.
If the operator Ki =∑
m,nKm,ni , where
Km,ni = Km,n
i (k1, . . . , km|l1, . . . , ln)a+k1 . . . a+kmal1 . . . aln ,
then the diagram of the operator Ki is a set of several stars, the
diagrams of the operators Km,ni .
The normal form of the operator K1 . . .Kr is represented then
by the collection of diagrams consisting of r stars — a diagram of
every operator K1, . . . ,Kr — and several lines connecting outgoing
vertices of the star diagram of one of the operators Ki with incoming
vertices of the star diagram of one of the operators Kj , j > i.
The diagram technique in Fig. 6.1 can be applied to reduce
to the normal form a product of operators expressed in terms
of creation and annihilation operators satisfying CCR. A similar
diagram technique can also be constructed in the case of CAR; the
only difference is in some signs in front of the diagrams. We do not
describe explicitly the rules for these sign factors; however, we will
formulate accurately the analog of Wick’s theorem for the CAR case.
Let A1, . . . , Am, B1, . . . , Bn be operators having the property that
the anticommutators [Aα, Bβ]+ are numbers. The number BβAα =
[Aα, Bβ]+ is called a contraction of the operators Aα and Bβ. The
product B1 . . . BnA1 . . . Am with k contractions Bβ1Aα1 , . . . , BβkAαk
is defined as the operator obtained by deleting from the product
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94 Mathematical Foundations of Quantum Field Theory
the operators Bβ1 , . . . , Bβk , Aα1 , . . . , Aαkand including instead of
them the contractions Bβ1Aα1 , . . . , BβkAαkand the sign factor
(−1)mn+λ+µ, where λ is the sum of the lengths of all contractions
and µ is the number of pairs of overlapping contractions (the length
of a contraction BβAα is defined as n−β+α; two contractions BβAα
and Bβ′Aα′ are called overlapping if (β′ − β)(α′ − α) > 0).
Given these definitions, one can formulate Wick’s theorem in the
form that is applicable in both situations.
The product A1 . . . AmB1 . . . Bn is equal to the product
B1 . . . BnA1 . . . Am with all possible contractions (we do not exclude
the case where the number of contractions is equal to zero).
For example, in the case when the anticommutators [Aα, Bβ]+ are
numbers, we obtain
An operator, acting on the Fock space F (B), where B = L2(X),
is said to be represented in normal form if it is written in the form
A =∑m,n
∫km,n(x1, . . . , xm|y1, . . . , yn)a+(x1) . . . a
+(xm)
× a(y1) . . . a(yn)dmxdny (6.22)
(here X is a measure space and a+(x), a(x) are operator generalized
functions satisfying the relation (6.7)). The diagram techniques
discussed above can also be used to represent in normal form
the product of operators written in the form (6.22). The only
modification that is necessary in the case in question is that the
role of the indices k ∈ M is played by the elements of the space
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Operators on Fock Space 95
X, correspondingly the summation with respect to M is replaced by
integration over the space X.8
Let us use the above results to prove some useful relations. For
definiteness, we will consider the case of CCR.
First of all, we will calculate the inner product⟨a+(f1) . . . a
+(fm)θ, a+(g1) . . . a+(gn)θ
⟩=⟨a(gn) . . . a(g1)a
+(f1) . . . a+(fm)θ, θ
⟩.
The operator a(gn) . . . a(g1)a+(f1) . . . a
+(fm) can be reduced to
normal form by means of Wick’s theorem; it is equal to the sum where
every summand is the product a+(f1) . . . a+(fm) × a(gn) . . . a(g1)
with some contractions. Remembering that the vacuum average of
an operator is equal to the constant term in the representation of the
operator in normal form, we see that the quantity we would like to
calculate is equal to the sum of the summands, where all operators
are contracted (i.e. m = n is equal to the number of contractions).
Noting that
a+(fi)a(gj) = [a(gj), a+(fi)] = 〈gj , fi〉 = 〈fi, gj〉,
thus, we obtain the necessary formula⟨a+(f1) . . . a
+(fm)θ, a+(g1) . . . a+(gn)θ
⟩= δnm
∑P
〈f1, gi1〉〈f2, gi2〉 . . . 〈fm, gim〉 (6.23)
8It is useful to note that an orthonormal basis φk, with k ∈ M , for the spaceB, specifies an isomorphism between the spaces B and L2(M). (The set M isconsidered as a measure space, equipped with the counting measure, where themeasure of a finite subset is equal to the number of its elements; then the integralof f(k) over the set M is equal to
∑k∈M f(k).) This remark allows us to say
that the representation of an operator in the form (6.20) is a particular case ofrepresentation in the form (6.22); correspondingly, the diagram technique for theoperators of the form (6.22) is a generalization of the techniques for operators ofthe form (6.20).
March 26, 2020 16:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch06 page 96
96 Mathematical Foundations of Quantum Field Theory
(where the sum is taken over all permutations P = (i1, . . . , im)). The
formula (6.23) can also be expressed in the form⟨a+(x1) . . . a
+(xm)θ, a+(y1) . . . a+(yn)θ
⟩= δnm
∑P
δ(x1, yi1) . . . δ(xm, yim), (6.24)
where a+(x), a(y) are operator generalized functions satisfying the
relations (6.7).
Using the statements proved in this section, we can find the
relation between the coefficient functionsKm,n(x1, . . . , xm|y1, . . . , yn)
in the representation of an operator A in the form (6.22) (normal
form) and the matrix elements of the operator A in the generalized
basis a+(x1) . . . a+(xm)θ. To find these relations, we can, for example,
write the operator Aa+(y1) . . . a+(yn)θ in normal form by means of
Wick’s theorem, and then we can calculate the matrix element
Am,n(x1, . . . , xm|y1, . . . , yn)
=⟨Aa+(y1) . . . a
+(yn)θ, a+(x1) . . . a+(xm)θ
⟩using the formula (6.24). As a result, we obtain the following formula:
Am,n(x1, . . . , xm|y1, . . . , yn) = Km,n(x1, . . . , xm|y1, . . . , yn)m!n!
+∑i,j
Km−1,n−1(x1, . . . , xi−1, xi+1, . . . , xm|y1, . . . , yj−1, yj+1,
. . . , yn)δ(xi − yj)(m− 1)!(n− 1)! + · · ·
+∑B,B′,α
Km−r,n−r(xi, i∈B|yj , j∈B′)
×∏i∈B
δ(xi − yα(i))(m− r)!(n− r)! + · · ·
(here B and B′ consist of r elements, B ⊂ 1, . . . ,m, B′ ⊂1, . . . , n, α is a one-to-one correspondence between B and B′,
and we are summing over all possible B,B′, α; Km−r,n−r(xi, i∈B|yj ,j∈B′) denotes the function Km−r,n−r with arguments xi and yjsatisfying i∈B, j∈B′).
March 26, 2020 16:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch06 page 97
Operators on Fock Space 97
To conclude this section, we will formulate some useful definitions.
Let K and L be operators represented in normal form as follows:
K =∑m,n
∫Km,n(k1, . . . , km|l1, . . . , ln)a+(k1)
. . . a+(km)a(l1) . . . a(ln)dmkdnl,
L =∑m,n
∫Lm,n(k1, . . . , km|l1, . . . , ln)a+(k1)
. . . a+(km)a(l1) . . . a(ln)dmkdnl.
Then the normal product of operators K and L in the case of CCR
is the operator
N(KL) =∑
m,n,r,s
∫Km,n(k1, . . . , km|l1, . . . , ln)Lr,s(p1,
. . . , pr|q1, . . . , qs)a+(k1) . . . a+(km)a+(p1) . . . a
+(pr)a(l1)
. . . a(ln)a(q1) . . . a(qs)dmkdnldrpdsq,
and in the case of CAR is the operator
N(KL) =∑
m,n,r,s
(−1)nr∫Km,n(k1, . . . , km|l1, . . . , ln)Lr,s(p1,
. . . , pr|q1, . . . , qs)a+(k1) . . . a+(km)a+(p1) . . . a
+(pr)a(l1)
. . . a(ln)a(q1) . . . a(qs)dmkdnldrpdsq
(instead of using the symbol N(KL) to denote normal product, one
can also use the symbol :KL:). In other words, the normal product
is obtained if we reduce the product KL to the normal form using
the prescriptions above and delete all summands containing at least
one contraction (i.e. we delete all diagrams having at least one non-
dotted line). The normal product of n operators K1, . . . ,Kn can be
defined by induction
N(K1, . . . ,Kn) = N(N(K1, . . . ,Kn−1)Kn).
March 26, 2020 16:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch06 page 98
98 Mathematical Foundations of Quantum Field Theory
The normal exponent of the operator K is the operator
N(expK) =∞∑n=0
1
n!N(Kn).
6.4 Diagram techniques
Let us consider the Hamiltonian H = H0 + gV (t), where
H0 =∑
ε(k)a+k ak, (6.25)
where
V (t) =∑m,n
∑k1,...,km
vm,n(k1, . . . , km|p1, . . . , pn|t)a+k1 . . . a+km
×ap1 . . . apn
(here, a+k = a+(φk), ak = a(φk) are operators in the Fock space F (B)
corresponding to the orthonormal basis φk ∈ B and the index k runs
over the set M). For concreteness, we consider the case of CCR;
however, the consideration of this section can also be performed
in the case of CAR (in the case of CAR, the Hamiltonian H is
assumed to be Fermi-even, i.e. it is assumed that every summand
in the operator H contains an even number of operators a+ and a
(m+ n is even)).
Using the representation of the operator S(t, t0) in the form of
T -exponent (4.4) and Wick’s theorem (Section 6.3), we can construct
diagram techniques for the calculation of the operator S(t, t0). More
precisely, we obtain the decomposition of the normal form of S(t, t0)
as a power series in g.
Let us recall that a star is a diagram consisting of a point with m
incoming lines and n outgoing lines. The lines belonging to a star are
depicted as dotted lines (Fig. 6.1 depicts a star with three incoming
and two outgoing lines).
A diagram of the operator S(t, t0) is a collection of several stars
and several edges that are depicted as directed lines. Every edge
starts with an out-vertex of some star and ends in an in-vertex of
another star. Let us assume that two lines belonging to a diagram
cannot have common internal points and can have only one common
March 26, 2020 16:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch06 page 99
Operators on Fock Space 99
vertex. Two vertices belonging to the same edge cannot belong to
the same star.
The vertices belonging to edges are called internal vertices. Other
vertices are called external vertices.
All vertices of a diagram should be numbered in such a way that
the vertices belonging to the same star are numbered by neighboring
numbers. Two diagrams are equivalent if there exists a topological
equivalence between these diagrams transforming a vertex with some
number to a vertex with the same number.
Some examples of diagrams are drawn in Fig. 6.2. The diagrams
(a) and (b) are topologically equivalent, but are distinct, while the
diagrams (a) and (c) are equivalent.
For every diagram, we can construct an operator in the following
way. Let us assume that to every vertex of the diagram we have
assigned an index ki ∈ M and a real number ti (time). We assign
the same time to every vertex belonging to one star. Every star
contributes a factor vm,n(k1, . . . , km|p1, . . . , pn|t), where t is the time
of this star and k1, . . . , km are indices of in-vertices of this star
and p1, . . . , pn are indices of the out-vertices of the star. An edge
contributes
exp(−iε(k1)(t1 − t2))δk2k1θ(t1 − t2),
where the index k1 ∈ M and the time t1 correspond to the
beginning of the edge and k2 ∈ M and t2 correspond to the end
of the edge. To every free in-vertex, we assign an operator a+k (t) =
exp(iε(k)t)a+k , and to every free out-vertex, we assign the operator
ak(t) = ak exp(−iε(k)t) (here k and t stand for the index and the
time corresponding to the vertex, respectively).
To every diagram, we assign an operator obtained in the following
way. First of all, we take the product of functions corresponding to
the stars and edges of the diagram and the operators corresponding
to the external in- and out-vertices. The operator is obtained from
this product by means of summing over the indices of all vertices and
integration over the times of all stars (calculating S(t, t0) we should
integrate over the interval [t0, t]). The operators corresponding to the
external vertices are taken in normal order. We include the factor1n!g
n in the operator corresponding to the diagram with n stars.
March 26, 2020 16:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch06 page 100
100 Mathematical Foundations of Quantum Field Theory
Fig. 6.2 Examples of diagrams.
Let us write down, for example, the operator corresponding to
Fig. 6.2(a):
1
3!g3
∑k1,...,k9
∫dt1dt2dt3v1,2(k1|k2, k3|t1)v2,1(k4, k5|k6|t2)
× v2,1(k8, k7|k9|t3) exp(−iε(k3)(t1 − t3))δk8k3θ(t1 − t3)
× exp(−iε(k2)(t1− t2))δk4k2θ(t1 − t2) exp(−iε(k6)(t2 − t3))
× δk7k6 (t2 − t3) exp(iε(k1)t1 + iε(k5)t2
− iε(k9)t3)a+k1a+k5ak9 .
March 26, 2020 16:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch06 page 101
Operators on Fock Space 101
Let us prove that the operator S(t, t0) can be expressed as a
sum of all different diagrams, or more precisely, as a sum of all the
operators corresponding to these diagrams. Of course, this statement
(and other statements of this kind) means only that the sum of the
diagrams gives a decomposition S(t, t0) as a series in g; we cannot
say anything about the convergence of this series.
Before giving the proof of this statement, let us make several
remarks about the structure of the diagram representation of the
operator S(t, t0) and consider an example. Let us note first that we
can omit all diagrams containing n stars joined in a cyclic order by
edges. (We say that n stars are connected in cyclic order by edges if
they can be ordered in such a way that for every i, 1 ≤ i ≤ n, there
exists an edge starting in a vertex of the ith star and ending in the
vertex of the (i+1)-th star; we identify the (n+1)th star with the first
star.) Let us denote the times of these stars by τ1, . . . , τn; then the
edges introduce the number θ(τ1− τ2) . . . θ(τn−1− τn)θ(τn− τ1), and
hence in the integral defining our diagram, the integrand is equal
to zero unless τ1 = τ2 = · · · = τn, which means that the integral
vanishes.
It is easy to check that two topologically equivalent diagrams
specify the same operator, therefore one usually draws only one of
the class of topologically equivalent diagram and this diagram should
be multiplied by the number of topologically equivalent diagrams.
Let us consider as an example the Hamiltonian of the system
of non-relativist identical bosons in a box of the volume L3. Here,
H = H0 + V , where
H0 =∑ k2
2ma+k ak,
V =1
2
(2π
L
)3 ∑k1,k2,p1,p2
W(k1 − p2)δk1+k2p1+p2 × a+k1a
+k2ap1ap2 ,
and k, k1, k2, p1, p2 run over a lattice in three-dimensional space with
a spacing of 2π/L.
Let us write down the summands corresponding to the diagrams
of Fig. 6.3 in the decomposition of the adiabatic S-matrix Sα. The
March 26, 2020 16:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch06 page 102
102 Mathematical Foundations of Quantum Field Theory
Fig. 6.3 More examples of diagrams.
first of these two diagrams represents the operator
1
2
(2π
L
)3 ∫ ∞−∞
∑k1,k2,p1,p2
W(k1 − p2)δk1+k2p1+p2 exp(−α|τ |)
× a+k1(τ)a+k2(τ)a+p1(τ)a+p2(τ)dτ
=1
2
(2π
L
)3 ∑k1,k2,p1,p2
W(k1 − p2)δk1+k2p1+p2
× 2α(k212m +
k222m +
p212m +
p222m
)2+ α2
a+k1a+k2ap1ap2 .
The operator corresponding to the second diagram has the form
1
2l
∫dτ1dτ2
∑k1,k2,p1,p2
∑k′1,k
′2,p′1,p′2
1
4
(2π
L
)6
W(k1 − p2)
× δk1+k2p1+p2 exp(−α|τ1|) exp
(−i p
21
2m(τ1 − τ2)
)δp1k′1θ(τ1 − τ2)
× exp
(−i p
22
2m(τ1 − τ2)
)θ(τ1 − τ2)δp2k2W(k′1 − p′2)δ
k′1+k′2
p′1+p′2
× exp(−α|τ2|)a+k1(τ1)a+k2
(τ1)ap′1(τ2)ap′2(τ2)
(where ki, pi and k′i, p′i are summed over a lattice).
March 26, 2020 16:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch06 page 103
Operators on Fock Space 103
The third diagram is equivalent to the second; we take this into
account by multiplying the second diagram by 2.
We can reduce the number of diagrams that should be calculated
if we note that the operator corresponding to a disconnected diagram
is equal to the normal product of the operators corresponding to each
connected component.
Let us start now the derivation of the diagram representation of
the operator S(t, t0). First of all, we note that for the representation
of the operator V (τ1) . . . V (τn) in normal form, we can use the
technique developed in Section 6.3 with some modifications. (This
follows from the formula
V (τ) = exp(iH0τ)V (τ) exp(−iH0τ)
=∑m,n
∑k1,...,km
vm,n(k1, . . . , km|p1, . . . , pn|τ)a+k1(τ)
. . . a+km(τ)ap1(τ) . . . apn(τ)
=∑m,n
∑k1,...,km
exp
i m∑j=1
ε(kj)−n∑j=1
ε(pj)
τ
×vm,n(k1, . . . , km|p1, . . . , pn|τ)a+k1 . . . a
+kmap1 . . . apn).
Let us describe the diagram representation of the operator
V (τ1) . . . V (τn) in more detail. The representation consists of dia-
grams that constitute the collection of n stars and some edges that
point in the direction of increasing order (recall that the vertices of
the diagram are ordered).
To every vertex of the diagram, we assign the index k ∈ M .
To every free in-vertex (out-vertex), we assign the operator a+k (τi)
(correspondingly the operator ak(τi)). To every edge, we assign the
function exp(−iε(k)(τi − τj))δkl , where k, l are indices of the vertices
at the beginning and end of the edge and i, j are the numbers of
stars that contain the beginning and end of the edge. The operator
corresponding to the diagram is obtained from the product of the
March 26, 2020 16:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch06 page 104
104 Mathematical Foundations of Quantum Field Theory
operators and functions that correspond to free vertices, stars, and
edges. Namely, one should sum over the indices of all vertices (the
operators are written in normal order).
It follows from the calculation of Section 6.3 that the operator
V (τ1) . . . V (τn) is the sum of operators corresponding to all the
different diagrams.
Using the diagram techniques for the normal form of the operator
V (τ1) . . . V (τn), we can get the diagram technique for representing
in normal form the operator T (V (τ1) . . . V (τn)), where all times are
different. The only difference in the techniques is that we should
consider arbitrary diagrams with n stars without restriction on the
direction of the edges and to every edge we should assign the function
exp(−iε(k)(τi − τj))δkl θ(τi − τj). This follows from the remark that
the factors θ(τi − τj) in the functions assigned to the edges are
equal to 1 in the diagrams where the edges go in the direction
of decreasing time and are equal to 0 in all other diagrams. This
means that the diagrams that are not equal to zero coincide with
the diagrams from the diagram representation for V (τi1) . . . V (τin),
where i1, . . . , in is a permutation where τi1 > · · · > τin . Noting that
V (τi1) . . . V (τin) = T (V (τ1) . . . V (τn)), we obtain the representation
that we have claimed.
To get a diagram representation of the operator S(t, t0), it is
sufficient to refer to the expression of this operator as a T -exponent
(4.4) and to note that in the integral over τ1, . . . , τn the value of the
integrand on the set where at least two variables τ1, . . . , τn coincide
is irrelevant.
The diagram representation of the operator S(t, t0) can be easily
obtained in the case when the index k runs over an arbitrary measure
space X. More precisely, let us consider the Fock space F (L2(X)),
where X is a measure space, and in this space we define the operator
generalized functions a+(k), a(k), obeying
[a(k), a(k′)] = [a+(k), a+(k′)] = 0,
[a(k), a+(k′)] = δ(k, k′); a(k)θ = 0.
March 26, 2020 16:10 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch06 page 105
Operators on Fock Space 105
Let us fix the Hamiltonian H = H0 + gV (t), where
H0 =
∫ε(k)a+(k)a(k)dk,
V (t) =∑m,n
∫vm,n(k1, . . . , km|p1, . . . , pn|t)a+(k1)
. . . a(pn)dmkdnp.
We would like to find the normal form of the operator S(t, t0)
for this Hamiltonian. The diagram technique to solve this problem
is analogous to the technique described above. The only difference is
that to every vertex of the diagram we should assign a point k ∈ Xinstead of k ∈M and we should replace the summation with respect
to M with integration with respect to X.
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March 26, 2020 16:33 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch07 page 107
Chapter 7
Wightman and Green Functions
7.1 Wightman functions
Let us consider a Hilbert space H, a Hamiltonian H with the ground
state Φ, and a family of operators a(λ)λ∈Λ closed with respect
to Hermitian conjugation. More precisely, let us assume that the
operators a(λ) are defined on a dense subspace D ⊂ H and for every
operator a(λ), there exists an operator a(λ) that satisfies the relation
〈a(λ)x, y〉 = 〈x, a(λ)y〉 for every x, y ∈ D. The symbol a(λ, t) will
denote the Heisenberg operators a(λ, t) = exp(iHt)a(λ) exp(−iHt).We will assume that the operators a(λ) and exp(iHt) transform
the set D into itself. Note that under these assumptions, the
operators a(λ1, t1) . . . a(λn, tn) also transform D into itself and that
(a(λ1, t1) . . . a(λn, tn))+ = a+(λn, tn) . . . a+(λ1, t1)
= a(λn, tn) . . . a(λ1, t1).
We will assume that the ground state Φ belongs to the set D; its
energy will be denoted by E0.
Definition 7.1. The Wightman function wn(λ1, t1, . . . , λn, tn) of the
Hamiltonian H is the expectation value of the operator
a(λ1, t1) . . . a(λn, tn)
in the ground state Φ of the Hamiltonian H:
wn(λ1, t1, . . . , λn, tn) = 〈a(λ1, t1) . . . a(λn, tn)Φ,Φ〉.
107
March 26, 2020 16:33 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch07 page 108
108 Mathematical Foundations of Quantum Field Theory
Everything in this chapter can also be easily proved in the case
when a(λ) is a generalized operator function of the parameter λ.
Naturally, in this case, the Wightman function will be a generalized
function of the variables λ1, . . . , λn.
It is easy to prove the following properties of Wightman functions:
(1) Invariance with respect to time translation:
wn(λ1, t1 + τ, . . . , λn, tn + τ) = wn(λ1, t1, . . . , λn, tn).
(7.1)
(2) Hermiticity:
wn(λ1, t1, . . . , λn, tn) = wn(λn, tn, . . . , λ1, t1). (7.2)
(3) Positivity: Let fn(λ1, t1, . . . , λn, tn) denote a sequence of func-
tions that doesn’t vanish only for finitely many values of n. Let
us also assume that the functions belonging to this sequence do
not vanish only on finitely many values of λ1, t1, . . . , λn, tn. Then∑m,n,λi,µj ,ti,τj
fm(λ1, t1, . . . , λm, tm)fn(µ1, τ1, . . . , µn, τn)
×wm+n(λ1, t1, . . . , λm, tm, µn, τn, . . . , µ1, τ1) ≥ 0
(7.3)
(the sum is taken over λi, µj ∈ Λ and m,n = 0, 1, . . . ; −∞ <
ti, τj <∞).
(4) Spectral property:∫exp(−iωa)wn(λ1, t1, . . . , λk, tk, λk+1, tk+1
+ a, . . . , λn, tn + a)da = 0 (7.4)
if the number ω + E0 does not belong to the spectrum of the
Hamiltonian H (in particular, the relation (7.4) is true for every
ω < 0).
The proof of property (1) follows immediately from exp(−iHt)Φ = exp(−iE0t)Φ. Property (2) follows from 〈a(λ1, t1) . . .
March 26, 2020 16:33 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch07 page 109
Wightman and Green Functions 109
a(λn, tn)Φ,Φ〉 = 〈Φ, a(λn, tn) . . . a(λ1, t1)Φ〉. Property (3) follows
from applying the inequality 〈AA+Φ,Φ〉 ≥ 0 to the operator
A =∑m,λi,ti
fm(λ1, t1, . . . , λm, tm)a(λ1, t1) . . . a(λm, tm),
where the sum is taken over m = 0, 1, . . . ; λi ∈ Λ; −∞ < ti <∞.
The fourth property follows from the relation∫exp(−iβa) 〈exp(iBa)Ψ1,Ψ2〉 da = 0,
which is correct if the number β does not belong to the spectrum of
the self-adjoint operator B (see Appendix A.5) (we should assume
that B = H, Ψ1 = UΦ, Ψ2 = V +Φ, where
U = a(λk+1, tk+1) . . . a(λn, tn), V = a(λ1, t1) . . . a(λk, tk)).
If the Hamiltonian H acts on a representation space of CR (on
the Fock space F (B), for example), we can consider as the family
of operators a(λ) the family a(k, ε) of annihilation and creation
operators that correspond to some orthonormal basis φk ∈ B (here,
k ∈M , ε = ±1, a(k, ε) = a(φk, ε)).
In other words, the Wightman functions of the Hamiltonian H
acting on a representation space of CR are functions
wn(k1, ε1, t1, . . . , kn, εn, tn)
= 〈a(k1, ε1, t1) . . . a(kn, εn, tn)Φ,Φ〉 .
In the case of CAR, the operators a(k, ε, t) are bounded and
hence the Wightman functions are well defined. In the case of CCR,
we assume that the ground state Φ belongs to the domain D of
operators a(k, ε) and that the domain D is invariant with respect to
the operators exp(iHt).
For Hamiltonians acting on a representation space of CR, we can
prove one more property of Wightman functions.
(5) Permutation of arguments: Let the functions w(i)n be the func-
tions obtained from wn by swapping (ki, εi, ti) and (ki+1, εi+1, ti+1)
March 26, 2020 16:33 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch07 page 110
110 Mathematical Foundations of Quantum Field Theory
(with a change of sign in the case of CAR). If ti = ti+1, then
w(i)n (k1, ε1, t1, . . . , kn, εn, tn)
= wn(k1, ε1, t1, . . . , kn, εn, tn) +Aεiεi+1δki,ki+1
wn−2
× (k1, ε1, t1, . . . , ki−1, εi−1, ti−1, ki+2, εi+2, ti+2, . . . , kn, εn, tn)
(7.5)
(here, Aεε′ is the matrix defined in formula (6.3)).
To check this equation, we note that from the commutation
relations in the form (6.3), we can obtain
[a(k, ε, t), a(k′, ε′, t)]∓ = [a(k, ε), a(k′, ε′)]∓ = Aεε′δkk′ .
Wightman functions can be easily calculated for the Hamiltonian
H0 =∑ω(k)a+(k)a(k) in Fock space because for this Hamiltonian,
we have a(k, ε, t) = exp(iεω(k)t)a(k, ε), and hence,
wn(k1, ε1, t1, . . . , kn, εn, tn)
= exp(i(ε1ω(k1)t1 + · · ·+ εnω(kn)tn)) 〈a(k1, ε1)
. . . a(kn, εn)Φ,Φ〉 .
In the case of CCR, the operator H0 is bounded from below
when ωk ≥ 0; then Φ = θ (see Section 6.2) and the functions
w2(k1, ε1, t1, k2, ε2, t2) do not vanish only in the case when ε1 =
−1, ε2 = 1; in this case, we have
w2(k1 − 1, t1, k2 − 1, t2)
= δk1,k2 exp(−iω(k1)t1 + iω(k2)t2).
In the case of CAR, the ground state is Φ =∏ω(k)<0 a
+(k)θ. In
this case, the function w2(k1, ε1, t1, k2, ε2, t2) does not vanish only if
ε2ω(k2) > 0, ε1ω(k1) < 0; then
w2(k1, ε1, t1, k2, ε2, t2) = δk1,k2 exp(iε1ω(k1)t1 + iε2ω(k2)t2).
In other words,
w2(k1, ε1, t1, k2, ε2, t2)
= θ(ε2ω(k2))θ(−ε1ω(k1))δk1k2
× exp(i(ε1ω(k1)t1 + ε2ω(k2)t2)).
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Wightman and Green Functions 111
For Hamiltonians acting on the space of representations CR(B),
one can give a more invariant definition of Wightman functions that
does not depend on the choice of basis in the space B. Namely, for the
Wightman function wn(f1, ε1, t1, . . . , fn, εn, tn), where fi ∈ B, εi =
±1,−∞ < ti < ∞, we can consider the expectation value of the
operator a(f1, ε1, t1) . . . a(fn, εn, tn) with respect to the ground state
Φ of the Hamiltonian H (wn is a linear functional of f1, . . . , fn). If
the space B = L2(X), then we can write the operator a(f, ε) in the
form a(f, ε) =∫f(x)a(x, ε)dx, where a(x, ε) is a generalized operator
function (x ∈ X, ε = ±1). It is easy to check that
wn(f1, ε1, t1, . . . , fn, εn, tn)
=
∫f1(x1) . . . fn(xn)wn(x1, ε1, t1, . . . , xn, εn, tn)dx1 . . . dxn,
where wn(x1, ε1, t1, . . . , xn, εn, tn) is the Wightman function con-
structed by means of the generalized operator function a(x, ε) and
the Hamiltonian H.
7.2 Green functions
Let us define Green functions for Hamiltonians acting on the space
of representations of canonical commutation or anticommutation
relations. As in Section 7.1, we use the notation a(k, ε) = a(φk, ε),
where φk is an orthonormal basis in B.
Let us define the T -product of Heisenberg operators T (a(k1, ε1, t1)
. . . a(kn, εn, tn)) as the product of the operators a(k1, ε1, t1) . . .
a(kn, εn, tn) in chronological order (in order of decreasing times). In
the case of CAR, this product is taken with a minus sign if we need
an odd permutation to put the operators in chronological order; if
we need an even permutation, we take it with a plus sign. In other
words,
T (a(k1, ε1, t1) . . . a(kn, εn, tn))
= (−1)γa(ki1 , εi1 , ti1) . . . a(kin , εin , tin), (7.6)
where P = (i1, . . . , in) is the permutation obeying ti1 > . . . tin , γ = 0
in the case of CCR and γ is equal to the parity of the permutation P
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112 Mathematical Foundations of Quantum Field Theory
in the case of CAR (here, ki ∈ M , εi = ±1 and all the times ti are
distinct).
For the case of n = 2, we can write
T (a(k1, ε1, t1)a(k2, ε2, t2))
= θ(t1 − t2)a(k1, ε1, t1)a(k2, ε2, t2)
± θ(t2 − t1)a(k2, ε2, t2)a(k1, ε1, t1),
where we use the upper sign in the case of CCR and we use the lower
sign in the case of CAR.
The Green function of the Hamiltonian H is defined as an
expectation value of the T -product T (a(k1, ε1, t1) . . . a(kn, εn, tn))
with respect to the ground state Φ of the Hamiltonian H:
Gn(k1, ε1, t1, . . . , kn, εn, tn)
= 〈T (a(k1, ε1, t1) . . . a(kn, εn, tn))Φ,Φ〉.
Green functions can be easily expressed in terms of Wightman
functions, e.g.,
G2(k1, ε1, t1, k2, ε2, t2) = θ(t1 − t2)w2(k1, ε1, t1, k2, ε2, t2)
± θ(t2 − t1)w2(k2, ε2, t2, k1, ε1, t1).
The general expression of Green functions in terms of Wightman
functions can be written in the following way:
Gn(k1, ε1, t1, . . . , kn, εn, tn)
=∑π
(−1)γ(π)θπ(t)wπn(k1, ε1, t1, . . . , kn, εn, tn), (7.7)
where the sum is taken over all permutations π = (i1, i2,
. . . , in), θπ(t) = θ(ti1−ti2)θ(ti2−ti3) . . . θ(tin−1−tin) and the functions
wπn are defined by the relation
wπn(k1, ε1, t1, . . . , kn, εn, tn)
= w(ki1 , εi1 , ti1 , . . . , kin , εin , tin),
γ(π) = 0 in the case of CCR, and γ is equal to the parity of the
permutation π in the case of CAR.
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Wightman and Green Functions 113
The definition of the functions Gn(k1, ε1, t1, . . . , kn, εn, tn) is
meaningful only when all the times t1, . . . , tn are distinct. The set
where a Green function is not defined has measure zero, therefore, the
integral of a Green function multiplied by a conventional function of
t1, . . . , tn is well defined. This means, for example, that the function
Gn can be considered as a well-defined generalized function of the
variables t1, . . . , tn. We will need to specify the definition of a Green
function in the case when some of the arguments t1, . . . , tn coincide
only in one situation in Section 7.4. We will do this by requiring that
when the time variables coincide, the corresponding operators in the
T -product are written in normal order (i.e. the T -product is defined
with respect to such a permutation of indices (i1, . . . , in), such that
ti1 ≥ · · · ≥ tin and in the case of tik = til , we have εik ≥ εil).Let us formulate the following simple properties of Green
functions:
(1) Symmetry: Green functions Gn(k1, ε1, t1, . . . , kn, εn, tn) do not
change under permutation of variables in the case of CCR. In
the case of CAR, the Green function changes only by a sign
under a permutation of (ki, εi, ti) and (kj , εj , tj).
(2) Invariance with respect to time translation:
Gn(k1, ε1, t1 + a, . . . , kn, εn, tn + a)
= Gn(k1, ε1, t1, . . . , kn, εn, tn).
(3) Gn(k1, ε1, t+ 0, k2, ε2, t, k3, ε3, t3, . . . , kn, εn, tn)
∓Gn(k1, ε1, t1, k2, ε2, t+ 0, k3, ε3, t3, . . . , kn, εn, tn)
= δk1,k2Aε1ε2Gn−2(k3, ε3, t3, . . . , kn, εn, tn).
The first property follows from the fact that we can transpose
factors under the sign of T -product. The third property follows from
commutation relations. As for Wightman functions, one can give a
definition of Green functions which does not depend on the choice of
basis, namely, we should set
Gn(f1, ε1, t1, . . . , fn, εn, tn)
= 〈T (a(f1, ε1, t1) . . . a(fn, εn, tn))Φ,Φ〉
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114 Mathematical Foundations of Quantum Field Theory
(here, fi ∈ B, εi = ±1,−∞ < ti <∞). If B = L2(X), then
Gn(f1, ε1, t1, . . . , fn, εn, tn)
=
∫f1(x1) . . . fn(xn)Gn(x1, ε1, t1, . . . , xn, εn, tn)dx1 . . . dxn,
where Gn(x1, ε1, t1, . . . , xn, εn, tn) is a Green function constructed by
means of the generalized operator function a(x, ε).
It is easy to construct a perturbation series for Green functions as
well as a diagrammatic representation for this series. Let us describe
this construction.
Suppose that the Hamiltonian H is written in the form H =
H0 + gV . The ground states of the Hamiltonians H and H0 will be
denoted by Φ and Φ0 correspondingly. Let us use the relation
Φ = lima→0
C1(α)Sα(0,−∞)Φ0 = lima→0
C2(α)Sα(0,+α)Φ0 (7.8)
that follows from (4.11) (our goal is to get the representation of Green
functions in terms of perturbation series, therefore it is sufficient to
know that the relation (7.8) is true in the framework of perturbation
theory in the case when the ground states Φ0 of the Hamiltonian H0
is not degenerate). If (i1, . . . , in) is a permutation obeying ti1 > · · · >tin , then
Gn(k1, ε1, t1, . . . , kn, εn, tn)
= limα→0
C1(α)C2(α)× 〈a(ki1 , εi1 , ti1) . . . a(kin , εin , tin)
×Sα(0,−∞)Φ0, Sα(0,+∞)Φ0〉.
Let us introduce the operators
a(k, ε, t) = exp(iH0t)a(k, ε) exp(−iH0t).
It is easy to check that
a(k, ε, t) = exp(iHt) exp(−iH0t)a(k, ε, t) exp(iH0t) exp(−iHt)
= S(0, t)a(k, ε, t)S(t, 0).
For fixed t and α→ 0, we can write the approximate equations
S(t, 0) ≈ Sα(t, 0), S(0, t) ≈ Sα(0, t),
a(k, ε, t) ≈ Sα(0, t)a(k, ε, t)Sα(t, 0).
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Wightman and Green Functions 115
Using these formulas, we obtain
Gn(k1, ε1, t1, . . . , kn, εn, tn)
= limα→0
C1(α)C2(α)
× 〈Sα(0, ti1)a(ki1 , εi1 , ti1)Sα(ti1 , 0)Sα(0, ti2) . . .
× a(kin , εin , tin)Sα(tin , 0)Sα(0,−∞)Φ0, Sα(0,+∞)Φ0〉
= limα→0
〈AαΦ0,Φ0〉〈Sα(∞,−∞)Φ0,Φ0〉
,
where
Aα = Sα(∞, ti1)a(ki1 , εi1 , ti1)Sα(ti1 , ti2)
. . . a(kin , εin , tin)Sα(tin ,−∞). (7.9)
We have used the group property
Sα(t′′, t) = Sα(t′′, t′)Sα(t′, t),
and the relation
1 = 〈Φ,Φ〉 = limα→0
C1(α)C2(α) 〈Sα(0,−∞)Φ0, Sα(0,+∞)Φ0〉
= limα→0
C1(α)C2(α) 〈Sα(∞,−∞)Φ0,Φ0〉
in the derivation of (7.9). Equation (7.9) can be rewritten in the
following way:
Aα = T (a(k1, ε1, t1) . . . a(kn, εn, tn))
× exp
(1
i
∫ ∞−∞
g exp(−α|τ |)V (τ)dτ
)
=
∞∑n=0
1
n!
(gi
)n×∫T (a(k1, ε1, t1) . . . a(kn, εn, tn)
× exp(−α|τ1| − · · · − α|τn|)V (τ1) . . . V (τn))dnτ.
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116 Mathematical Foundations of Quantum Field Theory
One can use the short-hand notation
Aα = T (a(k1, ε1, t1) . . . a(kn, εn, tn)Sα(∞,−∞))
for the operator Aα.
Let us assume now that the operator H = H0 + gV is written in
the form (6.26) with the functions Vm,n that do not depend on time.
For the operator Sα(∞,−∞) corresponding to this Hamiltonian,
one can construct the diagram representation using the results of
Section 6.3. Using this representation, we can also construct a dia-
gram representation for the operator Aα. The diagram representation
for the numbers 〈AαΦ0,Φ0〉 and 〈Sα(∞,−∞)Φ0,Φ0〉 is obtained from
the diagrams of the operators Aα and Sα(∞,−∞) if we exclude from
the diagrams of these operators the diagrams where there are vertices
that do not belong to any edges. Using the fact that disconnected
diagrams decompose as a product of diagrams corresponding to
the connected components, it is easy to construct the diagram
representation for the expression
〈AαΦ0,Φ0〉〈Sα(∞,−∞)Φ0,Φ0〉
. (7.10)
It consists of the diagrams 〈AαΦ0,Φ0〉 that do not contain as a
component a diagram for 〈Sα(∞,−∞)Φ0,Φ0〉. To get a diagram
representation for a Green function Gn(k1, ε1, t1, . . . , kn, εn, tn), we
should take the limits α tending to zero in the diagrams for the
expression (7.10).
7.3 Kallen–Lehmann representation
Let us consider the Wightman function wn of the Hamiltonian H
with respect to the family of operators a(λ). It is easy to express
this function in terms of matrix entries of the operator a(λ) in the
basis of the eigenvectors of the Hamiltonian H. Let us prove such
an expression (called the Kallen–Lehmann representation) for the
function w2. Let us assume first that the operator H has discrete
spectrum, i.e. there exists an orthonormal basis Φn of eigenvectors
of the operator H with the corresponding eigenvalues denoted by En.
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Wightman and Green Functions 117
Then
w2(λ1, t1, λ2, t2) = 〈a(λ1, t1)a(λ2, t2)Φ,Φ〉
=⟨a(λ2, t2)Φ, a(λ1, t1)Φ
⟩=∑n
〈a(λ2, t2)Φ,Φn〉⟨
Φn, a(λ1, t1)Φ⟩
=∑n
〈exp(iHt2)a(λ2) exp(−iHt2)Φ,Φn〉
×⟨
Φn, exp(iHt1)a(λ1) exp(−iHt1)Φ⟩
=∑n
exp(i(En − E0)t2) 〈a(λ2)Φ,Φn〉
× exp(i(E0 − En)t1)⟨
Φn, a(λ1)Φ⟩.
Introducing the notation ρλn = 〈Φn, a(λ)Φ〉, we obtain the
representation
w2(λ1, t1, λ2, t2) =∑n
exp[−i(En − E0)(t1 − t2)]ρλ1n ρλ2n .
If the operator H additionally has continuous spectrum, then the
complete system of eigenvectors consists of normalized eigenvectors
Φn and generalized eigenvectors Φγ (we assume that the vectors
Φn are orthonormal and Φγ are δ-normalized; the corresponding
eigenvalues are denoted by En and Eγ). Repeating the above
calculations and using the relation
〈x, y〉 =∑n
〈x,Φn〉 〈Φn, y〉+
∫〈x,Φγ〉 〈Φγ , y〉 dγ,
we see that
w2(λ1, t1, λ2, t2) =∑n
exp[−i(En − E0)(t1 − t2)]ρλ1n ρλ2n
+
∫exp[−(Eγ − E0)(t1 − t2)]ρλ1γ ρ
−λ2γ dγ,
(7.11)
where
ρλn = 〈Φn, a(λ)Φ〉 , ρλγ = 〈Φγ , a(λ)Φ〉 .
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118 Mathematical Foundations of Quantum Field Theory
It is clear from the above formulas that the asymptotic behavior
of the function w2 as t1− t2 →∞ is determined only by the discrete
spectrum because the contribution of the continuous spectrum for
t1 − t2 →∞ is an integral of quickly oscillating functions.1
Let us note that the statement about the asymptotic behavior
of the function w2 should be modified in the case when a(λ) is a
generalized operator function.
The function w2 can also be expressed in terms of
σλ1,λ2(µ) = 〈a(λ1Eµa(λ)Φ,Φ)〉 ,
where Eµ = eµ(H) is the spectral decomposition of the operator H
(recall that eµ(x) = 1 for x ≤ µ, eµ(x) = 0 for x > µ), namely,
w2(λ1, t1, λ2, t2) =
∫exp(i(µ− E0)(t2 − t1))dσλ1,λ2(µ).
Indeed,
w2(λ1, t1, λ2, t2) = 〈a(λ2, t2)Φ, a(λ1, t1)Φ〉
= 〈exp(iHt2)a(λ2) exp(−iE0t2)Φ, exp(iHt1)
× a(λ1) exp(−iE0t1)Φ〉
= 〈exp[i(H − E0)(t2 − t1)]a(λ2)Φ, a(λ1)Φ〉
=
∫exp[i(µ− E0)(t2 − t1)]d〈Eµa(λ2)Φ, a(λ1)Φ〉
=
∫exp[i(µ− E0)(t2 − t1)]dσλ1,λ2(µ).
The Green function G2 of the operator acting on the representa-
tion space of CR can be expressed in terms of the function w2 by
means of the simple formula
G2(k1, ε1, t1, k2, ε2, t2) = θ(t1 − t2)w2(k1, ε1, t1, k2, ε2, t2)
± θ(t2 − t1)w2(k2, ε2, t2, k1, ε1, t1).
1A rigorous proof of this statement can be given under the assumption ofabsolute continuity of the continuous spectrum.
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Wightman and Green Functions 119
Representations for the function G2 follow immediately from repre-
sentations of w2. For example, for operators with a discrete spectrum,
we have
G2(k1, ε1, t1, k2, ε2, t2)
= θ(t1 − t2)∑n
exp[−i(En − E0)(t1 − t2)]ρk1,−ε1n ρk2,ε2n
± θ(t2 − t1)∑n
exp[−i(En − E0)(t2 − t1)]ρk2,−ε2n ρk1,ε1n
(here, ρk,εn = 〈Φn, a(k, ε)Φ〉 and Φn are the stationary states of the
Hamiltonian H and En are the corresponding eigenvalues).
Let us also consider the functions
w2(k1, ε1, ω1, k2, ε2, ω2)
= (2π)−1
∫exp[i(ε1ω1t1 + ε2ω2t2)]
w2(k1, ε1, t1, k2, ε2, t2)× dt1dt2;
G2(k1, ε1, ω1, k2, ε2, ω2)
= (2π)−1
∫exp[i(ε1ω1t1 + ε2ω2t2)]
×G2(k1, ε1, t1, k2, ε2, t2)dt1dt2
(the Wightman and Green functions in energy representations).
The representations of w2 and G2 imply representations of the
functions w2 and G2. In particular, for operators with discrete
spectrum, we have
G2(k1, 1, ω1, k2,−1, ω2) = G(k1, k2, ω1)δ(ω1 − ω2),
where
G(k1, k2, ω) =
∫exp(iωτ)G2(k1, 1, τ, k2,−1, 0)dτ
= i∑n
ρk1,−1n ρk2,−1
n
ω − (En − E0) + i0∓∑n
ρk2,1n ρk1,−1n
ω + (En − E0)− i0.
(7.12)
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120 Mathematical Foundations of Quantum Field Theory
Hence, the poles of the function G with respect to the variable ω
correspond to points in the discrete spectrum. If H has continuous
spectrum, we should add an integral over the continuous spectrum
to the expression (7.12). In all physical examples, the contribution
of the continuous spectrum is a continuous function of ω.
7.4 The equations for Wightman and Green functions
Let us consider the Hamiltonian H acting on the representation space
of CR (B). We will represent this Hamiltonian in normal form:
H =∑m,n
∑ki,lj
Γm,n(k1, . . . , km|l1, . . . , ln)a+k1. . . a+
kmal1 . . . aln ,
where a+k = a+(φk), ak = a(φk), φk is an orthonormal basis in
B, the functions Γm,n are symmetric with respect to variables
ki and lj in the case of CCR and antisymmetric in the case of
CAR. Then the Heisenberg equations for the operators ak(t) =
exp(iHt)ak exp(−iHt), a+k (t) = exp(iHt)a+
k exp(−iHt) can be writ-
ten in the form1
i
dak(t)
dt= [H, ak(t)]
= −∑
m,n,k1,...,km−1,l1,...,ln
mΓm,n(k, k1, . . . , km−1|l1, . . . , ln)
× a+k1
(t) . . . a+km−1
(t)al1(t) . . . aln(t);
1
i
da+k (t)
dt= [H, a+
k (t)]
=∑
m,n,k1,...,km−1,l1,...,ln
nΓm,n(k, k1, . . . , km|l1, . . . , ln−1, k)
× a+k1
(t) . . . a+km
(t)al1(t) . . . aln−1(t).
These equations immediately imply equations for Wightman
functions
wn(k1, ε1, t1, . . . , kn, εn, tn).
Let us calculate, for example, the expression of the derivative of
the function wn with respect to the variable t1 in terms of Wightman
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Wightman and Green Functions 121
functions (the expressions for the derivatives of wn with respect to
ti have similar form). It is easy to see that
1
i
∂
∂t1wr(k1, ε1, t1, . . . , kr, εr, tr)
= −δε1−1
∑m,n,pi,qj
mΓm,n(k1, p1, . . . , pm−1|q1, . . . , qn)
×wr+m+n−1(p1, 1, t1, . . . , pm−1, 1, t1, q1,−1, t1,
. . . , qn,−1, t1, k2, ε2, t2, . . . , kr, εr, tr)
+ δε1+1
∑m,n,pi,qj
nΓm,n(p1, . . . , pm|q1, . . . , qn−1, k1)
×wr+m+n−1(p1, 1, t1, . . . , pm, 1, t1, q1,−1, t1,
. . . , qn−1,−1, t1, k2, ε2, t2, . . . , kr, εr, tr). (7.13)
It is easy to derive the equations for Green functions from (7.13) or
directly from Heisenberg equations. Let us calculate, for example,
∂
∂t1G2(k1, ε1, t1, k2, ε2, t2)
=∂
∂t1[w2(k1, ε1, t1, k2, ε2, t2)θ(t1 − t2)
±w2(k2, ε2, t2, k1, ε1, t1)θ(t2 − t1)]
= θ(t1 − t2)∂
∂t1w2(k1, ε1, t1, k2, ε2, t2)
± θ(t2 − t1)∂
∂t1w2(k2, ε2, t2, k1, ε1, t1)
+ δ(t1 − t2)[w2(k1, ε1, t1, k2, ε2, t2)
∓w2(k2, ε2, t2, k1, ε1, t1)].
Using CR (or property (5) of Wightman functions), we can
simplify the last summand:
δ(t1 − t2)(w2(k1, ε1, t1, k2, ε2, t2)∓ w2(k2, ε2, t2, k1, ε1, t1))
= Aε1ε2δ(t1 − t2)δk1,k2 .
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122 Mathematical Foundations of Quantum Field Theory
The first two summands in the formula for ∂G2∂t1
can be expressed
in terms of Green functions (one should use (7.13)). As a result, we
obtain the equation
1
i
∂
∂t1G2(k1, ε1, t1, k2, ε2, t2) =
1
iδ(t1 − t2)Aε1ε2δk1,k2
− δε1−1
∑m,n,pi,qj
mΓm,n(k1, p1, . . . , pm−1|q1, . . . , qn)
×Gm+n(p1, 1, t1, . . . , pm−1, 1, t1, q1,−1, t1,
. . . , qn,−1, t1, k2, ε2, t2)
+ δε1+1
∑m,n,pi,qj
nΓm,n(p1, . . . , pm|q1, . . . , qn−1, k1)
×Gm+n(p1, 1, t1, . . . , pm, 1, t1, q1,−1, t1,
. . . , qn−1,−1, t1, k2, ε2, t2).
Similarly, one can derive the equations for Green functions Gn. We
will not write down the form of these equations since, in more general
situations, they are very complicated.
March 27, 2020 8:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch08 page 123
Chapter 8
Translation-Invariant Hamiltonians
8.1 Translation-invariant Hamiltonians in Fock space
Let us consider operators acting on the Fock space F (B), where
B = L2(E3) is the space of square-integrable functions ψ(k) on the
Euclidean space E3 with variable k denoting the momentum.1
It is convenient to express these operators in terms of operator
generalized functions a+(k), a(k) satisfying the commutation rela-
tions
[a(k), a(k′)]∓ = [a+(k), a+(k′)]∓ = 0,
[a(k), a+(k′)]∓ = δ(k− k′).
These operator generalized functions were defined in Section 3.2.
The Hamiltonian H is called translation-invariant if it commutes
with the momentum operators P =∫ka+(k)a(k)dk. An example
of a translation-invariant Hamiltonian is the Hamiltonian for a
system of interacting non-relativistic identical particles considered
in Section 3.2.
We point out the following easy-to-check statement: the vac-
uum vector θ is always an eigenvector of a translation-invariant
Hamiltonian H.
1The case B = L2(E3) corresponds to identical spinless particles. If we considerparticles with spin, or if we have several types of particles, we should considerB = L2(E3 × B), where B is a finite set. It is easy to generalize all the resultsof the present chapter and the following chapters to this case, hence there is noneed to describe the generalization in detail. It is important to note, however, thatfermions always have spin, therefore, in the case of CAR, the situation B = L2(E3)that we consider does not correspond to any particle existing in nature.
123
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124 Mathematical Foundations of Quantum Field Theory
To prove this statement, we note that the momentum operator P
has precisely one (up to a factor) eigenvector θ that corresponds to
a zero eigenvalue (here, we consider conventional — normalized —
eigenvectors; of course, P has generalized eigenvectors). Since we
have assumed that the operator H commutes with the operator
P, it transforms the single eigenvector of the operator P into an
eigenvector of P. Therefore, we see that Hθ = λθ, which proves our
statement.
Let us assume that the Hamiltonian H is written in the normal
form
H =∑m,n
∫Γm,n(p1, . . . ,pm|q1, . . . ,qn)a+(p1) . . . a+(pm)a(q1)
. . . a(qn)dmpdnq. (8.1)
It is easy to check that it is translation-invariant only in the case
when the functions Γm,n have the form
Γm,n(p1, . . . ,pm|q1, . . . ,qn) = Λm,n(p1, . . . ,pm|q1, . . . ,qn)
× δ(p1 + · · ·+ pm − q1 − · · · − qn).
The statement we have proven implies that expression (8.1) cannot
define a self-adjoint operator on Fock space if for some m the function
Γm,0 6≡ 0 (in physical terms, the Hamiltonian H generates vacuum
polarization in this case).2
The above statement implies that we should consider translation-
invariant operators with vacuum polarization outside the Fock space
framework. It is not that we should consider these Hamiltonians to
be bad, but that Fock space is too narrow for these Hamiltonians.3
We see that we should consider translation-invariant Hamil-
tonian (8.1) as a formal expression, composed of the symbols
a+(k), a(k). We will show how we can construct various physical
2Indeed, in the case of vacuum polarization, the expression (8.1) does notdetermine an operator on Fock space at all (see Section 3.2 for details).
3To construct an operator on Fock space corresponding to a translation-invariantHamiltonian with vacuum polarization, we should make the volume or, in otherwords, the infrared cutoff (see below).
March 27, 2020 8:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch08 page 125
Translation-Invariant Hamiltonians 125
quantities for such Hamiltonians. In particular, in Chapter 9, we will
give a definition for the scattering matrix of a translation-invariant
Hamiltonian.
We will use the following construction. Let H be a translation-
invariant Hamiltonian, i.e. a formal expression of the form
H =∑m,n
∫Λm,n(k1, . . . ,km|l1, . . . , ln)
× δ(k1 + · · ·+ km − l1 − · · · − ln)
× a+(k1) . . . a+(km)a(l1) . . . a(ln)dmkdnl. (8.2)
We will assume that the expression for H is formally Her-
mitian (i.e. we assume that Λm,n(k1, . . . ,km|l1, . . . , ln) =
Λn,m(ln, . . . , l1|km, . . . ,k1)). Let us define a Hamiltonian H with
volume cutoff Ω. We will denote by Ω the cube with edge length
L in coordinate space (0 ≤ x ≤ L, 0 ≤ y ≤ L, 0 ≤ z ≤ L). Then,
we denote by BΩ = L2(Ω) the space of square-integrable functions
ψ(r), where r ∈ Ω, and by FΩ = F (BΩ), we denote the Fock space
constructed with the Hilbert space BΩ. In the space BΩ, we select an
orthonormal basis of functions φk = L−3/2 exp(−ikr), where k runs
over the lattice TΩ with step size 2πL (i.e. k = 2π
L n, where n is an
integer vector). The operators a+(φk), a(φk) in FΩ corresponding to
this basis will be denoted by a+k , ak.
The operator HΩ on the space FΩ will be defined by the formula
HΩ =∑m,n
∑ki,lj
(2π
L
) 32
(m+n−2)
Λm,n(k1, . . . ,km|l1, . . . , ln)
× δk1+···+km,l1+···+lna+k1. . . a+
kma+l1. . . a+
ln(8.3)
(in other words, we obtain the expression for HΩ from the expression
for H by replacing a+(k), a(k) with ( L2π )3/2a+k , (
L2π )3/2ak, replacing
the integration with summation over the lattice TΩ and multiply-
ing by (2πL )3, and replacing the function δ(k1 + · · ·+km−l1−· · ·−ln)
by ( L2π )3δk1+···+km,l1+···+ln).
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126 Mathematical Foundations of Quantum Field Theory
Let us assume that the formula (8.3) specifies a self-adjoint
operator on the space FΩ. If the functions Λm,n are “good” enough,
then this assumption holds. For example, in the case of CAR, one
can prove the following statements.
If the coefficient functions Λm,n tend to zero at infinity faster
than q−3(m+n)−ε where q = (k21 + · · · + k2
m + l21 + · · · + l2n)1/2, ε > 0,
and Λm,n = 0 for m + n ≥ s, then the formula (8.3) specifies a
bounded self-adjoint operator in the space FΩ. If the Hamiltonian H
is represented in the form H0 + V , where H0 =∫ω(k)a+(k)a(k)dk,
and the coefficient functions of the expression V tend to zero at
infinity sufficiently fast, then the formula (8.3) specifies a self-adjoint
operator4 in the space FΩ.
The proof of these statements can be obtained from the consid-
erations at the end of Section 6.2.
Note that the operator HΩ commutes with the operator PΩ =∑k∈TΩ
ka+k ak (the momentum operator).
We will define the physical quantities corresponding to the formal
Hamiltonian H by taking the limit Ω→∞.
For example, let us say that the number E belongs to the
spectrum (it is an energy level) of the formal Hamiltonian H, if
one can find such eigenvalues EΩ of the Hamiltonian HΩ, such that
limΩ→∞(EΩ − E0Ω) = E (here, E0
Ω denotes the energy of the ground
state ΦΩ of the Hamiltonian H). We will say that the Hamiltonian
H has an energy level E with momentum k, if there exist vectors
ΨΩ ∈ FΩ satisfying the conditions HΩΨΩ = EΩΨΩ;PΩΨΩ =
kΩΨΩ; limΩ→∞(EΩ − E0Ω) = E; limΩ→∞ kΩ = k.
Another way to analyze a translation-invariant Hamiltonian
is based on the construction of an operator realization of the
Hamiltonian.
Let us suppose that commuting self-adjoint operators H and P =
(P1, P2, P3) (energy operator and momentum operator) act on the
Hilbert space H. Let us assume further that the operator functions
a(k, ε, t) are generalized functions with respect to the variable k, obey
4In the case of CCR, with the given assumptions on the coefficient functions,one can prove only that the formula (8.3) defines a Hermitian operator.
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Translation-Invariant Hamiltonians 127
CR and act on the same space (here, k ∈ E3, ε = ±1, a(k, 1, t) =
a+(k,−1, t)).
We will say that the operators H, P, the operator generalized
functions a(k, ε, t), and the vector Φ ∈ H constitute an operator
realization of the formal Hamiltonian (8.2), if
(1) The operator generalized functions a(k, ε, t) satisfy the Heisen-
berg equations that formally correspond to the Hamilto-
nian (8.2):
i∂a(k,−1, t)
∂t=∑m,n
m
∫Λm,n(k,k1, . . . ,km−1|l1, . . . , ln)
× δ(k + k1 + · · ·+ km−1 − l1 − · · · − ln)
× a(k1, 1, t) . . . a(km−1, 1, t)a(l1,−1, t)
. . . a(ln,−1, t)dm−1kdnl; (8.4)
i∂a(k, 1, t)
∂t= −
∑m,n
n
∫Λm,n(k1, . . . ,km|l1, . . . , ln−1,k)
× δ(k1 + · · ·+ km − l1 − · · · − ln−1 − k)
× a(k1, 1, t) . . . a(km, 1, t)a(l1,−1, t)
. . . a(ln−1, 1, t)dmkdn−1l. (8.5)
(2) exp(iτH)a(k, ε, t) exp(−iτH) = a(k, ε, t+ τ);
exp(iαP)a(k, ε, t) exp(−iαP) = exp(iαkε)a(k, ε, t).
(3) The operators a(f, ε, t) =∫f(k)a(k, ε, t)dk, where f ∈ S(E3)
are defined on a dense subset D of the space H and transform
this subset into itself; if Ψ1,Ψ2 ∈ D, then 〈a(f, ε, t)Ψ1,Ψ2〉continuously depends on f ∈ S(E3). The operators a(f, ε, t) also
satisfy CCR for fixed t.
(4) The vector Φ is a ground state of the operator H and satisfies
the condition HΦ = PΦ = 0.
(5) The vector Φ is a cyclic vector of the family of operators a(f, ε, t).
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128 Mathematical Foundations of Quantum Field Theory
Note that it follows from condition (3) that expressions of the
form ∫f(k1, . . . ,kn)a(k1, ε1, t1) . . . a(kn, εn, tn)dk1 . . . dkn,
where f ∈ S(E3n) make sense as operators defined on the set D. This
can be proven by means of an operator analog of the kernel theorem
(see Appendix A.5). This remark allows us to define rigorously
the right-hand side of (8.5) in the case when the function Λ1,1(k)
is smooth and all of its derivatives grow no faster than a power
of k and the other functions Λm,n belong to the space S. The
derivative ∂a(k,ε,t)∂t on the left-hand side is understood as a weak
derivative.
The Wightman and Green functions for the operator realizations
of translation-invariant Hamiltonians can be defined by the formulas
wn(k1, ε1, t1, . . . ,kn, εn, tn) = 〈a(k1, ε1, t1) . . . a(kn, εn, tn)Φ,Φ〉;
Gn(k1, ε1, t1, . . . ,kn, εn, tn) = 〈T (a(k1, ε1, t1) . . . a(kn, εn, tn))Φ,Φ〉.
It would be more precise to say that the functions wn are Wightman
functions in (k, t)-representations. Their Fourier transforms with
respect to the variables k1, . . . ,kn are the functions
wn(x1, ε1, t1, . . . ,xn, εn, tn) = (2π)−32n
∫exp
(i∑
εjxjkj
)× wn(k1, ε1, t1, . . . ,kn, εn, tn)dnk
which are Wightman functions in (x, t)-representations. Their
Fourier transforms with respect to the variables t1, . . . , tn are
given by
wn(k1, ε1, ω1, . . . ,kn, εn, ωn) = (2π)−12n
∫exp
(i∑
εjωjtj
)× wn(k1, ε1, t1, . . . ,kn, εn, tn)dnt
which are Wightman functions in the (k, ω)-representation. Anal-
ogously, we can define Green functions in (x, t) and (k, ω)-
representations; we denote them by Gn and Gn.
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Translation-Invariant Hamiltonians 129
In the following sections of this chapter, we define the Wightman
and Green functions of translation-invariant Hamiltonians by means
of taking the limit of Wightman and Green functions in finite volume.
We will relate these definitions to the definitions of Wightman and
Green functions by means of operator realization of the translation-
invariant Hamiltonian. We will show also how one can construct the
operator realization by taking the limit when the volume tends to
infinity. In Chapter 11, we consider the construction of operator
realization in perturbation theory.
Note that for translation-invariant Hamiltonians, we can
write down the Kallen–Lehmann representation for functions
w2(k1, ε1, t1,k2, ε2, t2) and G2(k1, ε1, t1,k2, ε2, t2).
To derive this representation, we fix a system of generalized
eigenfunctions Ψλ of the operators H and P:
HΨλ = E(λ)Ψλ;
PΨλ = k(λ)Ψλ.
Let us prove, first, that the generalized function 〈a(k, ε, t)Φ,Ψλ〉 can
be represented by the formula
〈a(k, ε, t)Φ,Ψλ〉 = exp(iE(λ)t)δ(εk− k(λ))ρ(ε, λ). (8.6)
Indeed,
〈a(k, ε, t)Φ,Ψλ〉 = 〈exp(iHt)a(k, ε)Φ,Ψλ〉
= 〈a(k, ε)Φ, exp(−iHt)Ψλ〉
= exp(iE(λ)t) 〈a(k, ε)Φ,Ψλ〉 .The function 〈a(k, ε)Φ,Ψλ〉 satisfies the relation
exp(iεαk) 〈a(k, ε)Φ,Ψλ〉 = exp(iαk(λ)) 〈a(k, ε)Φ,Ψλ〉 (8.7)
that can be derived using the transformations
〈exp(iεαk)a(k, ε)Φ,Ψλ〉 = 〈exp(iαp)a(k, ε) exp(−iαp)Φ,Ψλ〉
= 〈a(k, ε)Φ, exp(−iαpΨλ)〉
= exp(iαk(λ)) 〈a(k, ε)Φ,Ψλ〉 .It follows from (8.7) that the function 〈a(k, ε)Φ,Ψλ〉 has the form
ρ(ε, λ)δ(εk− k(λ)); this gives the proof of the statement we need.
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130 Mathematical Foundations of Quantum Field Theory
Using (8.6), we can write down the following representation of
Wightman functions:
w2(k1, ε1, t1,k2, ε2, t2) = 〈a(k1, ε1, t1)a(k2, ε2, t2)Φ,Φ〉
= 〈a(k2, ε2, t2)Φ, a(k1,−ε1, t1)Φ〉
=
∫〈a(k2, ε2, t2)Φ,Ψλ〉〈a(k1,−ε1, t1)Φ,Ψλ〉dλ
=
∫exp(iE(λ)(t2 − t1))δ(ε1k1 + ε2k2)
× δ(ε1k1 + k(λ))ρ(ε2, λ)ρ(−ε1, λ)dλ. (8.8)
The relation
G2(k1, ε1, t1,k2, ε2, t2) = θ(t1 − t2)w2(k1, ε1, t1,k2, ε2, t2)
±θ(t2 − t1)w(k2, ε2, t2,k1, ε1, t1),
and (8.8) implies the representation of the function G2 in terms of the
functions ρ(ε, λ) (the Kallen–Lehmann representation). In particular,
G2(k1, 1, t1,k2,−1, t2) = G(k1, t1 − t2)δ(k1 − k2),
where
G(k, t) = θ(t)
∫exp(−iE(λ)t)δ(k + k(λ))|ρ(−1, λ)|2dλ
±θ(−t)∫
exp(iE(λ)t)δ(−k + k(λ))|ρ(1, λ)|2dλ. (8.9)
In the (k, ω)-representation, we have
G2(k1, 1, ω1,k2,−1, ω2) = G(k1, ω1)δ(ω1 − ω2)δ(k1 − k2),
where
G(k, ω) =
∫exp(iωt)G(k, t)dt
= i
∫|ρ(−1, λ)|2
ω − E(λ) + i0δ(k + k(λ))dλ
∓i∫
|ρ(+1, λ)|2
ω + E(λ)− i0δ(−k + k(λ))dλ. (8.10)
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Translation-Invariant Hamiltonians 131
8.2 Reconstruction theorem
The Wightman functions wn(k1, ε1, t1, . . . ,kn, εn, tn) of a translation-
invariant Hamiltonian H can be defined by the following relation:
wn(k1, ε1, t1, . . . ,kn, ε, tn)
= limΩ→∞
(L
2π
) 32n
wΩn (k1, ε1, t1, . . . ,kn, εn, tn) (8.11)
(here, wΩn (k1, ε1, t1, . . . ,kn, εn, tn) = 〈aε1k1
(t1) . . . aεnkn(tn)ΦΩ,ΦΩ〉 are
Wightman functions of the Hamiltonian HΩ, constructed with
the basis φk, where k ∈ TΩ). The relation (8.11) requires some
explanation because the functions of continuous argument are defined
as the limit of functions of arguments running over a lattice. We will
understand this limit in the sense of generalized functions: for every
test function φ(k1, . . . ,kn), we require∫φ(k1, . . . ,kn)wn(k1, ε1, t1, . . . ,kn, εn, tn)dk1 . . . dkn
= limΩ→∞
(2π
L
) 3n2 ∑
k1,...,kn∈TΩ
φ(k1, . . . ,kn)
×wΩn (k1, ε1, t1, . . . , bkn, εn, tn). (8.12)
We assume that the limit (8.11) exists (the existence of this limit
can be checked in the framework of perturbation theory).
The functions wn(k1, ε1, t1, . . . ,kn, εn, tn) will be considered as
generalized functions with respect to the variables k1, . . . ,kn and
conventional functions with respect to the variables t1, . . . , tn.
Green functions Gn(k1, ε1, t1, . . . ,kn, εn, tn) of the translation-
invariant Hamiltonian H are defined in the same way, as limits
of finite-volume Green functions. Namely, we use the following
definition:
Gn(k1, ε1, t1, . . . ,kn, εn, tn)
= limΩ→∞
(L
2π
) 3n2
GΩn (k1, ε1, t1, . . . ,kn, εn, tn),
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132 Mathematical Foundations of Quantum Field Theory
where GΩn (k1, ε1, t1, . . . ,kn, εn, tn) = 〈T (aε1k1
(t1) . . . aεnkn(tn))ΦΩ,ΦΩ〉are Green functions of the Hamiltonian HΩ constructed with the
basis φk, where k ∈ TΩ. The limit is understood in the same way as
the limit in the definition of Wightman functions.
Green functions Gn can be expressed in terms of Wightman
functions wn. Namely, taking the limit of Ω→∞ in the relation (7.7)
applied to the functions GΩn and wΩ
n , we obtain
Gn(k1, ε1, t1, . . . ,kn, εn, tn)
=∑π
(−1)γ(π)θπ(t)wπn(k1, ε1, t1, . . . ,kn, εn, tn) (8.13)
(the notation is the same as in Section 7.2).
The definition of Green functions in (x, t)-representation Gn and
in the (k, ω)-representation Gn is analogous to similar definitions for
Wightman functions.
It is easy to check that the Wightman functions wn(k1, ε1, t1,
. . . ,kn, εn, tn) of a translation-invariant Hamiltonian H have the
following properties:
(1) Invariance with respect to time translation:
wn(k1, ε1, t1, . . . ,kn, εn, tn)= wn(k1, ε1, t1+τ, . . . ,kn, εn, tn+τ).
(2) Hermiticity:
wn(k1, ε1, t1, . . . ,kn, εn, tn) = wn(kn,−εn, tn, . . . ,k1,−ε1, t1).
(3) Positive definiteness: For every sequence fn(k1, ε1, t1,
. . . ,kn, εn, tn) of test functions that do not vanish for only a
finite number of indices n, we have∑m,n
∑εα,σβ
∫fm(k1, ε1, t1, . . . ,km, εm, tm)fn(qn,−σn, τn,
. . . ,q1,−σ1, τ1)wm+n(k1, ε1, t1, . . . ,km, εm, tm,
q1, σ1, τ1, . . . ,qn, σn, τn)dmkdnqdmtdnτ ≥ 0.
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Translation-Invariant Hamiltonians 133
(4) Spectral property:∫exp(−iωa)wn(k1, ε1, t1, . . . ,kr, εr, tr,kr+1, εr+1, tr+1 + a,
. . . ,kn, εn, tn + a)da = 0, (8.14)
if ω < 0. The relation (8.14) is also valid under a weaker
condition — it is sufficient to assume that ω does not belong
to the spectrum of the Hamiltonian H.
(5) Symmetry with respect to the permutation of arguments: Let us
introduce the notation
w(i)n (k1, ε1, t1, . . . ,ki, εi, ti,ki+1, εi+1, ti+1, . . . ,kn, εn, tn)
= ±wn(k1, ε1, t1, . . . ,ki+1, εi+1, ti+1,ki, εi, ti,
. . . ,kn, εn, tn)
(here, the plus sign corresponds to the case of CCR and the
minus sign corresponds to the case of CAR). In other words, w(i)n
is obtained from wn by permuting ki, εi, ti with ki+1, εi+1, ti+1
(with a change of sign in the case of CAR).
If ti = ti+1, then
w(i)n (k1, ε1, t1, . . . ,kn, εn, tn)
= wn(k1, ε1, t1, . . . ,kn, εn, tn) +Aεε′δ(ki − k′i)
× wn−2(k1, ε1, t1, . . . ,ki−1, εi−1, ti−1,ki+2, εi+2, ti+2,
. . . ,kn, εn, tn) (8.15)
(the definition of the matrix Aε′ε can be found in Section 6.1).
(6) Translation-invariance:
wn(k1, ε1, t1, . . . ,kn, εn, tn)
= vn(k1, ε1, t1, . . . ,kn, εn, tn)δ(∑
εjkj
)(in the (x, t)-representation this property takes the form
wn(x1, ε1, t1, . . . ,xn, εn, tn) = wn(x1+a, ε1, t1, . . . ,xn+a, εn, tn)).
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134 Mathematical Foundations of Quantum Field Theory
All the above listed properties (with the exception of (6)) can
be easily proven if we recall that wn are obtained as limits of the
functions wΩn , which have similar properties proven in Section 7.1.
Theorem 8.1. (Reconstruction theorem.)
Existence: Given a family of functions wn(k1, ε1, t1, . . . ,
kn, εn, tn) obeying properties (1) − (6), one can construct a Hilbert
space H, four commuting self-adjoint operators H, P, operator func-
tions a(k, ε, t) acting on H that are generalized 5 functions with
respect to the variable k, and the vector Φ ∈ H in such a way that
the following requirements are met:
(a) wn(k1, ε1, t1, . . . ,kn, εn, tn) = 〈a(k1, ε1, t1) . . . a(kn, εn, tn) Φ,Φ〉.(b) exp(iτH)a(k, ε, t) exp(−iτH) = a(k, ε, t+ τ),
exp(iαp)a(k, ε, t) exp(−iαp) = exp(iεαk)a(k, ε, t);
(c) The operators a(f, ε, t) =∫f(k)a(k, ε, t)dk are defined on the
dense set D of the space H and transform the subset into itself;
the operators a(f, ε, t) for fixed t specify a representation of CR;
the expression 〈a(f, ε, t)Ψ1,Ψ2〉 continuously depends on f in the
topology of the space S for all Ψ1,Ψ2 ∈ D.
(d) The vector Φ is the ground state of the operator H and satisfies
the conditions HΦ = 0 and PΦ = 0.
(e) The vector Φ is a cyclic vector with respect to the operators
a(f, ε, t).
Uniqueness: If Hi, Hi,Pi, ai(k, e t) and Φi are two sets of objects
(i = 1, 2) satisfying the conditions (a), (b), and (c) of the
reconstruction theorem, then there exists a unitary operator U that
maps the space H1 onto the space H2 and obeys the conditions
UΦ1 = Φ2, UH1 = H2U,UP1 = P2U , as well as the relations
Ua1(f, ε, t) = a2(f, ε, t)U on some dense subset of the space H1 (in
other words, these two sets are isomorphic).
Let us start the proof from the second part of the theo-
rem. Let us consider the set N of sequences of functions f =
5We will consider the space of test functions to be S(E3) (i.e. all functions f inthe formulation of the theorem satisfy f ∈ S(E3)).
March 27, 2020 8:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch08 page 135
Translation-Invariant Hamiltonians 135
fn(k1, ε1, t1, . . . ,kn, εn, tn) such that (1) every function fn is a
linear combination of functions of the form
λ1(k1, ε1)δ(t1 − τ1) . . . λm(km, εm)δ(tm − τm),
where λi(ki, εi) are test functions and (2) only a finite number of
functions fn do not vanish. Let us assign the vectors Ψif , where i =
1, 2, to every sequence f ∈ N by the formula
Ψif =
∞∑n=1
∑ε1,...,εn
∫fn(k1, ε1, t1, . . . ,kn, εn, tn)
× ai(k1, ε1, t1) . . . ai(kn, εn, tn)dnkdnt.
The set of vectors Ψif , where f ∈ N , will be denoted by Di. The
inner product 〈Ψif ,Ψ
ig〉 of two vectors from the set Di can be easily
expressed in terms of the sequences f, g and Wightman functions:⟨Ψif ,Ψ
ig
⟩=∑m,n
∑εα,σα
∫fm(k1, ε1, t1, . . . ,km, εm, Lm)fn(q1, σ1, τ1, . . . ,
qn, σn, τn)wm+n(k1, ε1, t1, . . . ,km, εm, tm,qn,−σn,
τn, . . . ,q1,−σ1, τ1)dmkdnqdmtdnτ. (8.16)
It is easy to check that
ai(φ)Ψif = Ψi
φf (8.17)
(here, φ = φ(k, ε, t) = λ(k, ε)δ(t−τ), where λ(k, ε) is a test function,
ai(φ) =∑
ε
∫φ(k, ε, t)ai(k, ε, t)dkdt =
∑ε
∫λ(k, ε)a(k, ε, τ))dk,
and φf ∈ N is a sequence whose nth entry is equal to
φ(k1, ε1, t1)fn−1(k2, ε2, t2, . . . ,kn, εn, tn)). Condition (b) implies that
exp(−iHiτ)Ψif = Ψi
Vτf ; exp(−iαP)Ψif = Ψi
Wαf , (8.18)
where Vτf ∈ N and Wαf ∈ N are sequences of functions with the
nth entry given by
fn(k1, ε1, t1 − τ, . . . ,kn, εn, tn − τ),
exp(−i∑
αkjεj
)fn(k1, ε1, t1, . . . ,kn, εn, tn).
March 27, 2020 8:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch08 page 136
136 Mathematical Foundations of Quantum Field Theory
Let us now construct the operator U on the set D1 assuming that
UΨ1f = Ψ2
f .
It follows from (8.16) that this operator preserves inner products,
therefore it can be extended by continuity to the unitary operator
mapping H1 into H2. It follows from the relations (8.17) and (8.18)
that the constructed operator obeys the conditions we need.
The above proof of the second part of the theorem gives us a path
to proving the first part of the theorem.
Let us start with the sequence of Wightman functions w. Let us
introduce the inner product 〈f, g〉 on the set N by means of the
formula
〈f, g〉 =∑m,n
∑εα,σβ
∫fm(k1, ε1, t1, . . . ,kn, εn, tn)fn(q1, σ1, τ1, . . . ,
qn, σn, τn)wm+n(k1, ε1, t1, . . . ,km, εm, tm,qn,−σn, τn, . . . ,
q1,−σ1, τ1)dmkdnqdmtdnτ.
Then, it follows from (3) in Section 8.2 that 〈f, f〉 ≥ 0. The elements
f, g ∈ N we consider to be equivalent (f ∼ g), if 〈f − g, f − g〉 =
0. The set of equivalence classes will be denoted by D and the
equivalence class of the element f ∈ N will be denoted by Ψf . In
other words, the set D consists of symbols Ψf , where f ∈ N , and two
symbols Ψf ,Ψg specify the same element of the set D if f ∼ g. The
inner product of the elements Ψf ,Ψg ∈ D is defined by the formula
〈Ψf ,Ψg〉 = 〈f, g〉. An element of the set D can be represented in
different ways in the form Ψf , however the inner product in D does
not depend on the choice of the representative because from the
relation f ∼ f ′, g ∼ g′, it follows that 〈f, g〉 = 〈f ′, g′〉. (Similar
considerations can be applied to other operations in D.) A linear
combination of elements in D is defined by the formula
λΨf + µΨg = Ψλf+µg,
where the linear combination λf +µg of the sequences f, g is defined
in the usual way. Hence, the set D can be considered a pre-Hilbert
space.
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Translation-Invariant Hamiltonians 137
In the space D, we can define the operator generalized functions
a(f, ε, t) =∫f(k)a(k, ε, t)dk, assuming that a(f, ε, t)Ψg = Ψφg,
where φ(k, σ, τ) = f(k)δσ,εδ(t − τ), and we can define the family
of operators Vτ , Wα, satisfying VτΨg = ΨVτg, WαΨg = ΨWαg.
Let us now define the Hilbert space H as the completion of the
pre-Hilbert space D.
The operators Vτ , Wα map the set D into itself and preserve inner
products. Therefore, they can be extended by continuity to unitary
operators on the spaceH. In this way, we obtain a one-parameter and
a three-parameter group of unitary operators on H; the generators
of these groups will be denoted by H and P (in other words, Vτ =
exp(−iHτ), Wα = exp(−iαP)). The symbol Φ will denote the vector
Ψθ, where θ is a function sequence with f0 = 1, fn = 0, for n > 0.
The operator generalized functions a(f, ε, t) are defined on the dense
subset D ⊂ H.
Hence, starting with the Wightman functions, we have con-
structed the objects that were described in the reconstruction theo-
rem. It is easy to check that they have all the necessary properties.
The only point we will consider in detail is the proof that the vector
Φ is the ground state of the Hamiltonian H. We derive this fact from
the following lemma.
Lemma 8.1. The number ω does not belong to the spectrum of the
operator H if and only if for all Wightman functions we have∫exp(−iωτ)wn(k1, ε1, t1, . . . ,ki, εi, ti,ki+1, εi+1, ti+1 + τ,
. . . ,kn, εn, tn + τ)dτ = 0. (8.19)
Then, by property (4) of Wightman functions and the lemma, it
follows that the operator H is non-negative. Taking into account that
HΦ = 0, we see that Φ is the ground state.
To prove (2), we first note that the functions wn are Wightman
functions of the operator H with respect to the operator generalized
function a(k, ε, t) (in the sense of the definition in Section 7.1).
Therefore, it follows from property (4) (Section 7.1) that for every
ω that does not belong to the spectrum of the operator H, we
have (8.19). To prove the inverse statement, it is sufficient to check
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138 Mathematical Foundations of Quantum Field Theory
that for a smooth finite function χ(ω) that does not vanish only for
ω satisfying (8.19), we have∫χ(t) exp(iHt)dt = 0, where χ(t) =∫
exp(−iωt)χ(w)dω (see Appendix A.5). It is easy to check that∫χ(t)〈exp(iHt)Ψ1,Ψ2〉dt = 0,
if Ψi = a(f(i)1 , ε
(i)1 , t
(i)1 ) . . . a(f
(i)ni , ε
(i)ni , t
(i)ni )Φ; to prove this, we should
express∫χ(t) 〈exp(iHt)Ψ1,Ψ2〉 dt =
∫χ(ω)〈exp(i(H − ω)t)Ψ1,Ψ2〉dωdt
in terms of Wightman functions. By using the cyclicity of the
vector Φ, we can see that the relation we have proven implies∫χ(t) exp(iHt)dt = 0.
This finishes the proof of the reconstruction theorem.
It follows from the lemma that every point of the spectrum of
the operator H belongs to the spectrum of the translation-invariant
Hamiltonian H in the sense of Section 8.1. Analogously, one can
prove the following statement: if the point (k, E) belongs to the
joint spectrum of a family of commuting operators (P, H), then the
Hamiltonian H has an energy level E with momentum k.
In conclusion, we will check that under certain conditions, the
space H, the operators H, P, the operator generalized functions
a(k, ε, t), and the vector Φ, constructed as in the reconstruction theo-
rem, can be considered as an operator realization of the translation-
invariant Hamiltonian H in the sense of Section 8.1. Namely, we
will show that this statement is correct if in formula (8.2), that
specifies the Hamiltonian H, the function Λ1,1(k) is smooth, all
derivatives of this function do not grow faster than a polynomial,
and the remaining functions Λm,n belong to the space S (as noted
in Section 8.1, equations (8.4) and (8.5) have precise meaning in this
case). To give the proof, it is sufficient to check that the operator
generalized functions a(k, ε, t) in the reconstruction theorem satisfy
the Heisenberg equations (8.4) and (8.5) that correspond formally
to the Hamiltonian H (all other conditions in the definition of an
operator realization follow from the reconstruction theorem). It is
March 27, 2020 8:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch08 page 139
Translation-Invariant Hamiltonians 139
easy to give this proof, using the remark that the Wightman functions
of the Hamiltonian H satisfy
1
i
∂
∂t1wr(k1, ε1, t1, . . . ,kr, εr, tr)
= δε11
∑m,n
n
∫Λm,n(p1, . . . ,pm|q1, . . . ,qn−1,k1)δ(p1 + · · ·+ pm
−q1 − · · · − qn−1 − k1)wr+m+n−2(p1, 1, t1, . . . ,pm, 1, t1,q1,
−1, t1, . . . ,qn−1,−1, t1,k2, ε2, t2, . . . ,kr, εr, tr)dmpdn−1q
−δε1−1
∑m,n
m
∫Λm,n(k1,p1, . . . ,pm−1|q1, . . . ,qn)
× δ(k1 + p1 + · · ·+ pm−1 − q1 − · · · − qn)wr+m+n−2(p1, 1, t1,
. . . ,pm−1, 1, t1,q1,−1, t1, . . . ,qn,−1, t1,k2, ε2, t2,
. . . ,kr, εr, tr)dm−1pdnq (8.20)
and satisfy similar conditions for derivatives with respect to other
time variables. (Equation (8.20) can be obtained if we use the
equations for the functions wΩr that follow from the considerations
in Section 7.4 and take the limit Ω → ∞ in these equations. The
conditions we have imposed on the functions Λm,n imply that we
can take this limit because we have assumed that the functions wΩr
tend to wr in the sense of generalized functions.) Using (8.20), we
can derive the Heisenberg equations (8.4) and (8.5); it is sufficient to
express in terms of Wightman functions the quantity⟨d
dta(f, t)Ψα,Ψβ
⟩,
where α, β ∈ N , using equations similar to (8.20).
It follows from the above statement that the Wightman functions
for a translation-invariant Hamiltonian defined in Section 8.2 are
Wightman functions of the operator realizations of this Hamiltonian
in the sense of Section 8.1. A similar statement can be proven
for Green functions. In the following, we will use the definition in
Section 8.1, but use the terminology from Section 8.2.
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140 Mathematical Foundations of Quantum Field Theory
Let us define Schwinger functions that, together with Wightman
and Green functions, play an important role in quantum field theory.
Let us consider the operators exp(iHτ), where τ is a complex
number. It follows from the non-negativity of the operator H that
these operators are bounded for τ in the upper half plane (Im τ ≥ 0).
Therefore, it is natural to assume that under the condition Im τ ≥ 0,
the operators exp(iHτ) transform the set D into itself; then, in
particular, the operator exp(−Hσ), where σ ≥ 0 has this property.
Under this assumption, we can define Schwinger functions using the
formula
Sn(k1, ε1, t1, . . . ,kn, εn, tn) = 〈a(k1, ε1, 0) exp(−H(t1 − t2))
× a(k2, ε2, 0) . . . exp(−H(tn−1 − tn))
× a(kn, εn, 0)Φ,Φ〉
(we assume that t1 ≥ · · · ≥ tn). Let us note that Schwinger functions
can also be defined without any additional assumptions. Indeed by
the property of time-translation invariance, the Wightman function
wn can be written in the following form:
wn(k1, ε1, t1, . . . ,kn, εn, tn)
= vn(k1, ε1, . . . ,kn, εn, t2 − t1, . . . , tn − tn−1).
The functions vn(k1, ε1, . . . ,kn, εn, τ1, . . . , τn−1) can be analytically
continued with respect to the variables τ1, . . . , τn−1 in the domain
Imτ1 ≥ 0, . . . , Imτn−1 ≥ 0 (to prove this fact, we should represent vnin the form
vn (k1, ε1, . . . ,kn, εn, τ1, . . . , τn−1)
=
∫exp(i
∑j
ωjτj)vn(k1, . . . , εn, ω1, . . . , ωn−1)dn−1ω
and note that by the spectrum condition the support of the function
vn is contained in the set ω1 ≥ 0, . . . , ωn−1 ≥ 0). The analytic
continuation of the function vn will be denoted by the same symbol.
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Translation-Invariant Hamiltonians 141
Schwinger functions can then be defined by the formula
Sn(k1, ε1, t1, . . . ,kn, εn, tn)
= vn(k1, ε1, . . . ,kn, εn, i(t1 − t2), . . . , i(tn−1 − tn))
(where the definition assumes t1 ≥ · · · ≥ tn).
It is easy to formulate properties of Schwinger functions that are
analogous to the properties of Wightman functions and to prove
the analog of the reconstruction theorem for Schwinger functions;
see Osterwalder and Schrader (1973).
8.3 Interactions of the form V (φ)
In this section, we will consider an important class of translation-
invariant Hamiltonians. The Hamiltonians of this class can be
obtained by the quantization of classical systems with an infinite
number of degrees of freedom; they can be written in the form (8.2),
which allows us to apply the results of Section 8.1.
In the present section, it will be convenient not to use the
assumption that ~ = 1, made in the rest of the book.
Let us recall that by quantizing a classical mechanical system with
the Hamiltonian
H(p, q) =n∑k=1
p2k
2+ U(q1, . . . , qn), (8.21)
where pk are generalized momenta and qk are generalized coordinates,
we obtain a quantum mechanical system described by the Hamilto-
nian
H =n∑k=1
1
2p2k + U(q1, . . . , qn), (8.22)
where pk, qk are self-adjoint operators satisfying the canonical com-
mutation relations (CCR)
[pk, pl] = [qk, ql] = 0; [pk, ql] =~iδkl
March 27, 2020 8:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch08 page 142
142 Mathematical Foundations of Quantum Field Theory
(the operators pk, qk can be realized in the space of square-integrable
functions ψ(q1, . . . , qn) and defined by the formulas pkψ = ~i∂∂qk
ψ;
qkψ = qkψ).
The Heisenberg operators pk(t) = exp(iHt)pk exp(−iHt), qk(t) =
exp(iHt)qk exp(−iHt) satisfy the equations
dpk(t)
dt= −∂U
∂qi(q1(t), . . . , qn(t));
dqk(t)
dt= pk(t).
Let us now consider the analog of a classical system with the
Hamiltonian (8.21) in the case of an infinite number of degrees of
freedom. Namely, we assume that the classical system is described
by the Hamiltonian functional
H(π, φ) =1
2
∫π2(x)dx + V (φ),
V (φ) =∑n
∫Vn(x1, . . . ,xn)φ(x1) . . . φ(xn)dx1 . . . dxn,
where π(x) are the generalized momentum variables and φ(x) are the
generalized coordinates. For definiteness, we assume that x runs over
three-dimensional Euclidean space. We consider only translation-
invariant functionals (i.e. we assume that the function Vn(x1, . . . ,xn)
has the form vn(x1− xn, . . . ,xn−1− xn)). It is natural to conjecture
that by quantizing such a system we will obtain a quantum system
described by the Hamiltonian
H =1
2
∫π2(x)dx +
∑n
∫Vn(x1, . . . ,xn)φ(x1) . . . φ(xn)dx1 . . . dxn,
(8.23)
where π(x), φ(x) are Hermitian operators (more precisely, operator
generalized functions) that obey the commutational relations
[π(x), π(x′)] = [φ(x), φ(x′)] = 0,
[π(x), φ(x′)] =~iδ(x− x′). (8.24)
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Translation-Invariant Hamiltonians 143
However, in trying to correctly define this quantum system, we
encounter some difficulties that we have already encountered in con-
sidering a Hamiltonian of the form (8.2). Namely, there exist many
essentially different systems of operators satisfying (8.24). For simple
constructions of the operators π(x), φ(x), the expression (8.23) does
not specify a well-defined operator. These difficulties can be overcome
by means of the approach in Section 8.1. Here, we will consider only
the method of solving the Heisenberg equations formally written
using the Hamiltonian (8.23).
Namely, we will consider the Hamiltonian H of the form (8.23)
as a formal expression. We define the operator realization of the
Hamiltonian H as a Hilbert space H with the energy operator H,
the momentum operator P, the vector φ, and the operator functions
φ(x, t), that are generalized functions with respect to the variable x
and conventional functions with respect to the variable t, satisfying
the following conditions:
(1)
∂2φ(x, t)
∂t2= −
∑n
n
∫Vn(x,x1, . . . ,xn−1)
× φ(x1, t) . . . φ(xn−1, t)dx1 . . . dxn−1.
(8.25)
(2)
exp
(i
~τH
)φ(x, t) exp
(−i~τH
)= φ(x, t+ τ),
exp
(−i~ap
)φ(x, t) exp
(i
~ap
)= φ(x+ a, t).
(3) The operators φ(f, t) =∫f(x)φ(x, t)dx and π(f, t) = d
dtφ(f, t) =∫f(x) ∂∂tφ(x, t)dx, where f ∈ S(E3), are defined on a dense
subset D of the space H and transform the subset into itself;
if the function f is real, then these operators are Hermitian. The
expressions 〈φ(f, t)Ψ1,Ψ2〉 and 〈π(f, t)Ψ1,Ψ2〉 should depend
March 27, 2020 8:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch08 page 144
144 Mathematical Foundations of Quantum Field Theory
continuously on the functions f ∈ S(E3) in the topology of the
space S(E3) for all Ψ1,Ψ2 ∈ D. We assume further that for all
t, the conditions
[π(f, t), π(f ′, t)] = [φ(f, t), φ(f ′, t)] = 0,
[π(f, t), φ(f ′, t)] =~i
∫f(x)f ′(x)dx
are satisfied.
(4) The operators H, P1, P2, P3 commute. The vector Φ is the ground
state of the energy operator H and satisfies the conditions HΦ =
0, PΦ = 0.
(5) The vector Φ is a cyclic vector of the family of operators φ(f, t).
In some simple cases, one can define the precise meaning for
equation (8.25) by means of the operator analog of the kernel theorem
(as in Section 8.1).
For Hamiltonians of the form
H0 =1
2
∫π2(x)dx +
1
2
∫v(x− y)φ(x)φ(y)dxdy (8.26)
(free Hamiltonians), it is easy to construct an operator realization.
Let us assume that the function v(k) =∫
exp(−ikx)v(x)dx is
positive almost everywhere (if this condition is not satisfied, then
an operator realization of the Hamiltonian H0 does not exist). Let
H, H, P,Φ, φ(x, t) be an operator realization of the Hamiltonian H0.
Let us construct the operator generalized functions
a(k, ε, t) = ~−1/2
(1√2
√ω(k)φ(εk, t) +
iε√ω(k)
π(εk, t)
),
where ω(k) =√v(k), φ(k, t) = (2π)−3/2
∫exp(−ikx)φ(x, t)dx, and
π(k, t) = ∂∂t φ(k, t). It is easy to check that these operator generalized
functions, together with the operators H, P and the vector Φ, specify
an operator realization of the Hamiltonian ~∫ω(k)a+(k)a(k)dk
in the sense of Section 8.1. This statement prompts the following
construction of an operator realization of the Hamiltonian (8.26).
March 27, 2020 8:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch08 page 145
Translation-Invariant Hamiltonians 145
As the space H, we should take the Fock space F (L2(E3)), and the
energy operator H and the momentum operator P should be defined
by the formulas
H = ~∫ω(k)a+(k)a(k)dk,
P = ~∫
ka+(k)a(k)dk,
and the operator generalized functions φ(x, t) should be defined by
the relation
φ(x, t) = (2π)−3/2~1/2
∫(a+(k) exp(iω(k)t− ikx)
+ a(k) exp(−iω(k)t+ ikx))dk√2ω(k)
.
The ground state Φ coincides with Fock vacuum θ. It is easy to
check that the objects we have constructed satisfy the conditions for
an operator realization and that any other operator realization is
unitarily equivalent to the realization we have described.
Let us now consider an arbitrary Hamiltonian H of the
form (8.23). Let us express it in terms of the symbols a+(k), a(k)
satisfying CCR, assuming that
φ(x) = (2π)−3/2~1/2
∫(a+(k) exp(−ikx) + a(k) exp(ikx))
dk√2ω(k)
,
π(x) = (2π)−3/2~1/2
∫i√ω(k)√2
(a+(k) exp(−ikx)
−a(k) exp(ikx))dk, (8.27)
where ω(k) is an almost everywhere positive function. This expres-
sion can be written in normal form by means of CCR; we obtain
a quadratic expression plus a (possibly infinite) constant. We will
discard this constant and as a result we will obtain a formal
expression for H of the form (8.2).
It is easy to see that the problem of constructing an operator
realization of the Hamiltonian H is equivalent to the same problem
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146 Mathematical Foundations of Quantum Field Theory
for H. For example, if H, H, P,Φ, a(k, ε, t) is an operator realization
of the Hamiltonian H, then the operator realization of the Hamil-
tonian H can be obtained if we take the same Hilbert space H, the
same operators H, P, and vector Φ, and the operator generalized
functions φ(x, t) can be written in the form
φ(x, t) = (2π)−3/2~1/2
∫(a(k, 1, t) exp(−ikx)
+ a(k,−1, t) exp(ikx))dk√2ω(k)
.
This remark allows us to transfer to Hamiltonians of the form (8.23)
everything we know for Hamiltonians of the form (8.2).
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 147
Chapter 9
The Scattering Matrix forTranslation-Invariant Hamiltonians
9.1 The scattering matrix for translation-invariant
Hamiltonians in Fock space
In this chapter, we review the basic facts of scattering theory for
translation-invariant Hamiltonians. Some of these facts are proved,
however, the proofs are not fully rigorous (Sections 9.2, 9.5), while
other facts are conveyed almost without proof (Sections 9.1, 9.3 and
9.4). Most of the results in this chapter will be proved later on the
basis of axiomatic scattering theory (see Chapter 11).
To construct a scattering matrix for a translation-invariant
Hamiltonian, one cannot directly use the general construction of
formal scattering theory (Section 5.1). The first obstacle for applying
this construction is the fact that a translation-invariant Hamiltonian
can define a self-adjoint operator on Fock space only in the case
when vacuum polarization is absent (Section 8.1). However, even
translation-invariant Hamiltonians that specify an operator on Fock
space still have problems. These problems are related to the fact
that in the case under consideration, a natural representation of the
Hamiltonian H as a sum of “free” Hamiltonian H0 and “interaction”
V usually does not exist.
It is reasonable to take the Hamiltonian H0 to be of the form∫ε(k)a+(k)a(k)dk because such a Hamiltonian describes a system
of non-interacting identical particles. We assume, therefore, that
the translation-invariant Hamiltonian H specifying an operator on
147
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148 Mathematical Foundations of Quantum Field Theory
Fock space F (L2(E3)) is represented in the form H = H0 + V ,
where
H0 =
∫ε(k)a+(k)a(k)dk, (9.1)
V =∑m,n≥1
∫Vm,n(k1, . . . ,km|p1, . . . ,pn)a+(k1)
. . . a+(km)a(p1) . . . a(pn)dmkdnp (9.2)
(here,
Vm,n(k1, . . . ,km|p1, . . . ,pn) = vm,n(k1, . . . ,km|p1, . . . ,pn)
× δ(k1 + · · ·+ km − p1 − · · · − pn)).
(9.3)
However, such a representation of the Hamiltonian H has physical
meaning only under the condition vm,1 ≡ 0. Otherwise, the choice of
free and interaction parts can be performed in different ways and is
dictated only by convenience.
One can prove the following statement: if Vm,1 6≡ 0 (i.e. the
“interaction” contains summands with one annihilation operator),
then the Møller matrices S± cannot be defined by means of the
definitions of formal scattering theory (5.1) and (5.2). Indeed, if there
exists a limit entering the relations (5.1) and (5.2), then for every
vector x ∈ F , we have
limt1,t2→−∞;(t1,t2→+∞)
∥∥∥∥∫ t2
t1
exp(iHt)V exp(−iH0t)xdt
∥∥∥∥ = 0.
(We can check this using the relation
exp(itH)V exp(−itH0)x =1
i
dξ(t)
dt,
where ξ(t) = exp(iHt) exp(iH0t)x and hence∫ t2
t1
exp(iHt)V exp(−iH0t)xdt =1
i(ξ(t2)− ξ(t1)).
)
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 149
The Scattering Matrix for Translation-Invariant Hamiltonians 149
However, if x =∫f(k)a+(k)dkθ, then∥∥∥∥dξdt
∥∥∥∥ = ‖ exp(iHt)V exp(−iH0t)x‖ = ‖V exp(−iH0t)x‖
does not depend on t. This is clear because in this case
exp(iH0t)x =
∫exp(iε(k)t)f(k)a+(k)dkθ,
V exp(−iH0t)x =∑n
∫fn(k1, . . . ,kn|t)a+(k1) . . . a+(kn)dnkθ,
where
fn(k1, . . . ,kn|t) = vn,1(k1, . . . ,kn|k1 + · · ·+ kn)
∫(k1 + · · ·+ kn)
× exp(−itε(k1 + · · ·+ kn)).
Noting that
‖V exp(−iH0t)x‖ =∑n
n!
∫|fn(k1, . . . ,kn|t)|2dnk,
we see that ‖dξdt ‖ does not depend on t. Similar considerations show
that ‖d2ξdt2‖ = ‖ ddt(exp(iHt)×V exp(−iH0t)x)‖ = ‖ exp(−iHt)(HV −
V H0) × exp(−iH0t)x‖ = ‖(HV − V H0) exp(−iH0t)x‖ does not
depend on t. Now, to finish the proof, we should apply the following
lemma to the vector η(t) = dξdt : if ‖η(t)‖ does not depend on t
and ‖dηdt ‖ is bounded above, then∫ t2t1η(t)dt cannot tend to zero as
t1, t2 →∞.
This mathematical statement — the fact that it is impossible
to give the definition of a Møller matrix in the same way as in the
theory of potential scattering — has a clear physical background. The
problem is that the states a+(k)θ (one particle states) are eigenstates
of the Hamiltonian H0 but are not eigenstates of the full Hamiltonian
H. In Section 5.3, we introduced the notion of a particle (single-
particle state) for the Hamiltonian H as a generalized vector function
Φ(k) satisfying the conditions (5.20)–(5.22).
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150 Mathematical Foundations of Quantum Field Theory
Using this notion, we can reformulate the above statement in
the following way: the generalized vector function a+(k)θ is a
single-particle state of the Hamiltonian H0 (bare single-particle
state), however, it is not a single-particle state of the Hamiltonian
H (“dressed” single-particle state). The fact that the notions of bare
and dressed particles do not coincide explains the necessity to modify
the definition of a scattering matrix.
In the case when vm,0 ≡ vm,1 ≡ 0, bare particles coincide
with dressed ones, and therefore, it is not necessary to modify the
definition of a scattering matrix.
Let us sketch how to perform the necessary modification of
the definition of a scattering matrix for a translation-invariant
Hamiltonian that specifies an operator on Fock space.
We will give three definitions of Møller matrices S± that can
be used in the situation when dressed particles coincide with bare
particles (i.e. the Hamiltonian has the form (9.1) and vm,1 ≡ 0). We
will show how to modify these definitions for the case at hand (recall
that knowing the Møller matrices S±, we can define the scattering
matrix by the formula S = S∗+S−).
The first definition is the same definition that was used in the
formal scattering theory (see (5.1) and (5.2)). One can also define
Møller matrices S± as strong limits of the adiabatic Møller matrices
Sα± with α → 0. In Section 5.1, we described the conditions that
guaranteed that the second definition is equivalent to the first one
(see (5.4) and (5.5)). Finally, the third definition is based on the
consideration of the in- and out-operators
a inout
(k) = slimt→∓∞
exp(iε(k)t)a(k, t),
a+in
out
(k) = slimt→∓∞
exp(−iε(k)t)a+(k, t). (9.4)
If we know the in- and out-operators, then the Møller matrices can
be defined by the relations
S−ain(k) = a(k)S−, S−θ = θ,
S+aout(k) = a(k)S+, S+θ = θ (9.5)
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The Scattering Matrix for Translation-Invariant Hamiltonians 151
(the equivalence of this definition with the first one was discussed in
Section 5.3).
Let us start by generalizing the third definition. The limit in (9.4)
does not exist for vm,1 6≡ 0; this is clear from the fact that
the expression
exp(iHt) exp(−iH0t)
∫f(k)a+(k)θdk
=
∫f(k) exp(−iε(k)t)a+(k, t)θdk
does not have a limit as t→ ±∞ (as was proven already).
However, this fact does not mean that the expression
exp(iω(k)t)a(k, t) does not have a weak limit. Therefore, the in- and
out-operators (operator generalized functions) ain(k), a+in(k), aout(k),
a+out(k) will be defined by means of the following relations:
ain(k) = wlimt→−∞
Λ(k) exp(iω(k)t)a(k, t),
a+in(k) = wlim
t→−∞Λ(k) exp(−iω(k)t)a+(k, t),
aout(k) = wlimt→+∞
Λ(k) exp(iω(k)t)a(k, t),
a+out(k) = wlim
t→+∞Λ(k) exp(−iω(k)t)a+(k, t),
where a positive function ω(k) should be found from the condition of
the existence of the limit and the function Λ(k) from the condition
that the operators ain(k), a+in(k) and aout(k), a+
out(k) satisfy CR:
[a inout
(k), a inout
(k′)]∓ = [a+in
out
(k), a+in
out
(k′)] = 0,
[a inout
(k), a+in
out
(k′)]∓ = δ(k− k′).
It is easy to check (see Section 9.2) that the vector generalized
functions a+in(k)θ and a+
out(k)θ are single-particle states of the
Hamiltonian H and therefore conclude that the functions ω(k) has
meaning of the energy of a single-particle state:
Ha+in
out
(k)θ = ω(k)a+in
out
(k)θ.
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152 Mathematical Foundations of Quantum Field Theory
Møller matrices are defined as earlier in terms of in- and out-
operators by (9.5).
The second definition can be generalized in the following way.
The Møller matrices S± are defined as unitary operators that can be
represented in the form
S+ = slimα→0
Sα+U∗α, (9.6)
S− = slimα→0
Sα−Uα, (9.7)
where Uα is an operator of the form
exp
(i
α
∫r(k)a+(k)a(k)dk
), (9.8)
and the function r(k) is chosen from the condition of the existence
of the limits (9.6) and (9.7).
If we want to define directly the scattering matrix, we can use the
relation
S = slimα→0
UαSαUα, (9.9)
where Sα is the adiabatic S matrix, the operator Uα has the
form (9.8) and the function r(k) is chosen from the condition of
existence of the limit (9.9).
Finally, we give the following modification for the first definition:
Møller matrices S± are operators of the form
S± = slimt→±
exp(iHt)T exp(−iHast), (9.10)
where T is an operator satisfying the conditions
Tθ = θ,
Ta+(k)θ = Φ(k)
and having the form T = N(expB), where
B =∑n
∫bn(k1, . . . ,kn)δ
(∑i
ki − k
)a+(k1) . . .
a+(kn)a(k)dkdk1 . . . dkn,
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The Scattering Matrix for Translation-Invariant Hamiltonians 153
(Φ(k) denotes a singe-particle state of the Hamiltonian H,Has =∫ω(k)a+(k)a(k)dk, where ω(k) is the energy of the single-particle
state: HΦ(k) = ω(k)Φ(k)).1
In the relation (9.10), one can replace the operator T described
above by other operators. All operators D such that
S± = slimt→±∞
exp(itH)D exp(−itHas)
will be called dressing operators (this term is related to the fact
that they transform the bare single-particle state a+(k)θ into the
dressed single-particle state Φ(k)). One can construct a broad class
of dressing operators (see Sections 9.4 and 9.5).
All three definitions of Møller matrices are equivalent (under
certain conditions). They can be generalized further to translation-
invariant Hamiltonians generating the polarization of vacuum.
Naturally, there are additional complications related to the fact
that the Hamiltonian does not specify an operator in Fock
space.
For the definition by means of in- and out-operators, these
complications can be overcome by considering an operator realization
of the Hamiltonian H. Other definitions should be modified by
considering the volume cutoff Ω of the Hamiltonian H and taking the
limit Ω → ∞ (in this case one should define the scattering matrix
directly because Møller matrices cannot be defined in this case).
In more detail, we will study various definitions of the scattering
matrix for a translation-invariant Hamiltonian in other sections of
this chapter and in Chapter 11.
In Section 5.3, we noted that the definition of scattering matrix
should be modified even for the simplest Hamiltonians if there exist
bound states. Similar modifications are necessary in the situation at
hand if there exist bound states. The definition of scattering matrix
that can be used in the case when bound states exist is given in
Section 11.1.
1The definition of Møller matrices by means of (9.10) was suggested byI. Ya. Arefieva.
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154 Mathematical Foundations of Quantum Field Theory
9.2 The definition of scattering matrix by means of
operator realization of a translation-invariant
Hamiltonian
Let H denote a translation-invariant Hamiltonian and let us consider
its operator realization (H, H, P, a(k, ε, t),Φ) (see Section 8.1).
Let us define the in-operators ain(k, ε, t) and out-operators
aout(k, ε, t) as limits
a inout
(k, t) = a inout
(k,−1, t)
= wlimτ→∓∞
Λ∓(k) exp(iω(k)τ)a(k,−1, t+ τ),
a+in
out
(k, t) = a inout
(k, 1, t)
= wlimτ→∓∞
Λ∓(k) exp(−iω(k)τ)a(k, 1, t+ τ),
(9.11)
where ω(k) is found from the condition of the existence of limits and
Λ∓(k) is found from the condition that the operators a+in(k, t) and
ain(k, t) (and, correspondingly, a+out(k, t), aout(k, t)) satisfy CR for
fixed t. (Here, ω(k) is an almost everywhere positive function and the
limit is understood as a weak limit of operator generalized functions,
ε = ±1. Introducing the notation Λ∓(k,−1) = Λ∓(k),Λ∓(k, 1) =
Λ∓(k), we can say that for every function f ∈ S, we have
a inout
(f, ε, t) =
∫f(k)a in
out(k, ε, t)dk
= wlimτ→∓∞
∫f(k)Λ∓(k, ε) exp(−iεω(k)τ)a(k, ε, t+ τ)dk
in the sense of weak limits of operators.)
This definition of in- and out-operators differs from the definition
accepted in the theory of potential scattering (Section 5.3) by
replacing strong limits with weak limits. The factor Λ∓(k) is related
to this modification: strong limits preserve CR, but weak limits
do not.
The question of the existence of in- and out-operators (i.e. the
question of the existence of the functions ω(k) and Λ∓(k) such that
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The Scattering Matrix for Translation-Invariant Hamiltonians 155
the limit in (9.11) exists and satisfies CR) will be discussed in Chap-
ters 10 and 11. To denote the in- and out-operators simultaneously,
we will use the notation aex(k, ε, t).
Let us establish a few simple properties of in- and out-operators:
(1) aex(k, ε, t) = exp(iHt)aex(k, ε, 0) exp(−iHt) =
exp(iεω(k)t)aex(k, ε, 0);
(2) exp(iPα)aex(k, ε, t) exp(−iPα) = exp(iεkα)aex(k, ε, t);
(3) aex(k, t)Φ = 0;
(4) the generalized vector function Φ∓(k) = a+ex(k, 0)Φ is
δ-normalized and is an eigenfunction of the operators H and P:
HΦ∓(k) = ω(k)Φ∓(k),
PΦ∓(k) = kΦ∓(k)
(the functions Φ∓(k) satisfying this condition describe a single-
particle state; see Section 5.3).
We first note that (2) follows from the relation
exp(iPα)a(k, ε, t) exp(iPα) = exp(iεkα)a(k, ε, t).
Then,
aex(k, ε, t) = wlimτ→∓∞
Λ∓(k, ε) exp(iεω(k)τ)a(k, ε, t+ τ)
= wlimτ→∓∞
exp(iHt)Λ∓(k, ε) exp(iεω(k)τ)
× a(k, ε, τ) exp(−iHt)
= exp(iHt)( wlimτ→∓∞
Λ∓(k, ε) exp(iεω(k)τ)a(k, ε, τ))
× exp(−iHt) = exp(iHt)aex(k, ε, 0) exp(−iHt).
From the other side, substituting ρ instead of t + τ in (9.11), we
obtain
aex(k, ε, t) = wlimρ→∓∞
Λ∓(k, ε) exp(iεω(k)(ρ− t))a(k, ε, ρ)
= exp(−iεω(k)t)aex(k, ε, 0).
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156 Mathematical Foundations of Quantum Field Theory
It follows from (1) that
exp(iHt)aex(k, ε)Φ = exp(iHt)aex(k, ε) exp(−iHt)Φ
= aex(k, ε, t)Φ = exp(iεω(k)t)Φ,
hence,
Haex(k, ε)Φ = εω(k)aex(k, ε)Φ. (9.12)
From (9.12), it follows that aex(k,−1)Φ = 0 (if this condition is
not satisfied for k in the set K having non-zero measure, then the
generalized vector function aex(k,−1)Φ is a generalized eigenfunction
of the operator H, hence the number −ω(k), where k ∈ K, belongs
to the spectrum of the operator H; this is impossible because the
operator H is positive).
Applying CR and (3), we see that the generalized function
Φ∓(k) = aex(k, 1)Φ is δ-normalized (〈Φ∓(k) , Φ∓(k′)〉 =
〈aex(k′,−1)aex(k, 1)Φ,Φ〉 = 〈[aex(k′), a+ex(k)]∓Φ,Φ〉 = δ(k − k′)).
Formula (9.12) implies that Φ∓(k) is a generalized eigenfunction
of the operator H. In order to prove that Φ∓(k) is a generalized
eigenfunction of the operator P, we should recall that by (2), we have
exp(iPα)aex(k, ε, t) exp(−iPα) = exp(iεkα)aex(k, ε, t),
which implies that
exp(iPα)Φ∓(k)=exp(iPα)a+ex(k, 0) exp(−iPα)Φ=exp(εkα)Φ∓(k)
and therefore PΦ∓(k) = kΦ∓(k).
Remark 9.1. We have assumed that the function ω(k) in the
definition of in- and out-operators is almost everywhere positive. One
can replace this condition by the condition that the operator H has
a unique ground state. Then, in the case of CCR, we can modify the
above considerations to check that the function ω(k) is automatically
almost everywhere positive. In the case of CAR, one should introduce
new operators a+(k, t), a(k, t), a+ex(k, t), aex(k, t) also satisfying CAR,
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The Scattering Matrix for Translation-Invariant Hamiltonians 157
using the formulas
a(k, t) = θ(ω(k))a(k, t) + θ(−ω(k))a+(−k, t),
aex(k, t) = θ(ω(k))aex(k, t) + θ(−ω(k))a+ex(−k, t).
Then, it is clear that
aex(k, t) = wlimτ→∓∞
Λ′∓(k) exp(i|ω(k)|τ)a(k, t+ τ),
where the function |ω(k)| is almost everywhere positive.
Let us introduce the space of asymptotic states Has as the space
F (L2(E3)) of Fock representation of CR. The symbol b(k, ε) denotes
the operator generalized functions in this space that satisfy the
conditions
[b(k, ε), b(k′, ε′)]∓ = Aεε′δ(k− k′),
b(k,−1)θ = 0, b+(k,−1) = b(k, 1).
Møller matrices S− and S+ are defined as isometric operators
mapping the space Has into the space H and satisfying the equations
ain(k, ε)S− = S−b(k, ε), S−θ = Φ, (9.13)
aout(k, ε)S+ = S+b(k, ε), S+θ = Φ (9.14)
(it follows from the results of Section 6.1 that such operators exist
and are defined by conditions (9.13) and (9.14) uniquely).
The scattering matrix of a translation-invariant Hamiltonian H
is defined as the operator S = S∗+S−.
It is easy to verify that the scattering matrix S will be unitary if
and only if the spaces Hin = S−Has and Hout = S+Has coincide.
We will assume that H = Hin = Hout (in other words, not only is
the S-matrix unitary, but so are the Møller matrices S− and S+).
Some of the relations proven later, in particular (9.22), are correct
without this assumption.
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158 Mathematical Foundations of Quantum Field Theory
Let us define the operator Has (asymptotic Hamiltonian) and the
operator Pas in the space Has using the formulas
Has =
∫ω(k)b+(k)b(k)dk,
Pas =
∫kb+(k)b(k)dk
(as usual, b+(k) = b(k, 1), b(k) = b(k,−1)). It is easy to see that
b(k, ε, t) = exp(iHast)b(k, ε) exp(−iHast) = exp(iεω(k)t)b(k, ε),
exp(iPasα)b(k, ε, t) exp(−iPasα) = exp(iεαk)b(k, ε, t).
Using this relation and the properties (1) and (2) of the operators
aex, we can show that
HS± = S±Has , PS± = S±Pas (9.15)
(i.e. the operators S+ and S− specify unitary equivalences between
the operators H and Has , P and Pas). From (9.15), it follows that
the scattering matrix commutes with the operators Has and Pas :
SHas = HasS,
SPas = PasS.
Let us now prove that the scattering matrix defined above has the
following properties:
(1) Sθ = θ (the vacuum is stable),
(2) Sb+(k)θ = c(k)b+(k)θ,
where |c(k)| = 1 (single-particle states are stable). The second of
these statements will be proved only under certain restrictions on the
function ω(k); it is sufficient to assume that it is strongly convex.
The stability of the vacuum follows immediately from the relations
S−θ = Φ, S+θ = Φ. To prove that single-particle states are stable,
we note that
HasSb+(k)θ = SHasb
+(k)θ = ω(k)Sb+(k)θ,
PasSb+(k)θ = SPasb
+(k)θ = kSb+(k)θ.
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The Scattering Matrix for Translation-Invariant Hamiltonians 159
Hence, the generalized vector function Ψ(k) = Sb+(k)θ satisfies the
condition HasΨ(k) = ω(k)Ψ(k), PasΨ(k) = kΨ(k); it is easy to see
that this generalized vector function is δ-normalized. It follows that
Ψ(k) = c(k)b+(k)θ, where |c(k)| = 1 (see Section 5.3).
We will now consider an ambiguity in our definition of in- and out-
operators. It is not difficult to check that the functions Λ−(k) and
Λ+(k) are not specified uniquely by the requirement the operators
aex(k, t) obey CR. Namely, we can replace the functions Λ−(k) and
Λ+(k) by functions Λ′−(k) and Λ′+(k) that have the same absolute
value as the functions Λ−(k) and Λ+(k) and obtain the new in- and
out-operators a′in(k, t) and a′out(k, t):
a′in(k, t) = exp(iφ−(k))ain(k, t),
a′out(k, t) = exp(iφ+(k))aout(k, t) (9.16)
(here, exp(iφ∓(k)) = Λ′∓(k)Λ−1± (k) and φ∓(k) are real-valued
functions).
One can check that under the above assumptions, all possible in-
and out-operators can be represented as operators a′ex(k, t).
It is easy to check that the Møller matrices S′− and S′+ constructed
by means of the operators a′ex(k, t) are related to the Møller matrices
S− and S+ corresponding to the operators aex(k, t) by the formula
S′∓ = S∓U∓,
where U∓ = exp(i∫φ∓(k)b+(k)b(k)dk) (this follows from the
relation U∓b(k)U−1∓ = exp(iφ∓(k))b(k)).
Hence, the scattering matrix S′ = S′∗+S′− corresponding to the
new ex-operators is related to the old scattering matrix S = S∗+S−by the relation S′ = U−1
+ SU−.
Using this ambiguity in the definition of in- and out-operators,
one can strengthen the stability condition of single-particle states,
namely, we can require
Sb+(k)θ = b+(k)θ. (9.17)
In what follows, we always assume that the in- and out-operators are
chosen in such a way that condition (9.17) is satisfied.
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160 Mathematical Foundations of Quantum Field Theory
There is less ambiguity in the definition of in- and out-operators
if we make the following assumption: the relation between the
old and new ex-operators is given by the formulas (9.16), where
φ−(k) = φ+(k); correspondingly, in the formulas relating the new
and old Møller matrices and new and old scattering matrices, we
have U− = U+.
Let us check that the condition (9.17) is satisfied if and only
if the functions Λ−(k) and Λ+(k) are equal: Λ−(k) = Λ+(k) =
Λ(k). To check this, we consider the decomposition of gener-
alized vector functions a+(k)Φ with respect to the generalized
basis a+in(k1) . . . a+
in(kn)Φ = Φ−(k1, . . . ,kn). This decomposition has
the form
a+(k)Φ = ρ(1,k)Φ−(k) +∞∑n=2
1√n!
∫ρ(1,k1, . . . ,kn)
× δ
(k−
n∑i=1
ki
)Φ−(k1, . . . ,kn)dnk, (9.18)
where the function ρ(ε,k1, . . . ,kn) is defined by the equation
1√n!〈a(k, ε)Φ,Φ−(k1, . . . ,kn)〉 = ρ(ε,k1, . . . ,kn)δ
(εk−
∑i
ki
).
From (9.18), it follows that∫f(k) exp(−iω(k)t)a+(k, t)Φdk
=
∫f(k) exp(−iω(k)t) exp(iHt)a+(k)Φdk
=
∫f(k)ρ(1,k)Φ−(k)dk +
∑n≥2
1√n!
∫ρ(1,k1, . . . ,kn)
× δ
(k−
∑i
ki
)exp(i(ω(k1) + · · ·+ ω(kn)
− ω(k))t)Φ−(k1, . . . ,kn)dnk.
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The Scattering Matrix for Translation-Invariant Hamiltonians 161
For large t, the second summand contains a fast-oscillating factor
and therefore it weakly tends to zero as t→ ±∞. Hence, we see that
wlimt→±∞
∫f(k) exp(−iω(k)t)a+(k, t)Φdk
=
∫f(k)ρ(1,k)Φ+(k)dk. (9.19)
From the other side, from (9.11), we obtain
wlimt→∓∞
∫f(k) exp(−iω(k)t)Λ∓(k)a+(k, t)Φdk
=
∫f(k)a+
inout
(k)Φdk =
∫f(k)Φ∓(k)dk. (9.20)
Comparing Equations (9.19) and (9.20) and noting that by (9.17),
Φ−(k) = Φ+(k), we obtain
Λ−(k) = Λ+(k) = Λ(k) = (ρ(1,k))−1. (9.21)
Hence, we have proven the equality Λ−(k) = Λ+(k) and also related
these quantities with the function ρ(1,k). Using this relation and
the Kallen–Lehmann representation, we will prove that the functions
ω(k) and |Λ(k)| can be expressed in terms of the Green function
G2(p1, 1, ω1,p2,−1, ω2) = G(p1, ω1)δ(ω1 − ω2)δ(p1 − p2).
Namely, the function ω(p) specifies the location of poles of the
functions G(p, ω) (i.e. the poles of the function G(p, ω) with respect
to the variable ω, for fixed p, are located at the points ω(p) and
−ω(p)). The function |Λ(p)| is equal to Z(p)−1/2, where iZ(p) is
the residue of the function G(p, ω) at the pole ω(p).
To prove this statement, we note that we can take as a
generalized eigenbasis of the operators H and P the basis of
vectors a+in(k1) . . . a+
in(kn)Φ = Φ−(k1, . . . ,kn). Let us write down
the Kallen–Lehmann representation for G(p, w) (8.10) in this basis.
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 162
162 Mathematical Foundations of Quantum Field Theory
We obtain
G(p, ω) =i|ρ(−1,−p)|2
ω − ω(−p) + i0∓ i|ρ(+1,p)|2
ω + ω(p)− i0
+ i∑n≥2
∫|ρ(−1,k1, . . . ,kn)|2
ω − (ω(k1) + · · ·+ ω(kn)) + i0
× δ(p + k1 + · · ·+ kn)dk1 . . . dkn
∓i∑n≥2
∫|ρ(+1,k1, . . . ,kn)|2
ω + ω(k1) + · · ·+ ω(kn)− i0
× δ(−p + k1 + · · ·+ kn)dk1 . . . dkn.
In combination with (9.21), this relation gives a proof of the
statements we need.
The generalized functions
Sm,n(p1, . . . ,pm|q1, . . . ,qn)
=⟨Sb+(q1) . . . b+(qn)θ, b+(p1) . . . b+(pm)θ
⟩(the matrix entries of the operator S in the generalized basis
b+(p1) . . . b+(pm)θ) are called the scattering amplitudes. They can
be easily expressed in terms of the generalized vector functions
a+ex(p1) . . . a+
ex(pm)θ (that are called the in- and out-states), namely,
Sm,n(p1, . . . ,pm|q1, . . . ,qn)
=⟨S−b
+(q1) . . . b+(qn)θ, S+b+(p1) . . . b+(pm)θ
⟩=⟨a+
in(q1) . . . a+in(qn)Φ, a+
out(p1) . . . a+out(pn)Φ
⟩.
Knowing the scattering amplitudes Sm,n, we can calculate the
differential collision cross-section (see Section 10.5). It is often
convenient to represent the operator S in normal form
S =∑m,n
1
m!n!
∫σm,n(p1, . . . ,pm|q1, . . . ,qn)b+(p1) . . . b(qn)dmpdnq.
The functions σm,n are closely related to the scattering amplitudes.
This can be easily checked with Wick’s theorem (see Section 6.3); we
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 163
The Scattering Matrix for Translation-Invariant Hamiltonians 163
note here that under the condition pi 6= qj(1 ≤ i ≤ m, 1 ≤ j ≤ n),
we have
Sm,n(p1, . . . ,pm|q1, . . . ,qn) = σm,n(p1, . . . ,pm|q1, . . . ,qn).
We will now show how to express the functions σm,n in terms of the
Green functions of a translation-invariant Hamiltonian H.
We will prove the following Lehmann–Symanzik–Zimmermann
formula (LSZ):
σm,n(p1, . . . ,pm|q1, . . . ,qn)
= (i√
2π)m+nm∏i=1
Λ(pi)
n∏j=1
Λ(qj) limωi→ω(pi)
limσj→ω(qj)
m∏i=1
(ωi − ω(pi))
×n∏j=1
(σj − ω(qj))Gm+n(q1, 1, σ1, . . . ,qn, 1, σn|p1,−1,
ω1, . . . ,pm,−1, ωm), (9.22)
where Gm+n are the Green functions of the translation-invariant
Hamiltonian H in the (k, ω) representation.
We will give the proof of the LSZ formula in the CCR case (the
case of CAR differs only by signs). First, let us note that
aout(k1, ε1)T (a(k2, ε2, t2) . . . a(kn, εn, tn))
= wlimt1→+∞
a(k1, ε1, t1) exp(−iε1ω(k1)t1)Λ(k1, ε1)
× T (a(k2, ε2, t2) . . . a(kn, εn, tn))
= wlimt1→+∞
exp(−iε2w(k1)t1)
× Λ(k1, ε1)T (a(k1, ε1, t1) . . . a(kn, εn, tn)). (9.23)
Indeed, this follows from the remark that if t1 > t2, . . . , > tn, then
a(k1, ε1, t1)T (a(k2, ε2, t2) . . . a(kn, εn, tn))
= T (a(k1, ε1, t1) . . . a(kn, εn, tn))).
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164 Mathematical Foundations of Quantum Field Theory
Similarly,
T (a(k2, ε2, t2) . . . a(kn, εn, tn))ain(k1, ε1)
= limt1→−∞
Λ(k1, ε1) exp(−iε1ω(k1)t1)
× T (a(k1, ε1, t1) . . . a(kn, εn, tn)). (9.24)
Combining the relations (9.23) and (9.24), we obtain
aout(k1, ε1)T (a(k2, ε2, t2) . . . a(kn, εn, tn))
− T (a(k2, ε2, t2) . . . a(kn, εn, tn))ain(k1, ε1)
= Λ(k1, ε1)
∫ ∞−∞
∂
∂t1(exp(−iε1ω(k1)t1)T (a(k1, ε1, t1)
. . . a(kn, εn, tn)))dt1
=
∫ ∞−∞
L1T (a(k1, ε1, t1) . . . a(kn, εn, tn))dt1. (9.25)
Here, we used the notation
Lif(k1, ε1, t1, . . . ,kn, εn, tn) = Λ(ki, εi)∂
∂ti(exp(−iεiω(ki)t)
× f(k1, ε1, t1, . . . ,kn, εn, tn)).
It is useful to note the simplest form of (9.25)
aout(k, ε)− ain(k, ε) =
∫ ∞−∞
Λ(k, ε)∂
∂t(exp(−iεω(k)t)a(k, ε, t))dt
=
∫ ∞−∞
La(k, ε, t)dt
(Yang–Feldman equation).
In the case of CAR, we analogously have
aout(k1, ε1)T (a(k2, ε2, t2) . . . a(kn, εn, tn))
−(−1)n−1T (a(k2, ε2, t2) . . . a(kn, εn, tn))ain(k1, ε1)
=
∫ ∞−∞
Λ(k1, ε1)∂
∂t1(exp(−iε1ω(k1)t1)T (a(k1, ε1, t1)
. . . a(kn, εn, tn)))dt1
=
∫ ∞−∞
L1T (a(k1, ε1, t1) . . . a(kn, εn, tn))dt1.
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The Scattering Matrix for Translation-Invariant Hamiltonians 165
Applying the relation (9.25) several times, we can express the matrix
entries of the scattering matrix in terms of Green functions.
We will give another, shorter, proof. Let us introduce the oper-
ators c(k, ε, t) = S∗−a(k, ε, t)S−. We can easily express the Green
function of the Hamiltonian H in terms of these operators:
Gn(k1, ε1, t1, . . . ,kn, εn, tn)
= 〈T (a(k1, ε1, t1) . . . a(kn, εn, tn))S−θ, S−θ〉
= 〈T (c(k1, ε1, t1) . . . c(kn, εn, tn))θ, θ〉 .
Multiplying (9.25) on the left by S∗+ = SS∗− and on the right by S−and using the relations (9.13) and (9.14), we obtain
[S · T (c(k2, ε2, t2) . . . c(kn, εn, tn)), b(k1, ε1)]
= −∫ ∞−∞
L1S · T (c(k1, ε1, t1) . . . c(kn, εn, tn))dt1. (9.26)
Applying this formula several times, we see that
[. . . [S, b(k1, ε1)] . . . b(kn, εn)]
= (−1)n∫Ln . . . L1(S · T (c(k1, ε1, t1) . . . c(kn, εn, tn)))dt1 . . . dtn.
(9.27)
The expression for the functions σm,n can be obtained from the
remark that
σm,n(p1, . . . ,pm|q1, . . . ,qn)
= (−1)n⟨[. . . [[. . . [S, b+(q1)] . . . b+(qn)]b(p1)] . . . b(pm)]θ, θ
⟩.
(9.28)
Namely, from (9.27) and (9.28) and the relation Sθ = θ, it follows
that
σm,n(p1, . . . ,pm|q1, . . . ,qn)
= (−1)m∫Lm+n . . . L1Gm+n(q1, 1, t1, . . . ,qn, 1, tn,p1,−1,
tn+1, . . . ,pm,−1, tm+n)dt1 . . . dtm+n.
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166 Mathematical Foundations of Quantum Field Theory
In order to obtain (9.22) from the above formula, we should take the
Fourier transform over the variables t1, . . . , tm+n, using that∫Lf(k, ε, t)dt =
∫ ∞−∞
Λ(k, ε)d
dt(exp(−iεω(k)t)f(k, ε, t))dt
= iεΛ(k, ε) limω→ω(k)
(ω − ω(k))
×∫ ∞−∞
exp(−iεωt)f(k, ε, t)dt.
For a more rigorous proof of (9.22), see, for example, Robertson
et al. (1980). Such a proof can be obtained also from considerations
of Chapter 13.
9.3 The adiabatic definition of scattering matrix
Let us now provide a definition of scattering matrix of a translation-
invariant Hamiltonian that does not use the notion of operator
realization. Let H be a translation-invariant Hamiltonian of the form
H = H0 + V , where H0 =∫ε(k)a+(k)a(k)dk.
We will consider the adiabatic S-matrix SΩα = SΩ
α (∞,−∞) corre-
sponding to the pair of operators (HΩ, H0Ω) (the Hamiltonian HΩ is
defined as the Hamiltonian H with volume cutoff, see Section 8.1).
Definition 9.1. The scattering matrix for a Hamiltonian H is an
operator in the space Has that has the matrix entries
〈p1, . . . ,pm|S|q1, . . . ,qm〉 = limα→0
limΩ→∞
(L
2π
) 32
(m+n)
×⟨p1, . . . ,pm|SΩ
α |q1, . . . ,qm⟩ ⟨θ|SΩ
α |θ⟩m+n
2−1√∏m
i=1 〈pi|SΩα |pi〉
∏nj=1 〈qj |SΩ
α |qj〉(9.29)
in the generalized basis b+(p1) . . . b+(pn)θ. Here Has, as in Sec-
tion 9.2, denotes the space of Fock representations of CR (the
operator generalized functions b+(k), b(k) acting on Has satisfy CR
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The Scattering Matrix for Translation-Invariant Hamiltonians 167
and k runs over E3). We use the notation
〈p1, . . . ,pm|S|q1, . . . ,qn〉 =⟨Sb+(q1) . . . b+(qn)θ, b+(p1)
. . . b+(pm)θ⟩,⟨
p1, . . . ,pm|SΩα |q1, . . . ,qn
⟩=⟨SΩα a
+q1. . . a+
bqnθ, a+
p1. . . a+
pmθ⟩
(recall that the operator SΩα , like the operator HΩ, acts on the Fock
space FΩ and the symbols a+p , ap denote the operators a+(φp), a(φp),
where φp(x) = L−3/2 exp(ipx) and p runs over the lattice TΩ,
consisting of the vectors 2πL n).
We should explain the meaning of convergence in (9.29), since the
functions of continuous argument are defined as limits of functions
with arguments in a lattice. We dealt with a similar situation
in Section 8.2; here and in the rest of the book, we will deal
with similar limits analogously (i.e. in the sense of generalized
functions). Namely, the relation (9.29) means that for any test
function φ(p1, . . . ,pm|q1, . . . ,qn)∫φ(p1, . . . ,pm|q1, . . . ,qn) 〈p1, . . . ,pm|S|q1, . . . ,qn〉 dmpdnq
= limα→∞
limΩ→∞
(2π
L
) 32
(m+n) ∑pi,qj∈TΩ
φ(p1, . . . ,pm|q1, . . . ,qn)
×⟨p1, . . . ,pm|SΩ
α |q1, . . . ,qm⟩ ⟨θ|SΩ
α |θ⟩m+n
2−1√∏m
i=1 〈pi|SΩα |pi〉
∏nj=1 〈qj |SΩ
α |qj〉.
We can represent the definition of scattering matrix in a different
form, noting that⟨p1, . . . ,pm|SΩ
α |q1, . . . ,qm⟩ ⟨θ|SΩ
α |θ⟩m+n
2−1√∏m
i=1 〈pi|SΩα |pi〉
∏nj=1 〈qj |SΩ
α |qj〉
=⟨p1, . . . ,pm|UΩ
α SΩαU
Ωα |q1, . . . ,qm
⟩, (9.30)
where UΩα = exp[i(C +
∑k r
Ωα (k)a+
k ak)] (the sum is taken over
the lattice TΩ), C = i2 ln
⟨θ|SΩ
α |θ⟩, rΩα (k) = i
2 ln〈k|SΩ
α |k〉〈θ|SΩ
α |θ〉(the
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168 Mathematical Foundations of Quantum Field Theory
relation (9.30) follows from UΩα a
+k1. . . a+
knθ = exp[i(C+rΩ
α (k1)+· · ·+rΩα (kn))]a+
k1. . . a+
knθ). The relation (9.30) follows from the fact that
S = limα→0
limΩ→∞
UΩα S
ΩαU
Ωα ,
where the convergence of operators is understood in the sense of
the convergence of the matrix entries in the basis a+k1. . . a+
knθ to the
matrix elements in the generalized basis b+(k1) . . . b+(kn)θ (further
in the book, we will understand analogous situations concerning the
convergence of operators acting on the space FΩ to operators on the
space Has in the same way).
Using the results of Chapter 4 (see Sections 4.2 and 4.3), one can
prove that the S-matrix defined by the relation (9.29) satisfies the
conditions Sθ = θ, Sb+(k)θ = b+(k)θ (the stability of the vacuum
and single-particle states).
Indeed, let φΩ(λ) (correspondingly, φΩk (λ)) be a stationary state
of the Hamiltonian HΩ(λ) = H0Ω + λVΩ continuously depending on
the parameter λ and converging to θ for λ = 0 (correspondingly
a+k θ). Furthermore, we let εΩ(λ) and εΩ(k, λ) be the energy levels of
the Hamiltonian HΩ(λ)
ρΩ =1
α
∫ 1
0
1
µ(εΩ(µ)− εΩ(0))dµ,
ρΩ(k) =1
α
∫ 1
0
1
λ(εΩ(k, λ)− εΩ(k, 0))dλ.
It follows from (4.11) that
θ = limα→0
exp(2iρΩ)SΩα θ,
a+k θ = lim
α→0exp(2iρΩ(k))SΩ
α a+k θ.
We conclude that for α→ 0,⟨k|SΩ
α |k′⟩≈ exp(2iρΩ(k))δkk′ , (9.31)⟨
θ|SΩα |θ⟩≈ exp(2iρΩ), (9.32)
and therefore,
UΩα ≈ UΩ
α = exp
(−iρα − i
∑k
(ρΩ(k)− ρΩ)a+k ak
). (9.33)
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The Scattering Matrix for Translation-Invariant Hamiltonians 169
Using Equations (9.31) and (9.32), we see that
limα→0
UΩα S
ΩαU
Ωα θ = θ, (9.34)
limα→0
UΩα S
ΩαU
Ωα a
+k θ = a+
k θ. (9.35)
Multiplying the relations (9.34) and (9.35) by the vectors a+k1. . . a+
knθ
and taking the limit Ω→∞ in the resulting matrix elements of the
operator UΩα S
ΩαU
Ωα , we obtain a proof of the stability of the vacuum
and single-particle states.
Let us also note that from the previous reasoning, we can obtain
the “almost unitarity” of the operators UΩα when α → 0. More
precisely, the scattering matrix can be represented in the form
S = limα→0
limΩ→∞
UΩα S
Ωα U
Ωα , (9.36)
where UΩα = exp(−iρΩ − i
∑k(ρΩ(k) − ρΩ)a+
k ak) are unitary
operators.
Note that the proofs above are clearly not rigorous.
We can modify the scattering matrix definition given above by
introducing the notion of N -equivalent and S-equivalent operators,
acting in Fock space.
Two operators S and S′ on Fock space Tas are N -
equivalent if there exist unitary operators U and V of the form
exp[i(∫ν(k)b+(k)b(k)dk + C)], such that S′ = USV . If U = V −1,
then we say that the operators S and S′ are S-equivalent.
The definitions of N -equivalence and S-equivalence for operators
acting on the Fock space FΩ only differ by assuming that the
operators U and V take the form exp[i(∑
k νka+k ak + C)].
Definition 9.2. The operator S on the space Has is called the
scattering matrix of the Hamiltonian H if it satisfies the conditions
Sθ = θ, Sb+(k)θ = b+(k)θ and it can be represented in the form
S = limα→0
limΩ→∞
SΩα , (9.37)
where SΩα are operators that are N -equivalent to the operators SΩ
α .
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170 Mathematical Foundations of Quantum Field Theory
Let us recall that the limit in the expression of the form (9.37) is
understood in the sense of the convergence of the matrix entries of
SΩα in the basis a+
k1. . . a+
knθ to the matrix entries of the operator S
in the generalized basis b+(k1) . . . b+(kn)θ.
It is clear from the relation (9.36) that a scattering matrix
specified by Definition 7.1 is also a scattering matrix in the sense of
Definition 9.1. However, unlike Definition 7.1, Definition 9.1 does not
specify a scattering matrix unambiguously: using the ambiguity in
the choice of the operators SΩα , it is easy to see that an operator that
is S-equivalent to the scattering matrix is also a scattering matrix in
the sense of Definition 9.1. The converse of this observation holds as
well: two scattering matrices of a Hamiltonian H are S-equivalent.
Note that in Section 9.2 the scattering matrix was also defined up to
S-equivalence.
In the framework of perturbation theory, one can prove that
Definition 9.1 is equivalent to the definition of the scattering matrix
given in Section 9.2.2 However, to show this equivalence, we need to
make a few assumptions about the Hamiltonian H (it is sufficient
to require that the Hamiltonian belongs to the class M defined in
Section 11.1 and the function ε(k), entering the definition of the
Hamiltonian H0, satisfies the condition ε(k1 + k2) < ε(k1) + ε(k2)).
We consider the connection between adiabatic S-matrices and
scattering matrices in Section 10.6 (in the framework of axiomatic
scattering theory) and in Section 11.5 (for Hamiltonians that do
not generate vacuum polarization). The proof of the equivalence of
Definition 9.1 with other definitions of scattering matrices in the
framework of perturbation theory with partial summation can be
obtained by modifying the considerations in Section 10.6. Another
less rigorous proof can be found in Likhachev et al. (1972).
9.4 Faddeev’s transformation and equivalence
theorems
Let us consider a Hamiltonian H = H0 +V of the form (9.29). In the
case of vm,0 ≡ vm,1 ≡ 0, the scattering matrix of H can be defined
2More precisely, to show this equivalence, we should perform a partial summa-tion of the series in perturbation theory for SΩ
α .
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The Scattering Matrix for Translation-Invariant Hamiltonians 171
by using the standard relations of formal scattering theory (5.3) (see
Section 9.1). If the Hamiltonian H is arbitrary, then under certain
conditions, we may replace H with an equivalent, in a certain sense,
Hamiltonian H ′ = H ′0 + V ′ for which the scattering matrix can be
defined as
S = slimt→∞t0→−∞
exp(iH ′0t) exp(−iH ′(t− t0)) exp(−iH ′0t0). (9.38)
This scattering matrix can be regarded as the scattering matrix for
the original Hamiltonian H.
The following discussion will proceed in the framework of per-
turbation theory. This means that the Hamiltonian H0 + V will be
included in the family of Hamiltonians H0 + gV depending on a
parameter g (coupling constant) and the Hamiltonian H ′ will be
constructed as a series in powers of g.
To begin, let us prove a few helpful statements. Consider the
equation
[h0, x] = r, (9.39)
where h0, x, r are operators on the Fock space FΩ = F (L2(Ω))
corresponding to the finite-volume Ω:
h0 =∑k∈TΩ
ε(k)a+k ak, (9.40)
where the positive function ε(k) satisfies
ε(k1 + k2) < ε(k1) + ε(k2). (9.41)
Let us write down the unknown operator x and the known operator
r in normal form:
x =∑m,n
∑ki,pj
ξm,n(k1, . . . ,km|p1, . . . ,pn)a+k1
. . . a+kmap1 . . . apn ; (9.42)
r =∑m,n
∑ki,pj
ρm,n(k1, . . . ,km|p1, . . . ,pn)a+k1
. . . a+kmap1 . . . apn
(in these formulas, just as in the following analogous formulas, the
sums are taken over ki,pj ∈ TΩ). If we calculate the commutator
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 172
172 Mathematical Foundations of Quantum Field Theory
[h0, x], we obtain the relation between the unknown functions ξm,nand the known functions ρm,n:
(ε(k1) + · · ·+ ε(km)− ε(p1)− · · · − ε(pn))
× ξm,n(k1, . . . ,km|p1, . . . ,pn)
= ρm,n(k1, . . . ,km|p1, . . . ,pn). (9.43)
If the operator is presented in normal form, then the sum-
mands containing m creation operators and n annihilation oper-
ators will be called summands of type (m,n). Summands of type
(m, 0), (0,m), (n, 1), and (1, n), where m ≥ 1, n ≥ 2, will be called
bad summands.
Let us prove the following lemma.
Lemma 9.1. If an operator r commutes with the momentum
operator PΩ =∑
ka+k ak and only contains bad summands, then
equation (9.39) is solvable. If we also assume that the operator
x commutes with the momentum operator and contains only bad
summands, then equation (9.39) has a unique solution which will
be denoted by Γ(r). If the operator r is Hermitian, then so is the
operator Γ(r).
In order to prove this lemma, we first note that from the
condition (9.43), we can find the functions ξm,n in terms of the
functions ρm,n if and only if for all values of k1, . . . ,km,p1, . . . ,pnsatisfying ε(k1) + · · · + ε(km) = ε(p1) + · · · + ε(pn), we have
ρm,n(k1, . . . ,km|p1, . . . ,pn) = 0. If we assume ε(k1) + . . . +
ε(km) > 0, then for the functions ρm,0 and ρ0,m, we can find the
functions ξm,0 and ξ0,m. Furthermore, from the assumption that r
commutes with the momentum operator, it follows that the quantity
ρm,n(k1, . . . ,km|p1, . . . ,pn) is non-zero only if k1 + · · · + km =
p1 + · · ·+ pn. This allows us to say that the inequality
ε(k1) + · · ·+ ε(km) > ε(p),
which holds in the case of k1 + · · · + km = p,m > 1, guarantees
the solvability of (9.43) for m > 1, n = 1 and n > 1,m = 1 (this
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 173
The Scattering Matrix for Translation-Invariant Hamiltonians 173
follows from (9.41)). Therefore, under the conditions of the lemma,
equation (9.39) is solvable. The proofs of the rest of the statements
in the lemma are trivial (for example, the uniqueness follows from
the fact that under the condition ε(k1) + · · · + ε(km) 6= ε(p1) +
· · ·+ ε(pn) we can find a unique ξm,n(k1, . . . ,km|p1, . . . ,pn) for each
ρm,n(k1, . . . ,km|p1, . . . ,pn)).
Let us take a Hermitian operator h, acting on the Fock space
FΩ, that commutes with the momentum operator PΩ, and consider
the question of finding a unitary operator w such that the operator
h′ = whw−1 contains no bad summands. We will show that in the
framework of perturbation theory, the operator w can be constructed.
More precisely, we will show that for an operator of the form
h = h0 + gv, where h0 takes the form (9.40) considered above
and g is a scalar parameter (coupling constant), the operators h′
and w can be constructed as formal series in powers of g. The
constructed operators h′ and w will commute with the momentum
operator PΩ; this implies that the operator h′ can be represented
in the form h′ = c + h′0 + v′, where c is a constant [a summand
of type (0, 0)], h′0 =∑ωka
+k ak [a summand of type (1, 1)],
v′ =∑
m,n≥2
∑v′m,n(k1, . . . ,km|p1, . . . ,pn)δp1+···+pn
k1+···+kma+k1. . . a+
kmap1
. . . apn (a sum of summands of type (m,n) with m ≥ 2, n ≥ 2). We
will assume that the operator w takes the form
w = exp(−iα),
where α is a Hermitian operator that can be represented as a series
in powers of g:
α =∞∑n=1
gnαn.
We will use the formula3
h′ = exp(−iα)h exp(iα) =
∞∑n=0
in
n![. . . [h, α] . . . , α].
3It is easy to see that the operators C(t) = exp(−itα)b exp(itα) and D(t) =∑∞n=0
(it)n
n![. . . [b, α] . . . , α] satisfy identical equations i dC(t)
dt= [α,C(t)], i dD(t)
dt=
[α,D(t)] with the same initial condition C(0) = D(0) = b; henceC(t) = D(t).
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174 Mathematical Foundations of Quantum Field Theory
Equating the terms of the same degree with respect to g, we obtain
the equation
h′n = i[h0, αn] + qn,
where qn denotes the operator that can be expressed in terms of
h0, v, and αk with k ≤ n − 1, and h′ngn denotes an order-n term
in the expansion of the operator h′ in powers of g. We will show
by induction with respect to n that we can find operators αn and
h′n such that the operator h′n contains no bad summands. This can
be done if we take as h′n the sum of good summands in the normal
form of the operator qn. Then, the operator h′n − qn contains only
bad terms and satisfies the conditions imposed on the operator r in
the lemma. Therefore, the operator αn satisfying the above equation
exists; let us define it by the formula αn = Γ(h′n − qn). It is easy to
check that the operators qn are Hermitian; it follows that h′n and αnare Hermitian as well.
By this method, we can inductively construct the operators h′nand αn such that the operator h′ =
∑h′ng
n does not contain bad
summands and the operator w = exp(−iα) = exp(−i∑αng
n)
establishes a unitary equivalence between the operators h and h′
(i.e. h′ = whw−1).
Note that our construction for the operators h′ and w is unam-
biguous; we will call this construction the Faddeev construction and
the unitary operator w the Faddeev transformation.
It is worth noting that the operators h′ and w were constructed in
the framework of perturbation theory; in other words, these operators
were constructed in the form of formal series in powers of g. It is not
clear whether these series converge.
Moreover, as was done in the lemma and in the construction
of the Faddeev transformation, we used the term “operator” for
expression (9.42), but we did not prove that this formal expression
specifies an operator on Fock space. In the situations where we use
Faddeev construction, it is easy to check that the formal expressions
hn and αn specify Hermitian operators; the proof is based on the
considerations in Section 6.2.
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 175
The Scattering Matrix for Translation-Invariant Hamiltonians 175
Let us consider the Faddeev transformation from another point
of view.
Consider the operators
b+k = w−1a+kw; bk = w−1akw. (9.44)
It is clear that these operators satisfy the same commutation relations
as the operators a+k , ak:
[bk, bk′ ]∓ = [b+k , b+k′ ]∓ = 0, [bk, b
+k′ ]∓ = δk,k′ .
A transformation from the operators ak, a+k to the operators bk, b
+k
satisfying the same commutation relations is called a canonical
transformation. From the relation h = w−1h′w, it follows that the
operator h can be expressed in terms of the operators b+k , bk in
the same way that h′ can be expressed with a+k , ak; in this way,
representing the operator h with the operators b+k , bk in normal form,
we obtain an expression without bad summands.
Let us now return to the translation-invariant Hamiltonian H =
H0 + gV , where H0 and V satisfy the formulas (9.1) and (9.2).
Suppose that ε(k) is a smooth function, whose derivatives do not
grow faster than a power, and that wm,n belongs to the space S. Let
us also assume that the function ε(k) satisfies equation (9.41).
Let us now apply the Faddeev construction to the Hamiltonian
HΩ obtained from H by means of a volume cutoff. The Hamiltonian
obtained through this construction contains no bad summands and
will be denoted by H ′Ω. The operator establishing a unitary equiv-
alence between H ′Ω and HΩ will be denoted by WΩ = exp(−iAΩ).
It is easy to understand the behavior of the operators H ′Ω and AΩ
as Ω → ∞. Namely, the operator H ′Ω can be written in the form
H ′Ω = CΩ +H ′Ω0 + V ′Ω, where CΩ is a constant that grows linearly
as Ω→∞. We obtain
H ′Ω0 =∑
ωΩk a
+k ak,
V ′Ω =∑m,n
∑(L
2π
)−3(m+n2−1)
v′Ωm,n(k1, . . . ,km|p1, . . . ,pn)
× δp1+···+pnk1+···+km
a+k1. . . a+
kmap1 . . . apn ,
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 176
176 Mathematical Foundations of Quantum Field Theory
the functions ωΩk , v
′Ωm,n converge to a limit as Ω → ∞.4 Let us
introduce the notation
limωΩk = ω(k);
lim v′Ωm,n(k1, . . . ,km|p1, . . . ,pn)
= v′m,n(k1, . . . ,km|p1, . . . ,pn), (9.45)
H ′0 =
∫ω(k)a+(k)a(k)dk;
V ′ =∑m,n
∫v′m,n(k1, . . . ,km|p1, . . . ,pn)
× δ(k1 + · · ·+ km − p1 − · · · − pn)a+(k1)
. . . a+(km)a(p1) . . . a(pn)dmkdnp. (9.46)
The translation-invariant Hamiltonian H ′ = H ′0+V ′ does not contain
bad summands; we will say that this Hamiltonian is obtained from
the Hamiltonian H by means of Faddeev transformation. One can
say that
H ′ = limΩ→∞
(H ′Ω − CΩ) (9.47)
(the relation (9.47) can be understood as a shorthand version
of (9.46)).
The operator AΩ behaves the same way in the limit Ω → ∞,
namely, it can be written in the form
AΩ =∑m,n
(L
2π
)−3(m+n2−1)∑
αΩm,n(k1, . . . ,km|p1, . . . ,pn)
× δp1+···+pnk1+···+km
a+k1. . . a+
kmap1 . . . apn ,
4The functions ωΩk , v
′Ωm,n are expressed in the form of a series in powers of g;
we understand the convergence as Ω → ∞ in the sense of convergence for everypower of g. It is useful to note that we can find such functions νrm,n ∈ S and σr(k)that do not grow faster than a power function and that satisfy the conditions|(v′Ωm,n)r| ≤ νrm,n, |(ωΩ
k )r| ≤ σr(k) (by the symbol fr here, we denote a term oforder r in g in the expansion of the function f).
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 177
The Scattering Matrix for Translation-Invariant Hamiltonians 177
where the functions αΩm,n(k1, . . . ,km|p1, . . . ,pn) have a limit as
Ω→∞,
αm,n(k1, . . . ,km|p1, . . . ,pn) = limΩ→∞
αΩm,n(k1, . . . ,km|p1, . . . ,pn)
(9.48)
in each term in the series with respect to g. Using the formal
translation-invariant expression
A =∑m,n
∫αm,n(k1, . . . ,km|p1, . . . ,pn)
× δ(k1 + · · ·+ km − p1 − · · · − pn)a+(k1)
. . . a+(km)a(p1) . . . a(pn)dmkdnp,
we can write the relation (9.49) in the form A = limΩ→∞AΩ.
The behavior of the operator WΩ for large Ω is more complicated;
one cannot find a formal translation-invariant expression for this
operator in the limit Ω → ∞. By construction, the expression for
V ′ has no summands of type (m, 0) or (m, 1) and therefore the
formal expression for H ′ can specify an operator H ′ in Fock space.
In this case, the scattering matrix for the Hamiltonian H ′ can be
constructed like an S-matrix for the pair of operators (H ′, H ′0) (see
Section 9.1). In Chapter 11, we will show that the S-matrix for the
pair of operators (H ′, H ′0) can be constructed in the framework of
perturbation theory.
Faddeev proposed to define the scattering matrix for the Hamil-
tonian H as the S-matrix constructed for the pair of operators
(H ′, H ′0). It can be proved that Faddeev’s definition of the scattering
matrix is equivalent to the other definitions we have considered (the
proof can only be given in perturbation theory, since Faddeev’s
construction is based on perturbation theory). The proof of this
statement is provided in Section 11.3, while here we will instead
turn our attention to some generalizations of Faddeev’s construction.
First, we will represent this construction in a slightly different form.
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 178
178 Mathematical Foundations of Quantum Field Theory
It is easy to check that the S-matrix (9.38) can be written in the form
S = limt→∞,t0→−∞
limΩ→∞
exp(iH ′Ω0 t) exp(−i(H ′Ω − CΩ)(t− t0))
× exp(−iH ′Ω0 t0)
(the limit should be understood in the sense explained in Section 9.3).
Using the relation H ′Ω = WΩHΩ(WΩ)−1, we can write that
S = limt→∞,t0→−∞
limΩ→∞
exp(i(H ′Ω0 + CΩ)t)WΩ exp(−iHΩ(t− t0))
× (WΩ)−1 exp(−i(H ′Ω0 + CΩ)t0). (9.49)
The operator (WΩ)−1 is called a dressing operator, since it trans-
forms the bare vacuum θ and the bare single-particle state a+k θ into
stationary states of the Hamiltonian HΩ that can be described as
the dressed vacuum and dressed single-particle states (in fact, the
operator (WΩ)−1 transforms stationary states of the Hamiltonian
H ′Ω into stationary states of the Hamiltonian HΩ and θ and a+k θ are
clearly stationary states of H ′Ω).
The formula (9.49) suggests the following generalization of a
dressing operator (more precisely, a family of dressing operators DΩ
depending on the parameter Ω).
The family of operators DΩ acting on the space FΩ is called a
family of dressing operators for the Hamiltonian H if there exists a
Hamiltonian HΩkb of the form γΩ+
∑k ν
Ωk a
+k ak such that the S-matrix
for the Hamiltonian H can be written in the form
S = limt→∞,t0→−∞
limΩ→∞
exp(iHΩkbt)(D
Ω)−1
× exp(−iHΩ(t− t−))DΩ exp(−iHΩkbt0).
Formula (9.49) shows that the family of operators (WΩ)−1 is a family
of dressing operators in the sense of the above definition.
In Section 11.4, we will describe a broad class of families of
dressing operators that includes the operators (WΩ)−1.
The theorem establishing the equivalence of Faddeev’s definition
with other definitions of scattering matrices is a particular case
of a theorem of invariance of scattering matrices under canonical
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 179
The Scattering Matrix for Translation-Invariant Hamiltonians 179
transformations (theorems of this kind are called equivalence theo-
rems).
Let us give a general definition of a canonical transformation.
Let us express the symbols b+(k), b(k) in terms of the symbols
a+(k), a(k) by using the formulas
b(k) =∑m,n
∫σm,n(k1, . . . ,km|p1, . . . ,pn)
× δ(k + k1 + · · ·+ km − p1 − · · · − pn)a+(k1)
. . . a+(km)a(p1) . . . a(pn)dmkdnp; (9.50)
b+(k) =∑m,n
∫σm,n(k1, . . . ,km|p1, . . . ,pn)
× δ(k + k1 + · · ·+ km − p1 − · · · − pn)a+(p1)
. . . a+(p1)a(km) . . . a(k1)dmkdnp, (9.51)
where σm,n are smooth functions whose derivatives do not grow faster
than a power.
The transformation from the symbols a+(k), a(k) into the sym-
bols b+(k), b(k) is called a canonical transformation if the sym-
bols b+(k), b(k) obey the same commutation (or anticommutation)
relations as the symbols a+(k), a(k) (i.e. obey CR). In the case of
CCR, we can calculate the commutators for the symbols b+(k), b(k)
by using CCR for the operators a+(k), a(k), distributivity, and
the relation [A,B,C] = [A,C]B + A[B,C] (we can calculate the
anticommutator in the case of CAR analogously).
The above equivalence theorem can be reformulated (not quite
rigorously) in the following way.
Theorem 9.1. The canonical transformation (9.50) transforms the
translation-invariant Hamitlonian H of the form (9.1) into the
translation-invariant Hamiltonian H having the same scattering
matrix.
(The Hamiltonian H can be constructed in the following way:
in the expression of the Hamiltonian H with respect to the symbols
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180 Mathematical Foundations of Quantum Field Theory
a+(k), a(k), we substitute against the symbols b+(k), b(k), expressed
in terms of a+(k), a(k) using the formula (9.50); we then write
the obtained expression into normal form using CR, discarding any
infinite constants that arise.)
It is easy to see that the Hamiltonian H ′ obtained from the
translation-invariant Hamiltonian using Faddeev’s construction is
connected with the Hamiltonian H through the canonical transfor-
mation (this canonical transformation is obtained from the canonical
transformation (9.44) relating the operators b+k , bk with the operators
a+k , ak in the limit Ω→∞).
It is therefore clear from the equivalence theorem that the
scattering matrices for the operators H and H ′ coincide; this
demonstrates that Faddeev’s construction leads to a conventional
scattering matrix.
We will prove the equivalence theorem in the framework of
perturbation theory in Section 11.3.
Note that for linear canonical transformations, the considerations
in Section 11.3 prove the equivalence theorem outside the framework
of perturbation theory.
9.5 Semiclassical approximation
We have already seen that the Hamiltonians of quantum field theory
can be obtained through the quantization of classical systems with
an infinite number of degrees of freedom. We will leverage this fact
to obtain approximate solutions to problems in quantum field theory.
As in Section 8.3, we will not make the assumption that ~ = 1, as
we do in the rest of this book.
Let us consider a quantum mechanical system with a finite num-
ber of degrees of freedom that is described by the Hamiltonian (8.22).
In order to find weakly excited states (stationary states whose
energy is close to the energy of the ground state), we can apply
the following method. Let us find the minimum of the function
U(q1, . . . , qn) (potential energy); suppose the minimum occurs at the
point (q01, . . . , q
0n). Let us introduce the new variables xi = qi − q0
i
(these should be understood as the deviation from the classical
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 181
The Scattering Matrix for Translation-Invariant Hamiltonians 181
equilibrium state) and let us decompose the function U as a series in
powers of xi:
U(q1, . . . , qn) = U(q01, . . . , q
0n) +
1
2
∑i,j
cijxixj
+1
3!
∑i,j,k
cijkxixjxk + · · · .
If the matrix cij is positive definite, then the energy of a weakly
excited state can be determined by the following approximate
expression:
U(q01, . . . , q
0n) +
~2
∑ωi + ~
∑niωi, (9.52)
where ω21, . . . , ω
2n are eigenvalues of the matrix cij and ni = 0, 1, 2, . . .
are non-negative integers.
To check this, we will write the Hamiltonian (8.22) in terms of
the operators pi and xi = qi − q0i , satisfying the relations [pi, pj ] =
[xi, xj ] = 0, [pi, xj ] = ~i δij ; we obtain
H = H0 + V ;
H0 =1
2
∑p2i +
1
2
∑cij xixj + U(q0
1, . . . , q0n);
V =1
3!
∑cijkxixj xk + · · · .
The operators pi, xi satisfy the same commutation relations as pi, qi,
i.e. the transformation between these operators can be considered to
be a canonical transformation.
The eigenvalues of the operator H0 are specified by the expres-
sion (9.52) (see Section 2.6). The eigenvalues of the operator H =
H0 + V can be calculated in the framework of perturbation theory
by considering the operator V as a perturbation. It is easy to see
that the corrections to the energies of a weakly excited state are
small. This becomes evident if we express H0 and V in terms of the
operators a+i , ai, satisfying CCR (we rely on the considerations in
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 182
182 Mathematical Foundations of Quantum Field Theory
Section 2.6):
H0 = U(q01, . . . , q
0n) +
1
2~∑
ωi + ~∑
ωia+i ai;
V =∑
m+n≥3
~( 12
)(m+n)∑
i1,...,im,j1,...,jn
Γm,ni1,...,im,j1,...,jna+i1. . . ajn
(the perturbation series here can be rewritten as a series in powers
of ~1/2, hence the correction to the energy has order at least ~3/2).
We see that the analysis of weakly excited states of Hamiltonian
H is based on the canonical transformation (transforming to the
operators pi, xi) where we single out the quadratic part of H0 and
consider V as a perturbation.
Let us now consider a classical system specified by the
Hamiltonian
H(π, φ) =1
2
∑i
∫π2i (x)dx +
∑m
∑i1,...,im
∫Vi1,...,im(x1, . . . ,xm)
×φi1(x1) . . . φim(xm)dmx; (9.53)
Vi1,...,im(x1, . . . ,xm) = vi1,...,im(x1−xm, . . . ,xm−1−xm). Quantizing
this system leads to the Hamiltonian
H =1
2
∑i
∫π2i (x)dx
+∑m
∑i1,...,im
∫Vi1,...,im(x1, . . . ,xm)φi1(x1)
. . . φim(xm)dmx, (9.54)
where
[πi(x), πj(x′)] = [φi(x), φj(x
′)] = 0,
[πi(x), φj(x′)] =
~iδi,jδ(x− x′)
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 183
The Scattering Matrix for Translation-Invariant Hamiltonians 183
(for a slightly less general Hamiltonian, see (8.23) discussed in
Section 8.3; everything that is known for the Hamiltonian (8.23) can
be easily proven for the Hamiltonians (9.54)).
As in the case of a finite number of degrees of freedom, the analysis
of the Hamiltonian (9.54) can be performed with the canonical
transformation: instead of the symbols φi(x), we should introduce the
symbols ξi(x) = φi(x)− αi, where αi are real numbers; the symbols
πi(x), ξi(x) satisfy the same commutation relations as πi(x), φj(x)
(we could introduce the symbol ξi(x) by the more general formula
ξi(x) = φi(x) − αi(x), but since our Hamiltonian is translation-
invariant it is natural to assume that αi(x) do not depend on x).
Let us consider a function of n variables
ν(φ1, . . . , φn) =∑m
∑i1,...,im
νi1,...,imφi1,...,im ,
where
νi1,...,im =
∫vi1,...,im(x1, . . . ,xm−1)dm−1(x)
(the function ν has the physical meaning of energy density of the
classical system in the state π1(x) = · · · = πn(x) ≡ 0, φ1(x) ≡φ1, . . . , φn(x) ≡ φn). The numbers α1, . . . , αn specifying the canoni-
cal transformation will be found from the condition
minφ1,...,φn
ν(φ1, . . . , φn) = ν(α1, . . . , αn).
The point in n-dimensional space α = (α1, . . . , αn), where the
function ν(φ1, . . . , φn) achieves its minimum, will be called the
classical vacuum.
It is easy to check, under certain conditions, that one can justify
the choice of the numbers α1, . . . , αn by considering a limit of systems
with finite number of degrees of freedom.
Replacing φi(x) with ξi(x) +αi in (9.54), we obtain the Hamilto-
nian H ′ with the same scattering matrix by the equivalence theorem.
Extracting from H ′ the quadratic terms H0, we can represent the
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 184
184 Mathematical Foundations of Quantum Field Theory
Hamiltonian H ′ in the form H0 + V , where
H0 =1
2
∑i
∫π2i (x)dx +
∑i,j
1
2
∫cij(x− y)
× ξi(x)ξj(y)dxdy,
V =∑m≥3
∑i1,...,im
∫ci1,...,im(x1 − xm, . . . ,xm−1 − xm)
× ξi1(x1) . . . ξim(xm)dmx.
Using the ideas of Section 8.3, we can construct an operator
realization of the Hamiltonian H0. We see that H0 describes n types
of particles with energy dependence on the momentum, defined by
the formula Ej(p) = ~ωj(p~ ), where ω21(k), . . . , ω2
n(k) are eigenvalues
of the matrix cij(k) = (2π)−3∫cij(x) exp(ikx)dx. Understanding
V as a perturbation, we obtain the corrections to the energy of
single-particle states. By perturbation theory, we can also calculate
the scattering matrix corresponding to the Hamiltonian H ′ and,
therefore, to the Hamiltonian H.
Hence, the semiclassical approximation, as it is described in this
section, prompts a reasonable choice of an initial approximation in
the perturbation theory. However, one can show that semiclassical
approximation leads to important qualitative conclusions. Let us
consider, for example, the Hamiltonian
H =1
2
∫π2(x)dx +
1
2
∫(∇φ)2dx +
∫U(φ(x))dx, (9.55)
where U(φ) = a1φ+a2φ2 +a3φ
3 +a4φ4. (This Hamiltonian describes
a Lorentz-invariant theory (see Section 12.3). The Hamiltonian H
leads to ultraviolet divergences that should be treated by means of
renormalization; we do not analyze these divergences here.) It is easy
to see that for the Hamiltonian (9.55), ν(φ) = U(φ). If U(φ) =
a2φ2 + a4φ
4, a2 > 0, a4 > 0, then the minimum of ν(φ) is achieved
at φ = 0. This means that in the analysis of the Hamiltonian (9.55),
we should use the perturbation theory where H0 is specified in the
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 185
The Scattering Matrix for Translation-Invariant Hamiltonians 185
usual way as follows:
H0 =1
2
∫π2(x)dx +
1
2
∫(∇φ)2dx +
∫a2φ
2(x)dx. (9.56)
In general, it is unreasonable to specify H0 by (9.56) (this is clear
since the Hamiltonian (9.56), in the case a2 < 0, describes particles
with imaginary mass; it is impossible to construct its operator
realization). In order to make the correct choice for an initial
approximation, we should find
min ν(φ) = minU(φ) = min(a1φ+ a2φ2 + a3φ
3 + a4φ4).
If this minimum is achieved for φ = α, then, using the canonical
transformation, we get the Hamiltonian H ′ = H0 + V , where
H0 =1
2
∫π2(x)dx +
1
2
∫(∇ξ)2dx +
1
2µ2
∫ξ2(x)dx,
µ2 = U ′′(α).
Now, we can apply the perturbation theory to the Hamiltonian H ′
considering V as a perturbation. In the case when U(φ) = a2φ2 +
a4φ4, a2 < 0, a4 > 0, the minimum of the function U(φ) is achieved
at two points α = ±√
a22a4
(there are two different classical vacua).
Hence, the necessary canonical transformation and the Hamiltonian
H ′ can be constructed in two ways: H ′± = H0 + V±, where
V± = a4
∫ξ4(x)dx± 2
√2a2a4
∫ξ3(x)dx.
This allows us to conjecture that in the case at hand, one can
construct different operator realizations of the Hamiltonian H. In
fact, let us take an operator realization of the Hamiltonian H ′+assuming that V+ is a perturbation. This means that
lim~→0
⟨ξ(x, t)Φ,Φ
⟩= 0.
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186 Mathematical Foundations of Quantum Field Theory
Starting with the operator realization of the Hamiltonian H ′+, we get
an equivalent operator realization of the Hamiltonian H that gives
lim~→0
⟨φ(x, t)Φ,Φ
⟩=
√a2
2a4+ lim
~→0〈ξ(x, t)Φ,Φ〉
=
√a2
2a4> 0.
For an operator realization of the Hamiltonian H, constructed in a
similar way by means of H ′−, we get that
lim~→0〈φ(x, t)Φ,Φ〉 = −
√a2
2a4< 0,
and hence, we have constructed two essentially different operator
realizations.
Note that we have encountered an important phenomenon called
symmetry breaking. The initial Hamiltonian function is invariant
with respect to the transformation φ(x) → −φ(x); however, the
classical vacuum is not invariant with respect to this transformation
(it transforms one classical vacuum into another). A similar situation
arises in the quantum case: the Hamiltonian H is invariant with
respect to the transformation φ(x) → −φ(x) and therefore, from
one operator realization of the Hamiltonian H, we can get another
by replacing the operators φ(x, t) by the operators −φ(x, t). If
〈φ(x, t)Φ,Φ〉 6= 0, then this replacement gives an operator realization
that is not equivalent to the original.
Let us consider one more interesting example, the Hamiltonian
H =1
2
∑∫π2i (x)dx +
∫ (∑φ2i (x)− a2
)2dx. (9.57)
Here,
ν(φ1, . . . , φn) = (φ21 + · · ·+ φ2
n − a2)2
and hence, the minimum of the function ν is achieved at the points
(α1, . . . , αn) satisfying the condition∑α2i = a2 (the classical vacua
fill an (n−1)-dimensional sphere). The Hamiltonian (9.57), as well as
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 187
The Scattering Matrix for Translation-Invariant Hamiltonians 187
the corresponding classical Hamiltonian functional, is invariant with
respect to the transformations
φ′i(x) =∑
aijφj(x), π′i(x) =∑
aij πj(x),
where aij ∈ O(n) (here, O(n) is the group of orthogonal matrices).
However, in the operator realization of the Hamiltonian H, we have
a smaller symmetry group — the group O(n − 1). To check this
fact, we note that the operator realization of the Hamiltonian H
corresponds to a classical vacuum and, for every classical vacuum, the
transformations g ∈ O(n) that sends this vacuum to itself (satisfying
the condition gα = α) constitute a subgroup isomorphic to O(n−1).
Then, transformations that do not belong to this subgroup send an
operator realization to another that is not equivalent to the original
operator realization.
In this example, we obtain a symmetry breaking of a continuous
symmetry group. In this situation, we necessarily have particles with
energy tending to zero when the momentum tends to zero (Goldstone
particles). It is easy to verify this statement when one can apply
the semiclassical considerations of the present section. Indeed, in the
initial approximation, the energy of the particles is determined by the
eigenvalues of the matrix cij(k). This matrix continuously depends
on k, and therefore, it is sufficient to analyze the matrix cij(0) that
coincides with the matrix of second derivatives ∂2ν∂φi∂φj
of the function
ν at the minimum point. This matrix is necessarily degenerate,
since in our assumption the minimum cannot be achieved at one
isolated point. Hence, we conclude that in the zeroth approximation,
the Goldstone particles exist. It is easy to check that taking into
account the higher approximations of the perturbation theory with
respect to V does not change this statement. In Lorentz-invariant
theory, the energy of a particle with momentum p is equal to√p2 +m2, where m is the mass of the particle, and hence Goldstone
particles have zero mass.
We can conclude that the Hamiltonian (9.54) describes particles
with energy having the form ~ωj(~−1p) when ~ → 0. However,
aside from these particles, that can be called elementary, the
Hamiltonian (9.54) also describes the particles with energies having
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188 Mathematical Foundations of Quantum Field Theory
a finite limit as ~ → 0. These particles correspond to the so-
called “particle-like” solutions of classical equations. The simplest
particle-like solutions are solitons. A solution πi(x, t), φi(x, t) of the
Hamiltonian equations is called a soliton if the function φi(x, t)
can be represented in the form si(x − vt) (we assume that the
energy of this solution is finite). (Here, we assume that the energy is
measured as a deviation from the classical vacuum, i.e. H(π, φ) = 0
if πi(x) ≡ 0, φi(x) = αi, where (α1, . . . , αn) is a classical vacuum. If
the condition H(π, φ) = 0 is not satisfied, then we should replace the
Hamiltonian functional (9.54) with the functional
H(π, φ) =1
2
∑i
∫π2i (x)dx +
∑m
∑i1,...,im
∫Vi1,...,im(x1, . . . ,xm)
× (φi1(x1) . . . φim(xm)− αi1 . . . αim)dmx,
which leads to the same equations of motion.)
A soliton with a zero velocity (v = 0) is a solution of the equations
of motion that does not depend on time. Note that it follows
from translation-invariance of the Hamiltonian functional (9.53)
that the momentum P =∑
i
∫πi∂φi∂x dx is an integral of motion
of the Hamilton equations; the soliton with zero velocity has zero
momentum. In the Lorentz-invariant case, we can obtain solitons
with arbitrary momentum (and with aribtrary velocity that is less
than the speed of light) from the solitons of zero velocity by means
of Lorentz transformation.
One can check that there exists a quantum particle that corre-
sponds to a stable soliton (the solution φi(x, t), πi(x, t) is called stable
if every solution φ′i(x, t), π′i(x, t) that is close to the original at t = 0
stays close to it at any other moment in time t). More precisely,
if for every three-dimensional vector p, we have a stable soliton
of the Hamilton equations, having the momentum p and energy
ε(p), then under certain conditions, we can prove that among the
particles described by Hamiltonian (9.54) are particles with energy
tending to ε(p) as ~ → 0 (in other words, in the space of operator
realizations of the Hamiltonian (9.54), we have vector generalized
functions Φ~(p) that are δ-normalized and obey the conditions
HΦ~(p) = E~(p)Φ~(p),PΦ~(p) = pΦ~(p), where limE~(p) = ε(p);
March 27, 2020 9:30 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch09 page 189
The Scattering Matrix for Translation-Invariant Hamiltonians 189
for more, see Takhtadzhyan and Faddeev (1974), Faddeev (1975),
Tiupkin et al. (1975b), Polyakov (1974) and Tyupkin et al. (1975).
As an example, we can consider a classical system with the
Hamiltonian
H(π, φ) =1
2
∫π2(x)dx +
1
2
∫ (dφ
dx
)2
dx
+
∫(φ2(x)− a2)2dx,
where x ∈ E1. The Hamiltonian equations take the form
∂
∂tφ(x, t) = π(x, t),
∂
∂tφ(x, t) =
∂2
∂x2φ(x, t)− 4φ(x, t)(φ2(x, t)− a2).
If φ(x, t) = s(x), then the function s(x) satisfies the equation
d2s
dx2= 4s(s2 − a2). (9.58)
From the condition of finite energy, we can conclude that
limx→±∞ |s(x)| = a; using this relation and (9.58), we obtain, in
the case at hand, the solitons with zero velocity
s(x) = ±ath√
2ax,
they have energy µ = 4√
23 a3. The soliton with momentum p can be
obtained directly or by means of Lorentz transformation; the energy
of this soliton is equal to√
p2 + µ2. One can prove that there exist
quantum particles corresponding to these solitons.
Let us note in conclusion that one can use the semiclassical
approximation to analyze more general Hamiltonians (8.2). Namely,
we should introduce the complex functions ψ(x) = 1√2(φ(x) +
iπ(x)), ψ(x) = 1√2(φ(x)− iπ(x)) in place of the generalized momenta
π(x) and coordinates φ(x). An arbitrary Hamiltonian H(π, φ) can be
expressed in terms of ψ(x), ψ(x); let us assume that
H(ψ,ψ) =∑m,n
∫Hm,n(x1, . . . ,xm|y1, . . . ,yn)
× ψ(x1) . . . ψ(xm)ψ(y1) . . . ψ(yn)dmxdny.
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190 Mathematical Foundations of Quantum Field Theory
By the quantization of the functions ψ(x), ψ(x), we obtain the
operator generalized functions ψ+(x), ψ(x) obeying the commutation
relations
[ψ(x), (x′)] = [ψ+(x), ψ+(x′)] = 0,
[ψ(x), ψ+(x′)] = ~δ(x− x′),
and the Hamiltonian functionalH specifies the quantum Hamiltonian
H =∑m,n
∫Hm,n(x1, . . . ,xm|y1, . . . ,yn)
× ψ+(x1) . . . ψ+(xm)ψ(y1) . . . ψ(yn)dmxdny
(in the transition from the Hamiltonian functional H to the quantum
Hamiltonian H, we should answer the question about the ordering
of the operators ψ+, ψ in the expression of H; the choice we have
made corresponds to the so-called “Wick quantization”). If the
Hamiltonian functional H is translation-invariant, then using the
formulas
ψ+(x) = (2π)−3/2~1/2
∫exp(ikx)a+(k)dk,
ψ(x) = (2π)−3/2~1/2
∫exp(−ikx)a(k)dk
we can write down the Hamiltonian (9.35) in the form (8.2). This
allows us to apply the considerations of this section to the analysis
of the Hamiltonian (8.2).
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Chapter 10
Axiomatic Scattering Theory
10.1 Main assumptions and the construction of the
scattering matrix
Let us suppose that we have four commuting self-adjoint operators
H and P = (P1, P2, P3) defined on the Hilbert space H. The
operator H has the physical meaning of an energy operator and the
(vector) operator P has the meaning of the momentum operator.
The operator exp(iPa) is called the spatial shift operator or spatial
translation and the operator exp(−iHt) is the time shift operator
(time translation). Let us assume that the energy operator has the
unique ground state Φ and the state is invariant with respect to
space shifts and time shifts: HΦ = PΦ = 0. The vector Φ is called
the physical vacuum.
In the situation at hand, we can apply the definition of a particle
given in Section 5.3. We will formulate this definition in a slightly
different way.
Let us suppose that for every function f ∈ L2(E3) we have
a corresponding vector Φ(f) ∈ H that linearly depends on the
function f . This relationship can be understood as a linear operator
mapping from L2(E3) to H; however, we will prefer to view it as
a vector generalized function Φ(k), where the vector Φ(f) can be
written in the form Φ(f) =∫f(k)Φ(k)dk. The vector generalized
function Φ(k) is called a particle (a single-particle state), if for every
191
March 27, 2020 15:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch10 page 192
192 Mathematical Foundations of Quantum Field Theory
pair of functions f, g ∈ L2(E3) we have the relations
HΦ(f) = Φ(hf),
PΦ(f) = Φ(pf),
〈Φ(f),Φ(g)〉 = 〈f, g〉 ,
where
(hf)(k) = ω(k)f(k),
(pf)(k) = kf(k).
The vector Φ(f) represents the state of a particle with the wave
function f and the function ω(k) has the meaning of the energy
of the single-particle state (dispersion law). The relations above
are equivalent to the relations (5.20)–(5.22). The last one implies
that the operator taking the function f to the vector Φ(f) is an
isometry; this is equivalent to the relation (5.22), which shows
that Φ(k) is δ-function normalized. It is possible that there are
several types of particles (several types of generalized functions Φ(k))
that satisfy (5.20)–(5.22). A system of particles Φ1(k), . . . ,Φs(k)
is called complete, if every pair of particles is orthogonal (i.e.
〈Φi(k),Φj(k′)〉 = 0 for i 6= j) and there does not exist another
particle that is orthogonal to all the particles in the system.
One can show that every particle Φ(k) can be decomposed into
the particles of a complete system (i.e. we can find functions fi(k) so
that Φ(k) =∑s
i=1 fi(k)Φi(k)).
Let us fix a complete particle system Φ1(k), . . . ,Φs(k) with the
dispersion laws ω1(k), . . . , ωs(k), assuming that the functions ωi(k)
are strongly convex. (We suppose for definiteness that we have a finite
number s of particles, even though the case of an infinite number of
particles is not much more difficult.) The single-particle subspace H1
of the space H will be defined as the smallest subspace that contains
all vectors of the form Φi(f) =∫f(k)Φi(k)dk. The multi-particle
subspace M will be defined as the orthogonal complement of the
direct sum H0 +H1 in H, where H0 is the one-dimensional subspace
generated by the physical vacuum Φ.
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Axiomatic Scattering Theory 193
The spaces H0,H1,M are clearly invariant with respect to the
operators H,P. The joint spectrum of the system of commuting
operators H,P in the space H1 (in the space M) will be called the
single-particle (correspondingly, the multi-particle) spectrum and we
will denote it by∑
1 (correspondingly,∑M). The set
∑1 clearly
coincides with the union of the sets∑(i)
1 , where∑(i)
1 denotes the set
of points of the form (k, ωi(k)).
Let us assume that the single-particle spectrum does not intersect
the multi-particle spectrum and that there exists δ > 0, such that
the spectrum of the operator H, with the exception of the point 0,
corresponding to the physical vacuum, is contained in the ray [δ,+∞).
We will later show how this assumption can be relaxed.
Let us consider, along with the spaceH, the asymptotic state space
Has, which we will define as the Fock space F (L2(E3×N)), where N
is a finite set whose elements are in one-to-one correspondence with
the complete system of particles we have assumed previously. Let us
suppose that the operator generalized functions a+i (k), ai(k) obey
CCR (see Section 3.2) and act on the space Has; these functions
can be considered as the creation and annihilation operators of
particles of the ith type with momentum k ∈ E3. We will define
the asymptotic Hamiltonian Has and the momentum operator Pas
by the formulas
Has =s∑i=1
∫ωi(k)a+
i (k)ai(k)dk,
Pas =
s∑i=1
∫ka+
i (k)ai(k)dk.
If A is an operator on the space H, then A(x, t) will denote the
operator exp(i(Ht − Px))A exp(−i(Ht − Px)). Bounded operators
A and B, acting on the space H, will be said to be asymptotically
commuting if for every natural number n we can find real numbers
C and r such that
‖[A,B(x, t)]‖ ≤ C 1 + |t|r
1 + |x|n.
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194 Mathematical Foundations of Quantum Field Theory
A family of bounded operators A on the space H that contains
the conjugate operator A∗ for every operator A in A will be
called asymptotically Abelian if two arbitrary operators in A are
asymptotically commuting.
An asymptotically Abelian family that contains the identity
operator will be called an asymptotically Abelian algebra (or asymp-
totically commutative algebra) if for every pair of operators A, B in
M, the operators
λA+ µB, AB,
∫f(x, t)A(x, t)dxdt
(here, λ, µ are complex numbers, x ∈ E3, −∞ < t <∞, and f(x, t)
is a function in the space S of smooth, fast-decaying functions) are
in M as well. It is easy to show that every asymptotically Abelian
family is contained in an asymptotically Abelian algebra. (The proof
is based on the fact that adding any of the listed operators along with
their adjoints to an asymptotically Abelian family preserves asymp-
totic commutativity.) Therefore, without loss of generality, we will
consider asymptotically Abelian algebras instead of asymptotically
Abelian families.
Let us fix an asymptotically Abelian algebra A of operators on the
space H; let us assume that the physical vacuum Φ is a cyclic vector
of the algebra A.
With the definitions and assumptions above, we will now show
how we can construct a scattering matrix (S-matrix) corresponding
to an asymptotically Abelian algebra A (we can then define a scat-
tering matrix of an asymptotically Abelian family as the scattering
matrix corresponding to the asymptotically Abelian algebra that
contains this family). For the sake of simplicity of definition, we will
assume that the full particle system consists of a single particle Φ(k).
Let us first make several preliminary definitions.
An operatorB is called smooth if it can be written in the formB =∫f(x, t)A(x, t)dxdt, where A ∈ A and the function f belongs to the
space S. (This name comes from the observation that the operator
B(x, t), where B is a smooth operator, is infinitely differentiable in x
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Axiomatic Scattering Theory 195
and t in the sense of differentiation in norm.) The smooth operator
B is called good, if (1) B∗Φ = 0; (2) there exists a function φ such
that BΦ =∫φ(k)Φ(k)dk.
Definition 10.1. An isometric operator S− (S+), transforming the
space Has into the space H, is called a Møller matrix if for every
collection of good operators B1, . . . , Bn and for every collection of
smooth functions with compact support f1(p), . . . , fn(p)
limt→ −∞(+∞)
B1(f1, t) · · ·Bn(fn, t)Φ = S −(+)a+(f1φ1) · · · a+(fnφn)θ.
(10.1)
[Here, θ is a vacuum vector in the Fock space Has, φj(k) =
〈BjΦ,Φ(k)〉, the functions fj(x|t) are determined by the relation
fj(x|t) =
∫exp(−iω(p)t+ ipx)fj(p)
dp
(2π)3, (10.2)
and the operators Bj(fj , t) are given by the formula
Bj(fj , t) =
∫fj(x|t)Bj(x, t)dx.
](10.3)
The vector B1(f1, t) · · ·Bn(fn, t)Φ, constructed with good oper-
ators B1, . . . , Bn and the smooth functions with compact support
f1, . . . , fn, will be denoted by Ψ(B1, . . . , Bn|f1, . . . , fn|t). The limit of
this vector as t→ ±∞ will be denoted by Ψ±(B1, . . . , Bn|f1, . . . , fn).
Definition 10.2. The operator S = S∗+S−, acting on the space Has,
is called the scattering matrix, corresponding to the asymptotically
Abelian algebra A.
Let us prove the following statement.
With the previous assumptions, the operators S− and S+ satisfy-
ing the conditions of Definition 10.1, exist, and are uniquely specified
March 27, 2020 15:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch10 page 196
196 Mathematical Foundations of Quantum Field Theory
by these conditions. The following relations also hold :
S∓θ = Φ, S±a+(k)θ = Φ(k), (10.4)
HS∓ = S∓Has, PS∓ = S∓Pas. (10.5)
It follows from relation (10.5) that the scattering matrix S =
S∗+S− commutes with the Hamiltonian and the momentum operator
SHas = HasS, SPas = PasS,
and the relation (10.4) implies the stability of the vacuum and single-
particle states:
Sθ = θ,
Sa+(k)θ = a+(k)θ.
If the images of the operators S− and S+ coincide (S−Has = S+Has),
then the scattering matrix is unitary and can be written in the form
S = S−1+ S−.
If, in addition to the algebra A, we have a second asymptotically
Abelian algebra A′, for which the vector Φ is cyclic and the algebras
A and A′ asymptotically commute,1 then the Møller matrices S∓and the scattering matrix S, constructed with the algebra A, coincide
with the Møller matrix S′∓ and the scattering matrix S′, constructed
with the algebra A′.The proof of the listed statements is based on a sequence of
lemmas. Let us now formulate the necessary lemmas and then
prove the statements. The proofs of the lemmas are relegated to
Section 10.2.
Lemma 10.1. The set of vectors of the form BΦ, where B runs over
all good operators in the algebra A, is dense in the single-particle
space H1.
1The algebras A and A′ asymptotically commute if every operator in Aasymptotically commutes with every operator in A′.
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Axiomatic Scattering Theory 197
Lemma 10.2. Let ω(p) be a smooth, strongly convex function, f(p)
be a smooth function with compact support, and
f(x|t) =
∫exp(−iω(p)t+ ipx)f(p)
dp
(2π)3, (10.6)
then
supx|f(x|t)| ≤ C1|t|−3/2, (10.7)∫|f(x|t)|dx ≤ C2(|t|3/2 + 1). (10.8)
If U is a set of vectors of the form v(p) = ∇ω(p), where p ∈ suppf
and Uε is an ε-neighborhood of the set U , then for any n and ε, we
can find a constant D such that
|f(x|t)| ≤ D(1 + x2 + t2)−n (10.9)
whenever xt 6∈ Uε.
[The symbol supp f , as usual, denotes the closure of the set of
points where f(p) 6= 0.]
Before we formulate Lemma 10.3, we need to introduce the
concept of a truncated vacuum expectation value.
Let us define the vacuum expectation value 〈A〉 of the operator A
as the number 〈AΦ,Φ〉. The truncated vacuum expectation value
〈A1 . . . An〉T of the product of n operators A1, . . . , An is defined by
the following formulas for n = 1, 2, 3:
〈A1〉T = 〈A1〉,
〈A1A2〉T = 〈A1A2〉 − 〈A1〉T 〈A2〉T = 〈A1A2〉 − 〈A1〉 〈A2〉,
〈A1A2A3〉T = 〈A1A2A3〉 − 〈A1A2〉T 〈A3〉T − 〈A1〉T 〈A2A3〉T
− 〈A2〉T 〈A1A3〉T − 〈A1〉T 〈A2〉T 〈A3〉T .
For arbitrary n, the truncated vacuum expectation value is defined by
the recurrence relation
〈A1 . . . An〉 =
n∑k=1
∑ρ∈Rk
〈A(π1)〉T · · · 〈A(πk)〉T , (10.10)
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198 Mathematical Foundations of Quantum Field Theory
where Rk is the collection of all partitions of the set 1, . . . , n into
k subsets; π1, . . . , πk are the subsets that constitute the partition
ρ ∈ Rk; 〈A(π)〉T is a truncated vacuum expectation value of the
product of the operators Ai with indices in the set π (the operators
are ordered in ascending order by the index i).
Let us suppose that for every λ in the (finite or infinite) set Λ we
have a corresponding operator Aλ from the algebra A.
The expectation value
wn(λ1,x1, t1, . . . , λn,xn, tn) = 〈Aλ1(x1, t1) · · ·Aλn(xn, tn)〉
of the product Aλ1(x1, t1) · · ·Aλn(xn, tn) in the ground state Φ will
be called the (n-point) Wightman function (see Section 7.1). Let us
define the truncated Wightman function wTn (λ1,x1, t1, . . . , λn,xn, tn)
as the truncated vacuum expectation value of this product:
wTn (λ1,x1, t1, . . . , λn,xn, tn) = 〈Aλ1(x1, t1) · · ·Aλn(xn, tn)〉T .
In other words,
wn(λ1,x1, t1, . . . , λn,xn, tn) =
n∑k=1
∑ρ∈Rk
wTα1(π1) · · ·wTαk(πk).
(10.11)
Here, αr denotes the number of elements in the subset πr; wTαr(πr) is
a truncated Wightman function with the arguments λi1 , xi1 , ti1 , . . . ,
λiαr ,xiαr , tiαr , where i1, . . . , iαr ∈ πr (the order of the arguments is in
order of increasing indices i). For n = 1, 2, 3, the relations connecting
Wightman functions and truncated Wightman functions have the
form
w1(λ1,x1, t1) = wT1 (λ1,x1, t1),
w2(λ1,x1, t1, λ2,x2, t2) = wT2 (λ1,x1, t1, λ2,x2, t2)
+wT1 (λ1,x1, t1)wT1 (λ2,x2, t2),
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Axiomatic Scattering Theory 199
w3(λ1,x1, t1, λ2,x2, t2, λ3,x3, t3)
= wT3 (λ1,x1, t1, λ2,x2, t2, λ3,x3, t3)
+wT1 (λ1,x1, t1)wT2 (λ2,x2, t2, λ3,x3, t3)
+wT1 (λ2,x2, t2)wT2 (λ1,x1, t1, λ3,x3, t3)
+wT1 (λ3,x3, t3)wT2 (λ1,x1, t1, λ2,x2, t2)
+wT1 (λ1,x1, t1)wT1 (λ2,x2, t2)w
T1 (λ3,x3, t3).
Lemma 10.3. If all the operators Aλi are smooth, then the truncated
Wightman function
wTn (λ1,x1, t1, . . . , λn,xn, tn) = 〈Aλ1(x1, t1) . . . Aλn(xn, tn)〉T ,
for fixed t1, . . . , tn, tends to zero faster than any power function of
D = max1≤i,j≤n |xi − xj | as D →∞.
Lemma 10.4. Let f0, . . . , fn be finite smooth functions and let
A0, . . . , An be smooth operators belonging to the algebra A. Then for
n ≥ 1, we have the inequality
| 〈A0(f0, t) . . . An(fn, t)〉T | ≤ C(1 + |t|)−32
(n−1), (10.12)
where Aj(fj , t) =∫fj(x|t)Aj(x|t)dx. This inequality remains valid if
in the expression for 〈A0(f0, t) . . . An(fn, t)〉T some of the operators
Aj(fj , t) are replaced by their time derivatives Aj(fj , t) and the
operators (Aj(fj , t))∗ are replaced by (Aj(fj , t))
∗.
Let us now turn to the derivation of the statements made in the
beginning of this chapter from the lemmas we have formulated. First,
let us derive the relations (10.4). The first of these is self-evident; to
prove the second relation, let us note that
Ψ(B|f |t) = B(f, t)Φ
=
∫f(x|t)B(x, t)Φdx
=
∫f(x|t) exp(itH − iPx)BΦdx
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200 Mathematical Foundations of Quantum Field Theory
=
∫f(x|t) exp(itω(p)− ipx)φ(p)Φ(p)dxdp
=
∫f(p)φ(p)Φ(p)dp, (10.13)
i.e. Ψ(B|f |t) does not depend on t:
d
dtΨ(B|f |t) =
d
dtB(f, t)Φ = 0. (10.14)
This implies that
Ψ±(B, f) =
∫f(p)φ(p)Φ(p)dp,
i.e.
S∓(
∫f(p)φ(p)dpθ) =
∫f(p)φ(p)Φ(p)dp. (10.15)
It follows from Lemma 10.1 that vectors of the form
BΦ =
∫φB(p)Φ(p)dp,
where B runs over a set of good operators, constitute a dense set
in the space of single-particle states. This implies that the functions
φB(p), corresponding to good operators B, constitute a dense set
in L2(E3), since the correspondence between the function φ(p) and
the vector∫φ(p)Φ(p)dp is an isometry between the space L2(E3)
and the space of single-particle states. It is clear that the functions
f(p)φB(p), where f(p) is a smooth function, also form a dense set
in L2(E3). The necessary relation then follows from (10.15).
Let us now show that the limit in the relation (10.1) exists. To
show this, we will prove the inequality⟨dΨ
dt,dΨ
dt
⟩≤ C|t|−3, (10.16)
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Axiomatic Scattering Theory 201
where Ψ(t) = Ψ(B1, . . . , Bn|f1, . . . , fn|t). This inequality implies the
convergence of the limit in (10.1) since
‖Ψ(t2)−Ψ(t1)‖ ≤∫ t2
t1
∥∥∥∥dΨ
dt
∥∥∥∥ dt ≤ ∫ t2
t1
C|t|−3/2dt→ 0,
for t1 ≤ t2 and t1 → +∞ or t2 → −∞.
In order to prove the inequality, let us write the expansion⟨dΨ
dt,dΨ
dt
⟩=∑i,j
〈B1(f1, t) . . . Bi(fi, t)
. . . Bn(fn, t)Φ, B1(f1, t) . . . Bj(fj , t) . . . Bn(fn, t)Φ〉
=∑i,j
〈(Bn(fn, t))∗ . . . (Bj(fj , t))
∗ . . . (B1(f1, t))∗
×B1(f1, t) . . . Bi(fi, t) . . . Bn(fn, t)Φ,Φ〉
as a sum of the products of truncated vacuum expectation values
using the relation (10.10).
Every factor in an arbitrary term of this expansion has the form
Ik,l(t) = 〈(Bi1(fi1 , t))∗ . . . (Bik(fik , t))
∗
× Bj1(fj1 , t) . . . Bjl(fjl , t)〉T ,
where the dots above the operators denote differentiation with
respect to time, which can enter in one of the first k operators and in
one of the last l operators. We can suppose without loss of generality
that k ≥ 1, l ≥ 1 (otherwise, Ik,l = 0 by the relation B∗Φ = 0). If
each factor in the given term has k = l = 1, then the term is equal
to zero because one of the factors either has the form
〈(Bi(fi, t))∗Bj(fj , t)〉T = 〈(Bi(fi, t))∗Bj(fj , t)Φ,Φ〉
or the form
〈(Bi(fi, t))∗Bj(fj , t)〉T = 〈(Bi(fi, t))∗Bj(fj , t)Φ,Φ〉
and both expressions are zero by the relation (10.14). Hence, all
non-zero terms in the sum above contain either a factor Ik,l with
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202 Mathematical Foundations of Quantum Field Theory
k + l ≥ 4 or at least two factors Ik,l with k + l = 3 (a term where
all factors except one have k + l = 2 and the remaining factor has
k+ l = 3 is not possible since the total number of operators is even).
The inequality
|Ik,l(t)| < C|t|−32
(k+l−2),
follows from Lemma 10.4. In particular, we have
|I1,1(t)| < C,
|Ik,l(t)| < Ct−3/2,
for k + l = 3, and
|Ik,l(t)| < Ct−3
for k + l ≥ 4. From these inequalities and the statements shown
above, we obtain the inequality (10.16) for⟨dΨdt ,
dΨdt
⟩and therefore
the existence of the limit in (10.1) follows.
Since the limit in (10.1) exists, the relation (10.1) specifies the
operators S∓ on some subspace L of the space Has. However, it is
not clear if the action of the operators S∓ is uniquely defined, i.e.
whether the vector S∓ξ = Ψ∓(B1, . . . , Bn|f1, . . . , fn) remains the
same if we write ξ in two different forms
ξ = a+(φ1f1) . . . a+(φnfn)θ,
where the functions fi are smooth and finite and the functions φicorrespond to good operators.
Therefore, let us now prove that the relation (10.1) defines S∓uniquely and, furthermore, this operator can be uniquely extended
to a linear isometric operator defined on the whole space Has. To
show this, we will use the following general statement.
Let us suppose that on the total2 subset L of the Hilbert space
H, we have a multi-valued isometric mapping α taking values in H′(i.e. to every point ξ ∈ L corresponds a set of points α(ξ) ⊂ H′ in
such a way that for any selection of points ξ1, ξ2 ∈ L, x1, x2 ∈ H′satisfying x1 ∈ α(ξ1), x2 ∈ α(ξ2) we have 〈x1, x2〉 = 〈ξ1, ξ2〉). Then
2A subset is called total if the set of its linear combinations is dense in H.
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Axiomatic Scattering Theory 203
the mapping α is in fact single-valued (i.e. every set α(ξ) consists of
a single point) and can be uniquely extended to a linear isometric
operator defined on the whole space H.
To prove this statement, let us first extend the mapping α by
linearity to the multi-valued mapping α, defined on the set L of linear
combinations of points in the set L. More precisely, we will assume
that x ∈ α(ξ) if we can write ξ and x in the forms ξ =∑n
i=1 λiξi,
x =∑n
i=1 λixi with xi ∈ α(ξi). It is easy to see that the mapping
α inherits linearity: if ξ = λ1ξ1 + λ2ξ2, x1 ∈ α(ξ1), x2 ∈ α(ξ2), then
λ1x1 + λ2x2 ∈ α(ξ).
The mapping α is also a multi-valued isometric mapping (if x(1) ∈α(ξ(1)), x(2) ∈ α(ξ(2)) then ξ(1) =
∑λiξ
(1)i , x(1) =
∑λix
(1)i , ξ(2) =∑
µjξ(2)j , x(2) =
∑µjx
(2)j , where x
(1)i = α(ξ
(1)i ), x
(2)j = α(ξ
(2)j ), and
therefore⟨x(1), x(2)
⟩=∑
λiµj
⟨x
(1)i , x
(2)j
⟩=∑
λiµj
⟨ξ
(1)i , ξ
(2)j
⟩=⟨ξ(1), ξ(2)
⟩).
Let us now prove that the mapping α is in fact single-valued. Let
us suppose that x1 ∈ α(ξ), x2 ∈ α(ξ). It follows from the isometry of
the mapping that
〈x1, x1〉 = 〈x2, x2〉 = 〈x1, x2〉 = 〈ξ, ξ〉 .
From this equation, we obtain that x1 = x2.
Therefore, α is a well-defined isometric linear mapping defined on
the set L. Since the set L is total and the set L is dense in H, we can
continuously extend α to an isometric linear operator defined on all
of H.
To apply the statement above to the proof of the Møller matrix
properties, we should check that the set L of vectors of the form
a+(φ1f1) . . . a+(φnfn)θ, on which the operators S∓ are defined
by (10.1), is total in the space Has, and furthermore, we should
check that for good operators B1, . . . , Bn, B′1, . . . , B
′m and for smooth
functions with compact support f1, . . . , fn, f′1, . . . , f
′m we have⟨
Ψ∓(B1, . . . , Bn|f1, . . . , fn),Ψ∓(B′1, . . . , B′m|f ′1, . . . , f ′m)
⟩=⟨a+(φ1f1) · · · a+(φnfn)θ, a+(φ
′1f′1) · · · a+(φ
′nf′n)θ⟩. (10.17)
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204 Mathematical Foundations of Quantum Field Theory
The totality of the set L follows from the fact that functions of
the form φB(k)f(k), where f(k) is a smooth function with compact
support and B is a good operator, are dense in L2(E3) (as we have
noted above, this fact follows from Lemma 10.1).
To prove relation (10.17), let us represent the left side of this
relation in the form⟨Ψ∓(B1, . . . , Bn|f1, . . . , fn),Ψ∓(B′1, . . . , B
′m|f ′1, . . . , f ′m)
⟩= lim
t→∓∞
⟨Ψ(B1, . . . , Bn|f1, . . . , fn|t),Ψ(B′1, . . . , B
′m|f ′1, . . . , f ′m|t)
⟩= lim
t→∓∞
⟨(B′m(f ′m, t))
∗ · · · (B′1(f ′1, t))∗B1(f1, t) · · ·Bn(fn, t)
⟩(10.18)
and then expand the expression under the limit sign in terms
of truncated Wightman functions. Every term of the resulting
expansion consists of a product of factors of the form
Rk,l(t) =⟨(B′i1(fi1 , t))
∗ · · · (B′ik(fik , t))∗Bj1(fj1 , t) · · ·Bjl(fjl , t)
⟩T.
It follows from Lemma 10.4 that for k + l ≥ 3 the factors Rk,l(t)
converge to zero as t→ ±∞. On the other hand, the factors Rk,l are
equal to zero if k = 0 or l = 0, by the relation BΦ = 0. Therefore,
the terms that differ from zero in the limit t → ∓∞ are the ones
where all the factors have the form R1,1(t). It is easy to check that
R1,1(t) =⟨(B′i(f
′i , t))
∗Bj(fj , t)⟩T
=⟨(B′i(f
′i , t)
∗Bj(fj , t))⟩
=⟨Bj(fj , t)Φ, B
′i(f′i , t)Φ
⟩=
⟨∫fj(k)φj(k)Φ(k)dk,
∫f ′i(k)φ′i(k)Φ(k)dk
⟩=⟨fjφj , f
′iφ′i
⟩=⟨a+(f jφj)θ, a
+(f′iφ′i)θ⟩
and that R1,1 in fact does not depend on t [in the calculations we
have used the formula (10.13)].
The statements proven above imply that the expression (10.18)
is equal to δnm∑∏n
s=1 〈φsfs, φ′isf ′is〉 [the sum is taken over all
permutations (i1, . . . , in)]. This justifies equation (10.17).
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Axiomatic Scattering Theory 205
Thus, we have proven that relation (10.1) specifies S∓ as an
isometric mapping on a total set and we have shown that one can
uniquely extend this mapping to an isometric operator defined on
the space Has.
To prove relation (10.5), note that the equation
exp(−iHτ)B(f, t) exp(iHτ)
=
∫exp(−iHτ)f(x|t)B(x, t) exp(iHτ)dx
=
∫f(x|t)B(x, t− τ)dx
=
∫ (∫exp(−iω(p)t+ ipx)f(p)
dp
(2π)3
)B(x, t− τ)dx
=
∫ (∫exp(−iω(p)(t− τ) + ipx)f(p) exp(−iω(p)τ)
dp
(2π)3
)×B(x, t− τ)dx,
holds; this implies
exp(−iHτ)B(f, t) exp(iHτ) = B(f τ , t− τ), (10.19)
[f τ denotes the function f τ (p) = f(p) exp(−iω(p)τ)]. It follows from
equation (10.19) that
exp(−iHτ)Ψ(B1, . . . , Bn|f1, . . . , fn|t)
= Ψ(B1, . . . , Bn|f τ1 , . . . , f τn |t− τ). (10.20)
Taking the limit t→ ∓∞ in (10.20) and using the fact that
exp(−iHasτ)a+(φ1f1) . . . a+(φnfn)θ
= a+(φ1fτ1) . . . a+(φnf
τn)θ,
we obtain
exp(−iHτ)S∓ξ = S∓ exp(−iHasτ)ξ
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206 Mathematical Foundations of Quantum Field Theory
for any vector ξ ∈ L. By the totality of the set L, we have
exp(−iHτ)S∓ = S∓ exp(−iHasτ)
and, therefore, the first relation in (10.5) follows (the second can be
proven by the same method, but is even easier).
To prove the final statement, let us note that Møller matrices
do not change when we replace an asymptotically Abelian algebra
by a larger one. Indeed, if the algebra A, satisfying the necessary
conditions, is contained in an asymptotically Abelian algebra A, then
in the construction of the operators S∓ corresponding to the algebra
A, we can use the good operators that belong to the algebraA; hence,
it is clear that Møller matrices corresponding to the algebrasA and Aare identical. If the algebras A and A′ are asymptotically commuting,
then the family A∪A′ is asymptotically Abelian and can be included
in an asymptotically Abelian algebra A. Since A ⊂ A and A′ ⊂ A,
the Møller matrices S∓ and S′∓, constructed with the algebras A and
A′, coincide with the Møller matrices constructed with the algebra
A, and therefore coincide with each other. The coincidence of Møller
matrices clearly entails the coincidence of the scattering matrices
constructed with the algebras A and A′.In conclusion, let us note that the Møller matrices and the
scattering matrix do not depend on the choice of complete particle
system (recall that there exists an isomorphism between any two
choices of complete particle system for the space Has; therefore,
it makes sense to discuss the coincidence of scattering matrices
constructed under different particle systems). To prove this, note
that for any particle system with a single particle Φ(k), any other
particle can be written in the form
Φ′(k) = exp(iα(k))Φ(k),
where α(k) is a real-valued function.
The operators a+(k), a(k) in the space Has, corresponding to
the particle Φ(k), are related to the operators a′+(k), a′(k) on Has,
corresponding to Φ′(k), by the equations
a′(k) = exp(−iα(k))a(k),
a′+(k) = exp(iα(k))a+(k).
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Axiomatic Scattering Theory 207
It is easy to check, using these relations, that for any operator Bj
a′+(f jφ′j) = a+(f jφj), (10.21)
where
φ′j(k) =⟨BjΦ,Φ
′(k)⟩, φj(k) = 〈BjΦ,Φ(k)〉 .
Applying (10.21) and noting that the left part of (10.1) does not
depend on the choice of the function Φ(k) representing the particle,
we obtain that the Møller matrices S± do not depend on the choice
of the function Φ(k).
10.2 Proof of lemmas
Proof of Lemma 10.1. Let λ(ω,p) be a function of four parame-
ters with support in the set ∆. We will consider the operator λ(H,P)
as a function of the commuting self-adjoint operators H,P. It is easy
to check that if the set ∆ does not intersect the spectrum of the
operators (H,P), then the operator λ(H,P) = 0, and if the set ∆
does not contain the origin and points belonging to the multi-particle
spectrum∑M, then the set of values of the operator λ(H,P) belongs
to the single-particle subspace.
Using this remark, we can construct a good operator by the
following method.
Let us consider a function α(t,x) ∈ S(E4) with the property that
its Fourier transform
α(ω,p) =
∫α(t,x) exp(i(ωt− px))dxdt
has support that does not intersect the multi-particle spectrum and
the half-space ω ≤ 0. Then the operator
B =
∫α(t,x)A(x, t)dxdt,
where A ∈ A, is good.
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208 Mathematical Foundations of Quantum Field Theory
Indeed, the operator B is smooth. The vector
BΦ =
∫α(t,x)A(x, t)Φdxdt
=
∫α(t,x) exp(iHt− iPx)A exp(−iHt+ iPx)Φdxdt
=
∫α(t,x) exp(i(Ht−Px))AΦdxdt = α(H,P)AΦ
belongs to the single-particle subspace, by the remark above. The
vector
B∗Φ =
∫α(t, x) exp(i(Ht−Px))A∗Φdxdt
= α(−H,−P)A∗Φ = 0,
since the support of the function α(−ω,−p) does not intersect the
spectrum of the operators H, P.
Let us now show that vectors BΦ, where B is a good operator,
constructed by the method above, are dense in the single-particle
subspace.
To prove this, let us consider the vector Φ(λ) =∫λ(k)Φ(k)dk,
where λ(k) is a smooth function with compact support, in H1.
We will construct a sequence of good operators Bn such that
BnΦ → Φ(λ) (this is enough to prove the statement above since
smooth functions with compact support λ(k) are dense in L2(E3)
and therefore the corresponding vectors Φ(λ) =∫λ(k)Φ(k)dk are
dense in H1).
By the cyclicity of the vacuum vector there exists a sequence of
operators An ∈ A so that AnΦ → Φ(λ). The necessary sequence of
good operators can now be constructed by setting Bn =∫α(t,x)
An(x, t)dxdt, where
α(t,x) =
∫α(ω,p) exp[−i(ωt− px)]
dωdp
(2π)4,
the properties of the supports of the functions α ensure that the
operators Bn are good, and the function α satisfies the condition
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Axiomatic Scattering Theory 209
α(ω(k),k) = 1 for all k, belonging to the support of the function
λ(k). Indeed, we have
limn→∞
BnΦ = limn→∞
α(H,P)AnΦ
= α(H,P)Φ(λ)
=
∫λ(k)α(H,P)Φ(k)dk
=
∫λ(k)α(ω(k),k)Φ(k)dk
=
∫λ(k)Φ(k)dk = Φ(λ).
This completes the proof of Lemma 10.1.
Before we prove Lemma 10.2, we will conduct some non-rigorous
but intuitive discussion. The integral (10.6) for large t can be
approximated by the method of stationary phase. The stationary
point p0 of the phase ω(p)t − px is determined by the equation
v(p0)t = x; denoting by w(ξ) the solution of the equation v(p) = ξ,
we can write the stationary point in the form p0 = w(xt ). Applying
the method of stationary phase to f(x|t) for large t, we obtain
f(x|t) ≈ Ct−3/2f(w(x
t
))exp
(−iω
(w(x
t
))t+ iw
(x
t
)),
where
C = (2πi)−3/2| det γik|−1/2,
γik =∂2
∂pi∂pkω(p)|p=w(x
t).
Hence, f(x|t), for large t, decays as t−3/2. If xt 6∈ Uε, then
f(w(xt )) = 0 and the stationary point is not contained in the support
of f(p); in this case, we should expect the function f(x|t) to be
very small. Unfortunately, the simple considerations here cannot be
applied to estimate the function f(x|t) uniformly in x; we will provide
accurate but more cumbersome proofs for these inequalities.
Proof of Lemma 10.2. Let us first estimate f(x|t) when xt 6∈ Uε.
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210 Mathematical Foundations of Quantum Field Theory
We will use the formula∫exp(−iσ(p))χ(p)dp =
∫exp(−iσ(p))(Lnχ)(p)dp, (10.22)
where χ(p) is a smooth function with compact support, (Lχ)(p) =
−div(u(p)χ(p)); u(p) = ∇σ(p)|∇σ(p)|2 (see, for example, Fedoryuk (1971)).
In the case at hand, we have
σ(p) = ω(p)t− px, χ(p) = f(p),
u(p) =v(p)t− x|v(p)t− x|2
=1
t·
v(p)− xt
|v(p)− xt |2
.
If xt 6∈ Uε, it is easy to check that
supp∈suppχ
|D(α)u(p)| ≤ C
|t|
(here, D(α) = ∂|α|
∂pα11 p
α22 p
α33
, |α| = α1 + α2 + α3, and C, here and in
the following, denotes a quantity independent of x and t, though it
may depend on other parameters, for example, here it depends on α
and ε). The inequality
supp∈suppχ
|(Lnχ)(p)| ≤ C|t|−n (10.23)
follows. The formula (10.22) and the inequality (10.23) lead to
the inequality |f(x|t)| ≤ C1+|t|n , where x
t 6∈ Uε (n is an arbitrary
number).3 To prove the inequality (10.9), we should consider the
cases |x| ≤ at and |x| > at, where a = 2 supp∈supp f |v(p)|, separately.
In the first case, the inequality (10.9) follows from (10.23); in the
second case, we can use the fact that
supp∈supp f|x|>at
|D(α)u(p)| ≤ C
|x|.
3In the proof, we have assumed that the function χ(p) is smooth. One can relaxthis condition assuming that this function is m times continuously differentiable.Then we have the inequality (10.23) for n = m.
March 27, 2020 15:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch10 page 211
Axiomatic Scattering Theory 211
To prove inequality (10.7), let us take a smooth function µ(x)
equal to zero for |x| ≤ ν1 and equal to one for |x| ≥ ν2.
Let us split the integral representing the function f(x|t) in two
as follows:
I1 =
∫exp(−iω(p)t+ ipx)µ
(tρ(v(p)− x
t
))f(p)dp; (10.24)
I2 =
∫exp(−iω(p)t+ ipx)
(1− µ
(tρ(v(p)− x
t
)))f(p)dp,
(10.25)
where the number ρ is in the interval 512 < ρ < 1
2 . The integral I1 can
be estimated with the formula (10.22). Let us introduce the notation
‖φ‖n = supp∈Γ|α|≤n
|D(α)φ (p)|,
where Γ denotes the set of points p such that |v(p) − xt | ≥ ν1|t|−ρ
(outside of Γ the integrand in the integral I1 is zero). It is easy to
check that ‖u‖n ≤ C|t|−1+(n+1)ρ.
Furthermore, we have
‖Lχ‖n ≤ C‖uχ‖n+1 ≤ C∑
α+β=n+1
‖u‖α‖χ‖β,
which implies that
‖Lrχ‖n ≤ C∑
α1+···+αr+β=n+r
‖u‖α1 . . . ‖u‖αr‖χ‖β
≤ C∑β
|t|−r+(n+2r−β)ρ‖χ‖β.
To estimate the integral I1, we should introduce
χ(p) = µ(tρ(v(p)− x
t
))into formula (10.22). Then, ‖χ‖β ≤ C|t|βρ implies
|I1| ≤ C‖Lrχ‖0 ≤∑β
C|t|−r+(2r−β)ρ‖χ‖β ≤ C|t|−r+2rρ.
When r ≥ 1.5−2ρ+1 , we can show that |I1| ≤ C|t|−3/2.
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212 Mathematical Foundations of Quantum Field Theory
In integral I2, let us replace ω(p) with
ω1(p) = ω(p0) + v(p0)(p− p0) +∑i=1
∑j=1
1
2
∂2ω(p0)
∂pi∂pj
× (pi − p0i)(pj − p0j),
where p0 is any point with |v(p0)− xt | ≤ ν2|t|−ρ. We will show that
the error ∆ resulting from this change does not exceed C|t|−3/2. To
show this, let us use the formula w = v(p)− xt to make a substitution
in the integral
∆ =
∫(exp(−iω(p)t+ ipx)− exp(−iω1(p)t+ ipx))
×(
1− µ(t−ρ(v(p)− x
t
)))f(p)dp.
Since the function ω(p) is strongly convex and the function f(p) has
compact support, it follows that the Jacobian of this transformation
is bounded above and below by positive constants.
By the properties of the functions µ, we can assume that the
integral ∆ is taken over the domain Γ1, where |w| ≤ ν2|t|−ρ; the
volume of this domain equals C|t|−3ρ. Over the domain Γ1, we can
use the estimate
|(exp(−iω(p)t+ ipx)− exp(−iω1(p)t+ ipx))f(p)|
≤ C|t||p− p0|3 ≤ C1|t||w|3 ≤ C2|t|−3ρ+1,
and we then obtain that |δ| ≤ C|t|1−6ρ ≤ C|t|1−6· 512 = C|t|−3/2.
In the remaining part of the proof, we need to estimate the
expression
I2 −∆ =
∫exp(−iω1(p)t+ ipx)f(p)dp
−∫
exp(−iω1(p) + ipx)µ(t−ρ(v(p)− x
t
))dp.
The second term in this expression can be estimated in the same way
as the integral I1. The first term can be estimated if we rewrite it
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Axiomatic Scattering Theory 213
using the Fourier transform, in the form∫G(x− y, t)f(y)dy,
where
f(x) =
∫f(p) exp(ipx)
dp
(2π)3,
G(x, t) =
(2π
it
)3/2
|det γjk|−1/2 exp
i
2
∑j,k
λjkxjxk
,γjk =
∂2ω(p0)
∂pj∂pk,
λjk is the inverse matrix of the matrix γjk.
The inequality (10.8) follows from the inequality (10.7) in the case
x ∈ Uεt and from the inequality (10.9) in the case x 6∈ Uεt.
Remark 10.1. With some extra conditions on the behavior of the
function ω(p) at infinity, we can prove Lemma 10.2 for any smooth,
rapidly decaying function f(p).
Proof of Lemma 10.3. Let us begin by proving the simple but
important identity that connects the truncated vacuum expectation
value
〈AB(t)〉T = 〈AB(t)Φ,Φ〉 − 〈AΦ,Φ〉 〈BΦ,Φ〉
with the vacuum expectation value of the commutator [A,B(t)]. To
prove this, let us consider the smooth function h(ω), satisfying the
conditions h(ω) = 0 for ω ≤ 0 and h(ω) = 1 for ω ≥ δ [the number
δ > 0 is chosen such that all points of the spectrum of the operator
H, with the exception of 0, lie on the ray (δ,+∞)]. Let us note that
h(H) = 1− P0,
h(−H) = 0,(10.26)
where P0 is a projection operator on the vacuum vector Φ (these
equations can be checked by using the isomorphism between the
spaces H and L2(M), which sends the operator H to an operator
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214 Mathematical Foundations of Quantum Field Theory
of multiplication by a function). Let us assign to a function f ∈ Sthe function
fh(t) =
∫h(ω) exp(iω(τ − t))f(τ)
dωdτ
2π.
It is easy to check that the function fh(t) also belongs to the space S(the operator mapping f to the function fh is in fact a composition of
the Fourier transform over t, the operator of multiplication by h(ω),
and the inverse Fourier transform; all these operators transform Sinto itself). If the function f only depends on t and not on the other
variables, then the function fh is constructed by means of the Fourier
transform over t (other variables are considered as parameters). If
the function f(t, x1, . . . , xn) belongs to the space S(En+1), then the
function fh belongs to it as well.
Let us show that for any function f ∈ S, we have∫f(t) 〈AB(t)〉T dt =
∫fh(t) 〈[A,B(t)]〉 dt. (10.27)
Indeed, we have∫fh(t) 〈AB(t)〉 dt =
∫fh(t) 〈A exp(iHt)B〉 dt
= 〈Ah(H)f(H)B〉 = 〈A(1− P0)f(H)B〉
=
∫f(t) 〈A(1− P0) exp(iHt)B〉 dt =
∫f(t) 〈AB(t)〉T dt,
(10.28)∫fh(t) 〈B(t)A〉 dt =
∫fh(t) 〈B exp(−iHt)A〉 dt
= 〈Bh(−H)f(−H)A〉 = 0. (10.29)
Here, we have used the relation (10.26) and the equations∫fh(t) exp(iωt)dt = h(ω)f(ω),∫f(t) exp(iωt)dt = f(ω).
Combining (10.28) and (10.29), we obtain (10.27).
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Axiomatic Scattering Theory 215
Let us now prove the lemma for the case n = 2. We will suppose
that A1, A2, C ∈ A, α ∈ S, A2 =∫α(x, t)C(x, t)dxdt and introduce
the notation D =∫αh(x, t)C(x, t)dxdt. From the relation (10.27), it
is easy to see that
〈A1A2(x, t)〉T = 〈[A1, D(x, t)]〉 .
Since D ∈ A and all operators in A asymptotically commute, it
follows that
|wT2 (x1, t1,x2, t2)| = | 〈A1(x1, t1)A2(x2, t2)〉T |
= | 〈[A1, D(x2 − x1, t2 − t1)]〉 | ≤ C(1 + |t2 − t1|r)1 + |x2 − x1|n
.
Now that we have proved the lemma for the case n = 2, let us
give the proof in the case of arbitrary Wightman functions. Let us
consider the function
wn(x1, t1, . . . ,xn, tn) = 〈A1(x, t1) . . . An(xn, tn)〉
and the two functions
w(1)k (xi1 , ti1 , . . . ,xik , tik) = 〈Ai1(xi1 , ti1) . . . Aik(xik , tik)〉 ,
w(1)l (xj1 , tj1 , . . . ,xjl , tjl) = 〈Aj1(xj1 , tj1) . . . Ajl(xjl , tjl)〉 ,
obtained by splitting the operatorsA1, . . . , An, entering the definition
of the function wn, into two groups Ai1 , . . . , Aik and Aj1 , . . . , Ajk(here, k + l = n, the indices i1, . . . , ik and j1, . . . , jl are in ascending
order, and the operators Ai are assumed to be smooth).
The set of indices i1, . . . , ik, the indices of the operators in the
first group, will be denoted by K; the set of indices j1, . . . , jl of the
second group of operators will be denoted by L; clearly K ∪ L =
1, 2, . . . , n.It can be shown that when all the points xi, where i ∈ K, are
far from the points xj , where j ∈ L, the following approximation
holds:
wn(x1, t1, . . . ,xn, tn)
≈ w(1)k (xi1 , ti1 , . . . ,xik , tik)w
(2)l (xj1 , tj1 , . . . ,xjl , tjl).
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216 Mathematical Foundations of Quantum Field Theory
More precisely, for fixed t1, . . . , tn, the absolute value of the
function
wn(x1, t1, . . . ,xn, tn)− w(1)k (xi1 , ti1 , . . . ,xik , tik)w
(2)l
× (xj1 , tj1 , . . . ,xjl , tjl)
does not exceed Cd−p. Here, d = mini∈K,j∈L |xi−xj |; p is an arbitrary
number; C is a constant depending on p, but not on x1, . . . ,xn.
(This property of Wightman functions is often called asymptotic
factorization or the cluster property.)
To prove the asymptotic factorization property, we will first
consider the case when the set K consists of the points (1, 2, . . . , k)
and the set L consists of the points (k + 1, . . . , n). In this case, the
difference we aim to estimate can be written in the form
〈RU〉T = 〈RU〉 − 〈R〉 〈U〉 ,where
R = A1(x1, t1) . . . Ak(xk, tk),
U = Ak+1(xk+1, tk+1) . . . An(xn, tn).
Recall that the operators A1, . . . , An are smooth and can therefore
be written in the form
Ai =
∫fi(x, t)Bi(x, t)dxdt,
where fi ∈ S, Bi ∈ A. Noting that
U =
∫fk+1(ξ1, τ1) . . . fn(ξn−k, τn−k)Bk+1(xk+1 + ξ1, tk+1 + τ1)
. . . Bn(xn + ξn−k, tn + τn−k)dn−kξdn−kτ
=
∫α(t, ξ1 − xk+1, . . . , ξn−k − xn, σ1, . . . , σn−k−1)
× exp(iHt)Bk+1(ξ1, tk+1)Bk+2(ξ2, tk+2 + σ1)
. . . Bn(ξn−k, tn + σn−k−1) exp(−iHt)dtdn−kξdn−k−1σ,
where
α(t, ξ1, . . . , ξn−k, σ1, . . . , σn−k−1)
= fk+1(ξ1, t)fk+2(ξ2, t+ σ1) · · · fn(ξn−k, t+ σn−k−1),
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Axiomatic Scattering Theory 217
we can apply the relation (10.27). This leads to
〈R,U〉T =
∫αh(t, ξ1 − xn+1, . . . , ξn−k − xn, σ1, . . . , σn−k−1)
×〈[R,Bk+1(ξ1, t+ tk+1)Bk+2(ξ2, t+ tk+2 + σ1)
. . . Bn(ξn−k, t+ tn + σn−k−1)]〉 dtdn−kξdn−k−1σ. (10.30)
The commutator on the right-hand side of equation (10.30) can be
written in the form∑1≤i≤k
k+1≤j≤n
Cij [Ai(xi, ti), Bj(ξj−k, t+ tj + σj−k−1)]Dij ,
where Cij , Dij are operators whose norms are bounded by a constant
not depending on xi, ti, ξi, σi. Using this fact, asymptotic commu-
tativity, and formula (10.30), we obtain the asymptotic factorization
property for the caseK = 1, . . . , k, L = k+1, . . . , n. We can show
that the general case can be reduced to this special case. Suppose the
set K consists of the points (i1, . . . , ik) and the set L consists of the
points (j1, . . . , jl), where i1 < · · · < ik, j1 < · · · < jl. If the quantity
d = mini∈K,j∈L |xi − xj | is large, we can use the approximation
〈A1(x1, t1) . . . An(xn, tn)〉 ≈ 〈Ai1(xi1 , ti1) . . . Aik(xik , tik)
×Aj1(xj1 , tj1) . . . Ajl(xjl , tjl)〉 , (10.31)
whose error does not exceed Cd−m, where m is an arbitrary number;
C is a constant depending on m,n, t1, . . . , tn. We can justify this
approximation by using the relation
〈EAi(xi, ti)Aj(xj , tj)F 〉 ≈ 〈EAj(xj , tj)Ai(xi, ti)F 〉 ,
for i ∈ K, j ∈ L, that we have proved already (here, E, F are
compositions of the operators Aα(xα, tα), where α ∈ K ∪ L).
From the relation (10.31) and the asymptotic factorization prop-
erty, we can now obtain the asymptotic factorization property in the
general case by the same method as in the special case shown earlier.
Let us now show how to derive Lemma 10.3 from the asymptotic
factorization property of Wightman functions. We will suppose
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218 Mathematical Foundations of Quantum Field Theory
that the lemma holds for functions wTn−1 and then show it for
functions wTn . However, we will first prove the following geometric
lemma.
Let N be a set consisting of the points x1, . . . ,xn in Euclidean
space and let D be the diameter of N (i.e. D = maxxi,xj∈N |xi−xj |).Then, we can partition N into two subsets P and Q in such a way
so that ρ(P,Q) = minxi∈P,xj∈Q |xi − xj | ≥ D2n .
We will prove this fact by induction on n. Let D = |xα−xβ|. Let
us delete from N the point xγ which does not coincide with xα or
with xβ, obtaining as a result the set N ′, consisting of n − 1 points
and having diameter D. By the inductive hypothesis, we can split the
set N ′ into two sets P ′ and Q′ so that ρ(P ′, Q′) = minxi∈P,xj∈Q |xi−xj | ≥ D
2n−1 . By the triangle inequality, the point xγ cannot satisfy
the inequalities ρ(P ′,xγ) = minxi∈P ′ |xi − xγ | < D2n and ρ(Q′,xγ) =
minxi∈Q′ |xi − xγ | < D2n simultaneously; for concreteness, suppose
that ρ(P ′,xγ) ≥ D2n . Then, it is clear that the necessary partition
can be constructed by taking P to be P ′ and taking Q to be
Q′ ∪ xγ.Let us now suppose that Lemma 10.3 holds for functions wTk with
k < n. To prove the lemma for functions wTn , we need to estimate
wTn (x1, t1, . . . ,xn, tn) under the condition max1≤i≤j≤n |xi − xj | = D
(with the times t1, . . . , tn fixed). Using the geometric lemma we just
proved, let us partition the set 1, . . . , n into the subsets K and L
such that mini∈K,j∈L |xi − xj | ≥ D2n . To estimate the functions wTn ,
let us note that in the right-hand side of the relation (10.11) (the
recurrence relation defining the functions wTn ) all factors have the
form wαr(πr) with the set πr containing both elements of K and L.
By induction, we can obtain an estimate of the form CD−m, where
m is arbitrary and C depends on m. It therefore follows that every
term on the right-hand side of (10.11) contains at least one factor of
the form just described and therefore admits an estimate of the form
CD−m, since the factors wTαi(πi) do not exceed a constant, depending
only on ‖A1‖, . . . , ‖An‖ and the number n (this statement becomes
clear if we note that
| 〈A1(x1, t1) . . . An(xn, tn)〉 | ≤ ‖A1‖ . . . ‖An‖).
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Axiomatic Scattering Theory 219
Hence, with error not exceeding CD−m, we can write the
approximation
wn(x1, t1, . . . ,xn, tn) ≈ wTn (x1, t1, . . . ,xn, tn)
+n∑k=2
∑ρ
wTα1(π1) . . . wTαk(πk), (10.32)
where the sum is taken over partitions ρ of the set 1, . . . , n such
that every set πj is completely contained in either K or L.
It is easy to check that the sum in (10.32) equals the product
wk(xi1 , ti1 , . . . ,xik , tik)wl(xj1 , tj1 , . . . ,xjl , tjl), where (i1, . . . , ik) are
elements of K and (j1, . . . , jl) are elements of L.
Hence, we see that
wTn (x1, t1, . . . ,xn, tn)
≈ wn(x1, t1, . . . ,xn, tn)
−wk(xi1 , ti1 , . . . ,xik , tik)wl(xj1 , tj1 , . . . ,xjl , tjl).
To finish the proof of Lemma 10.3, we apply the asymptotic
factorization property of Wightman functions.
Remark 10.2. Because the operators Ai are smooth, the Wightman
functions wn(x1, t1, . . . ,xn, tn) and the truncated Wightman func-
tions wTn (x1, t1, . . . ,xn, tn) are smooth, and hence the derivatives of
these functions can be viewed as Wightman and truncated Wightman
functions constructed with different smooth operators. (This is clear
if we note that for a smooth operator A =∫f(τ,x)B(x, τ)dξdτ ,
where f ∈ S(E4), the following relation holds:
D(α)A(x, t) = D(α)
∫f(τ − t, ξ − x)B(ξ, τ)dξdτ
=
∫g(τ − t, ξ − x)B(ξ, τ)dξdτ = C(x, t),
where D(α) = ∂|α|
∂tα0∂xα11 ∂x
α22 ∂x
α33
is a differential operator; g(t,x) =
(−1)|α|D(α)f(t,x); C is a smooth operator defined by C =∫g(τ, ξ)B(ξ, τ)dξdτ ; operator derivatives are understood as norm
derivatives.) Using this fact and the statement of Lemma 10.3,
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220 Mathematical Foundations of Quantum Field Theory
we can say that for fixed t1, . . . , tn and fixed xn, the function
wTn (x1, t1, . . . ,xn, tn), constructed with smooth operators, belongs to
the space S(E3(n−1)).
The requirement that the operators A1, . . . , An be smooth in
Lemma 10.3 can be relaxed. However, if we completely remove this
requirement, then we have to weaken the statement of the lemma.
Namely, one can prove that without the smoothness condition on the
operators A1, . . . , An, the function
ν(x1, t1, . . . ,xn, tn)
=
∫γ(x1 − ξ1, t1 − τ1, . . . ,xn − ξn, tn − τn)
×wTn (ξ1, τ1, . . . , ξn, τn)dnξdnτ,
where γ ∈ S(E4n), for fixed t1, . . . , tn and fixed xn, belongs to the
space S(E3(n−1)).
Proof of Lemma 10.4. To prove this lemma, we need to estimate
the quantity
I = 〈A0(f0, t) . . . An(fn, t)〉T
=
∫f0(x0|t) . . . fn(xn|t) 〈A0(x0, t) . . . An(xn, t)〉T dx0 . . . dxn.
(10.33)
Let us change variables in integral (10.33) to the new variables
x0, ξ1, . . . , ξn, where ξj = xj − x0, and apply the estimate
| 〈A0(x0, t) . . . An(xn, t)〉T | ≤C
1 + ‖ξ‖m
=C
1 + (ξ21 + · · ·+ ξ2
n)m/2,
following from Lemma 10.3 (here, m is arbitrary and C depends on
m). Applying this estimate, we obtain
|I| ≤∫|f0(x0|t)|
(C
1 + |t|3/2
)n C
1 + ‖ξ‖mdx0d
nξ
≤ const(1 + |t|)−32n
∫|f0(x0|t)|dx0
≤ const(1 + |t|)−32
(n−1). (10.34)
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Axiomatic Scattering Theory 221
In the derivation above, we have used the inequalities |fj(x|t)| ≤C(1 + |t|3/2)−1, for j ≥ 1, and
∫|f0(x|t)|dx ≤ C(1 + |t|)3/2, which
follow from Lemma 10.2; the number m is chosen to be large enough
so that the integral∫
(1 + ‖ξ‖)−mdnξ converges.
The inequality (10.34) gives the necessary estimate of the quan-
tity I. To finish the proof of Lemma 10.4, we need an analogous
estimate in the case when in the expression 〈A1(f1, t) . . . An(fn, t)〉T
some of the operators Aj(fj , t) are replaced by the operators
Aj(fj , t) = ddtAj(fj , t).
Let us note that
Aj(fj , t) =d
dt
∫fj(x|t)Aj(x, t)dx
=
∫ (d
dtfj(x|t)
)Aj(x, t)dx +
∫fj(x|t)
d
dtAj(x, t)dx
=
∫gj(x|t)Aj(x, t)dx +
∫fj(x|t)Kj(x, t)dx,
where we have used the notation
K = i[H,A],
gj(x|t) =d
dtfj(x|t)
=
∫exp(−iω(p)t+ ipx)gj(p)
dp
(2π)3,
gj(p) = −iω(p)fj(p).
Hence, the operator Aj(fj , t) can be written in the form
Aj(fj , t) = Aj(gj , t) +Kj(fj , t), (10.35)
where gj is a smooth function with compact support; Kj is a smooth
operator. [The smoothness of Kj follows if we note that if the
operator Aj is represented in the form Aj =∫φj(τ, ξ)Bj(ξ, τ)dξdτ ,
the operator Kj is equal to −∫
( ddτ φj(τ, ξ))Bj(ξ, τ)dξdτ .]
By formula (10.35), the proof of the lemma in the case when
some of the operators Aj(fj , t) are replaced by the operators Aj(fj , t)
clearly reduces to the case already considered. The proof of the
lemma in the case when some of the operators Aj(fj , t) are replaced
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222 Mathematical Foundations of Quantum Field Theory
with the operators (Aj(fj , t))∗ or (Aj(fj , t))
∗ requires no new
ideas.
10.3 Asymptotic fields (in- and out-operators)
In Section 10.1, we defined the Møller matrices S± corresponding to
the asymptotically Abelian algebra A. With the Møller matrices, one
can define the in- and out-operators (asymptotic fields) a+in
out
(k) and
a inout
(k) by the relations
S−a+(k) = a+
in(k)S−,
S−a(k) = ain(k)S−,
S+a+(k) = a+
out(k)S+,
S+a+(k) = aout(k)S+.
These equations clearly define a+in(k), ain(k) as operator gener-
alized functions on the space Hin = S−Has and a+out(k), aout(k) as
operator generalized functions on the space Hout = S+Has.
In this section, we will show that in- and out-operators describe
the asymptotic behavior of the operators A(x, t), where A ∈ A, as
t→ ±∞ (this justifies the name asymptotic fields).
Before we begin precise formulations, let us introduce the follow-
ing definition.
Let us assign to every function f ∈ S(E3) the set U(f) ∈ E3
consisting of points of the form ∂ω(k)∂k , where k ∈ supp f (i.e. k
belongs to the support of f).
A family of functions f1, . . . , fn will be called non-overlapping if
the sets U(fj) do not intersect pairwise (i.e. the sets U(fi) and U(fj)
do not have common points if i 6= j).4
4Under the assumption of strong convexity on the functions ω(k), the familyf1, . . . , fn will be non-overlapping if and only if the supports of these functionssuppfj are pairwise non-intersecting. Therefore, in the case at hand, the definitionof non-overlapping family can be simplified. However, in the case when we havemultiple particles, the definition in the main text is necessary.
March 27, 2020 15:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch10 page 223
Axiomatic Scattering Theory 223
The following remark elucidates the physical meaning of the new
notion.
If f1, . . . , fn is a non-overlapping family of functions, then the
dynamics on the space Has as t → ±∞ transform the vector
a+(f1) . . . a+(fn)θ into a set of far apart particles. More precisely,
we have
exp(−iHast)a+(f1) . . . a+(fn)θ = a+(f t1) . . . a+(f tn)θ
= (2π)32n
∫f1(x1|t) . . . fn(xn|t)a+(x1) . . . a+(xn)dx1 . . . dxnθ,
where
f tj (k) = fj(k) exp(−iω(k)t),
fj(x|t) =
∫f tj (k) exp(ikx)
dk
(2π)3
=
∫fj(k) exp(−iω(k)t+ ikx)
dk
(2π)3.
It follows from Lemma 10.2 that the function fj(x|t), for large t, is
small outside the set tUε(fj), where the symbol Uε(fj) denotes an
ε-neighborhood of the set U(fj). Under the assumed conditions, for
small enough ε and large enough t, the sets tUε(fj) are far apart.
Let us consider vectors of the form Ψ−(B1, . . . , Bn|f1, . . . , fn)
[correspondingly, of the form Ψ+(B1, . . . , Bn|f1, . . . , fn)], where
B1, . . . , Bn are good operators and f1, . . . , fn are functions with
compact support. We will call them non-overlapping in-vectors (out-
vectors) if f1, . . . , fn is a non-overlapping family of functions. The
set of linear combinations of non-overlapping in-vectors (out-vectors)
will be denoted by Din (correspondingly, Dout); Din and Dout are
linear subspaces that are dense in Hin and Hout, respectively.
Let us now prove a theorem that describes the asymptotic
behavior of operators A(x, t), where A ∈ A, in terms of in- and
out-operators. For concreteness, we will formulate the theorem in
the case t→ −∞.
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224 Mathematical Foundations of Quantum Field Theory
If A1, . . . , An are good operators, Ψ1 ∈ Din, then
limt→−∞
A1(f1, t, σ1) . . . An(fn, t, σn)Ψ1
= ain(φ1f1, σ1) . . . ain(φnfn, σn)Ψ1. (10.36)
If A1, . . . , An are smooth operators in the algebra A, satisfying the
condition 〈AjΦ,Φ〉 = 0, Ψ1 ∈ Din, Ψ2 ∈ Din, then
limt→−∞
〈A1(f1, t, σ1) . . . An(fn, t, σn)Ψ1,Ψ2〉
=⟨ain(φ1f1, σ1) . . . ain(φnfn, σn)Ψ1,Ψ2
⟩(10.37)
under the condition that the σj, corresponding to smooth operators
Aj that are not good, coincide.
In (10.36) and (10.37), the functions fj are smooth and have
compact support. The functions φj are given by the relation φj(k) =
〈AjΦ,Φ(k)〉 and A(f, t, σ) are defined by the formulas
A(f, t, 1) = A(f, t) =
∫f(x|t)A(x, t)dx,
A(f, t,−1) = A∗(f, t) =
∫f(x|t)A∗(x, t)dx.
(In other words, for arbitrary smooth operators A1, . . . , An, we have
weak convergence of the operators A1(f1, t, σ1) . . . An(fn, t, σn) to the
operators ain(φ1f1, σ1) . . . ain(φnfn, σn) in the limit as t → −∞ on
the setDin under the condition that σ1 = · · · = σn; for good operators
A1, . . . , An, one can prove that the convergence is strong on Din and
the condition σi = σj is not necessary.)
Let us begin the proof of the theorem.
Lemma 10.5. Let A1, . . . , An ∈ A, f1, . . . , fn be smooth functions
with compact support and σ1, . . . , σn = ±1. If for at least one pair of
indices i, j, where 1 ≤ i ≤ j ≤ n, the sets U(fi) and U(fj) do not
intersect and the σi = σj, then for any m
|〈A1(f1, t, σ1) . . . An(fn, t, σn)〉T | ≤ C(1 + |t|m)−1.
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Axiomatic Scattering Theory 225
This statement holds if we replace some of the operators
Ai(fi, t, σi) with their time derivatives ddtAi(fi, t, σi) in the expression
〈A1(f1, t, σ1) . . . An(fn, t, σn)〉T .
Proof. Let us write the expression
〈A1(f1, t, σ1) . . . An(fn, t, σn)〉T (10.38)
in the form∫f1(p1, σ1) exp(−iσ1ω(p1)t) . . . fn(pn, σn)
× exp(−iσnω(pn)t)wTn (p1, σ1, . . . ,pn, σn)dnp.
[Here, we have introduced the notation
wTn (p1, σ1, . . . ,pn, σn)
=
∫exp
(i∑
pjxj
)〈A1(x1, t, σ1) . . . An(xn, t, σn)〉T dnx,
A(x, t, 1) = A(x, t), A(x, t,−1) = A∗(x, t).]
It follows from Lemma 10.3 in Section 10.1 that the function
wTn has the form wTn (p1, σ1, . . . ,pn, σn) = νn(p2, . . . ,pn,
σ1, . . . , σn)δ(p1 + · · ·+ pn), where νn is a smooth function [one can
check that νn ∈ S(E3(n−1))].
Let us assume for concreteness that σ1 = σ2 and U(f1) does not
intersect U(f2). It then becomes convenient to write (10.38) in the
form ∫exp(−iΩ(p2, . . . ,pn)t)χ(p2, . . . ,pn)dp2 . . . dpn,
where χ ∈ S(E3(n−1));
Ω(p2, . . . ,pn) = σ1ω(−p2 − · · · − pn)
+σ2ω(p2) + σ3ω(p3) + · · ·+ σnω(pn).
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226 Mathematical Foundations of Quantum Field Theory
Let us note that
∂Ω
∂p2= −σ1
∂ω
∂p
∣∣∣∣p=−p2−···−pn
+ σ2∂ω
∂p
∣∣∣∣p=p2
= σ2
(∂ω
∂p
∣∣∣∣p=p2
− ∂ω
∂p
∣∣∣∣p=p1
),
from which it is clear that
∂Ω
∂p26= 0
on the support of the function χ.
We can therefore use formula (10.22), assuming that all variables,
except for p2, are fixed. In the case at hand, we have
u(p) =1
t
(∂Ω
∂p2
) ∣∣∣∣ ∂Ω
∂p2
∣∣∣∣−2
,
and therefore
|(Lrχ)(p)| ≤ C|t|−r. (10.39)
The inequality (10.39) gives us the inequality we seek.
The proof in the case when the operators Ai(fi, t, σi) are replaced
by their derivatives is similar to the proof of Lemma 10.4.
Lemma 10.6. If f1, . . . , fn are non-overlapping smooth functions
with compact support and B1, . . . , Bn are good operators, then for
all m
limt→−∞
tm‖Ψ−(B1, . . . , Bn|f1, . . . , fn)
−Ψ(B1, . . . , Bn|f1, . . . , fn|t)‖ = 0.
Proof. It is enough to show that for any m, we have∣∣∣∣⟨ d
dtΨ(B1, . . . , Bn|f1, . . . , fn|t),
d
dtΨ(B1, . . . , Bn|f1, . . . , fn|t)
⟩∣∣∣∣≤ C(1 + |t|m)−1.
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Axiomatic Scattering Theory 227
This inequality can be obtained in the same way as the inequality
with m = 3 in Section 10.1, by using the just-proved Lemma 10.5
instead of Lemma 10.4.
Lemma 10.7. The relations (10.36) and (10.37) hold in the case of
Ψ1 = Φ.
Proof. To prove this lemma, we first note that the norm of the
vector ξ(t), defined by the formula
ξ(t) = A1(f1, t, σ1) . . . An(fn, t, σn)Φ,
is bounded above by a constant not depending on t. Indeed,
〈ξ(t), ξ(t)〉 can be expanded in terms of truncated Wightman
functions, and by Lemma 10.4 from Section 10.1, all the resulting
truncated Wightman functions are bounded. To prove the lemma, it
is enough to consider the case when the vector Ψ2 has the form
Ψ2 = Ψ−(B1, . . . , Bm|g1, . . . , gm) = limt→−∞
Ψ2(t),
where
Ψ2(t) = Ψ(B1, . . . , Bm|g1, . . . , gm|t).
It follows from the boundedness of ‖ξ(t)‖ and ‖Ψ2(t)‖ that
limt→−∞
〈ξ(t),Ψ2〉 = limt→−∞
〈ξ(t),Ψ2(t)〉.
The inner product
〈ξ(t),Ψ2(t)〉
= 〈A1(f1, t, σ1) . . . An(fn, t, σn)Φ, B1(g1, t) . . . Bm(gm, t)Φ〉
can be expanded in terms of truncated Wightman functions. By
Lemma 10.4 of Section 10.1, as t→ −∞, the products that differ from
zero contain two-point truncated Wightman functions. It therefore
follows that
limt→+∞
〈ξ(t),Ψ2(t)〉
=⟨ain(φ1f1, σ1) . . . ain(φnfn, σn)Φ, a+
in(ψ1, g1) . . . a+in(ψmgm)Φ
⟩=⟨ain(φ1f1, σ1) . . . ain(φn, fn, σn)Φ,Ψ−(B1, . . . , Bm|g1, . . . , gm)
⟩(10.40)
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228 Mathematical Foundations of Quantum Field Theory
(here, the symbol ψi denotes a generalized function defined by the
formula BiΦ =∫ψi(k)Φ(k)dk).
In the derivation of (10.40), we should use the formula
limt→∞〈Ai(fi, t, σi)Aj(fj , t, σj)〉T
=⟨ain(φif i, σi)ain(φjf j , σj)
⟩,
which holds if at least one of the operators Ai, Aj is good or is adjoint
to a good operator or, without this assumption, if we have σi = σj (in
particular, this formula can be applied in the case Aj = Bj , fj = gj ;
then φj = ψj).
The relation (10.40) implies equation (10.37) for Ψ1 = Φ.
In the case when A1, . . . , An are good operators, we can, as in
Section 10.1, prove that the vector ξ(t) has a limit as t→ −∞.
It follows from the statements we have already proved that
the vector ξ − η, where ξ = limt→−∞ ξ(t), η = ain(φ1f1, σ1)
. . . ain(φnfn, σn)Φ, is orthogonal to the space Hin. On the other
hand, a simple calculation, based on expansion in terms of truncated
Wightman functions, shows that 〈ξ, ξ〉 = 〈η, η, 〉. Since η ∈ Hin, this
implies that ξ = η. Therefore, relation (10.36) holds for Ψ1 = Φ,
which finishes the proof of Lemma 10.6 in this case.
Let us now prove the relations (10.36) and (10.37) in the general
case.
Let Ψ1 = Ψ−(E1, . . . , Er|h1, . . . , hr), where E1, . . . , Er are good
operators, h1, . . . , hr are non-overlapping functions, and Ψ1(t) =
Ψ(E1, . . . , Er|h1, . . . , hr|t).It follows from (10.8) that ‖Ai(f, t, σ)‖ ≤ C(1 + |t|3/2) and
therefore from Lemma 10.3 it follows that for t→∞ we have
‖A1(f1, t, σ1) . . . An(fn, t, σn)(Ψ1 −Ψ1(t))‖ → 0.
Therefore, we have
limt→−∞
A1(f1, t, σ1) . . . An(fn, t, σn)Ψ1
= limt→−∞
A1(f1, t, σ1) . . . An(fn, t, σn)Ψ1(t)
= limt→−∞
A1(f1, t, σ1) . . . An(fn, t, σn)E1(h1, t, 1) . . . Er(hr, t, 1)Φ.
To finish the proof, it remains to apply Lemma 10.3.
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Axiomatic Scattering Theory 229
Remark 10.3. Relation (10.37) holds in the case when Ψ2 is an
arbitrary vector in Hin.
10.4 Dressing operators
In this section, we will prove that the scattering matrix construction
provided in Section 10.1 is equivalent to a different construction
which is more closely connected to the physical picture of scattering.
Let us begin by introducing the notion of a dressing operator.
An operator D, acting from space Has to the space H, will be
called in-dressing (out-dressing), if
S− = slimt→−∞
exp(iHt)D exp(−iHast), (10.41)
S+ = slimt→+∞
exp(iHt)D exp(−iHast). (10.42)
If both relations are satisfied, then the operator D will be called
a dressing operator.
It is useful to note that it follows from relation (10.4) that both
in-dressing and out-dressing operators satisfy the conditions
Dθ = Φ, (10.43)
Da+(k)θ = Φ(k). (10.44)
However, these conditions are not sufficient for an operator to be
a dressing operator. Note that, in particular, the S− operator is an
in-dressing operator and the S+ operator is an out-dressing operator.
Let us now give sufficient conditions for an operator D to be in-
dressing. Namely, let us prove the following theorem.
Theorem 10.1. Let us suppose that (a) the operator D has norm 1,
(b) for all non-overlapping families of functions with compact support
f1(k), . . . , fn(k) and for all smooth operators A1, . . . , An ∈ A, we
have, for t < 0,
〈Da+(ft1) . . . a+(f
tn)θ,A1(g1) . . . An(gn)Φ〉
≈∑π∈P
⟨Φ(f t1), Ai1(gi1)Φ
⟩. . .⟨Φ(f tn), Ain(gin)Φ
⟩(10.45)
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230 Mathematical Foundations of Quantum Field Theory
with error not exceeding cνn(g)|t|−N (here, f ti (k) = fi(k) exp
(−iω(k)t), P is the set of permutations π = (i1, . . . , in) of indices
(1, . . . , n), ν(g) = maxi(sup |gi(x)| +∫|gi(x)|dx); gi ∈ S(E3); N is
an arbitrary number; C is a constant depending on N but not on the
functions gi). Then the operator D is in-dressing.
The condition for an operator to be out-dressing can be obtained
from the theorem above by replacing t < 0 with t > 0.
To prove the above theorem, let us consider the vector x = S−y,
where x = Ψ−(B1, . . . , Bn|f1, . . . , fn); y = a+(φ1f1) . . . a+(φnfn)θ;
B1, . . . , Bn are good operators and f1, . . . , fn are non-overlapping
functions, and
x(t) = Ψ−(B1, . . . , Bn|f t1, . . . , f tn)
= S−a+(φ1f
t1) . . . a+(φnf
tn)θ.
It is clear that
exp(−iHast)y = a+(φ1ft1) . . . a+(φnf
tn)θ (10.46)
and therefore, by the relation (10.5), we have
x(t) = exp(−iHt)x. (10.47)
For large t, the vector x(t) negligibly differs from the vector ξ(t) =
Ψ(B1, . . . , Bn|f t1, . . . , f tn|0) = B1(f t1, 0) . . . Bn(f tn, 0)Φ (to prove this
statement, note that for t→ ±∞ we have
‖x(t)− ξ(t)‖ = ‖ exp(iHt)(x(t)− ξ(t))‖
= ‖Ψ−(B1, . . . , Bn|f1, . . . , fn)
−Ψ(B1, . . . , Bn|f1, . . . , fn|t)‖ → 0). (10.48)
On the other hand, by relation (10.46), we have that
limt→−∞
〈D exp(−iHast)y, ξ(t)〉
= limt→−∞
〈Da+(φ1ft1) . . . a+(φnf
tn)θ,B1(f t1, 0) . . . Bn(f tn, 0)Φ〉
= limt→−∞
∑π∈P
⟨Φ(φ1f
t1), Bi1(f ti1 , 0)Φ
⟩. . .⟨Φ(φnf
tn), Bin(f tin , 0)Φ
⟩= 〈y, y〉 . (10.49)
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Axiomatic Scattering Theory 231
Recalling that ‖D exp(−iHast)y‖ ≤ ‖y‖ and that limt→−∞ ‖ξ(t) −x(t)‖ = 0, we obtain from (10.49) the relation
limt→−∞
〈D exp(−iHast)y, x(t)〉 = 〈y, y〉 , (10.50)
from which it follows by (10.47) that
limt→−∞
〈exp(iHt)D exp(−iHast)y, S−y〉
= limt→−∞
〈exp(iHt)D exp(−iHast)y, x〉
= limt→−∞
〈D exp(−iHast)y, x(t)〉 = 〈y, y〉 = 〈S−y, S−y〉.
Here, we apply the following simple lemma.
If lim 〈βi, η〉 = 〈η, η〉 and ‖βt‖ ≤ ‖η‖, then limβt exists and equals
η (to prove this statement, we write 〈βt, η〉 in the form
〈βt, η〉 = ‖βt‖ · ‖η‖ cosφt,
where φt is the angle between βt and η, and note that it follows from
the inequalities ‖βt‖ ≤ ‖η‖, | cosφt| ≤ 1 that ‖βt‖ cosφt converges to
‖η‖ only in the case when lim ‖βt‖ = ‖η‖, lim cosφt = 1).
Applying the lemma to the vectors βt = exp(iHt)D exp(−iHast)y,
η = S−y, we see that limt→−∞ exp(iHt)D exp(−iHast)y exists and
equals S−y.
Since the vectors y = a+(φ1f1) . . . a+(φnfn)θ, which we have
shown to satisfy the relation
limt→−∞
exp(iHt)D exp(−iHast)y = S−y,
constitute a total set in the space Has, it is clear that the relation
holds for a dense set of vectors and hence for any vector y ∈ Has (see
Appendix A.5).
We have thus obtained sufficient conditions for an operator D to
be in-dressing and out-dressing. These conditions can be relaxed by
changing the requirement that ‖D‖ = 1 with
limt→−∞
(t→+∞)
‖Da+(ft1) . . . a+(f
tn)θ‖ = ‖a+(f
t1) . . . a+(f
tn)θ‖ (10.51)
(the proof remains the same).
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232 Mathematical Foundations of Quantum Field Theory
The above statements also give conditions guaranteeing that
the operator D is dressing (since the operator is dressing if it is
simultaneously in- and out-dressing). These conditions can be easily
formulated in a slightly different form. Namely, we have the following
theorem.
Theorem 10.2. Let us suppose the operator D has norm 1 and
for all functions f1(x), . . . , fn(x) ∈ S(E3), whose supports are far
from each other, and for all smooth operators A1, . . . , An ∈ A, the
following approximation holds:⟨Da+(f1) . . . a+(fn)θ,A1(g1) . . . An(gn)Φ
⟩≈∑π∈P〈Φ(f1), Ai1(gi1)Φ〉 . . . 〈Φ(fn), Ain(gin)Φ〉 (10.52)
with error not exceeding Cνn(f)νn(g)d−N (here, fi(x) =∫fi(k)
exp(ikx) dk(2π)3
; the functions gi ∈ S(E3); d is the minimal distance
between the supports of the functions fi(x); N is an arbitrary number;
C is a constant depending on N but not depending on the func-
tions f1, . . . , fn, g1, . . . , gn; the sum is taken over all permutations
π = (i1, . . . , in)). Then the operator D is a dressing operator.
Indeed, relation (10.52) implies relation (10.45) for t ≤ 0 as
well as for t ≥ 0. To show this, note that by Lemma 10.2 from
Section 10.1 the “essential supports” of the functions fi(x|t) =∫exp(ikx)f ti (k) dk
(2π)3are far from each other in the limit t → ±∞.
(By this lemma, the function fi(x|t) is small outside the set tU(i)ε ,
where U(i)ε is an ε-neighborhood of the set U (i) consisting of points of
the form ∂ω(k)∂k , where k runs over suppfi. Hence, we can say that the
set tU(i)ε is the “essential support” of the function fi(x|t), recognizing
that the values of the function outside of this set are very small.)
To provide more rigorous reasoning, let us introduce smooth
functions hi(x) ∈ S, equal to 1 on the set U(i)ε and 0 outside
the set U(i)2ε . Then, the functions tri = hi(tx)fi(x|t) will have
far away supports as t → ±∞ and the quantity 〈Da+(tr1) . . .
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Axiomatic Scattering Theory 233
a+(trn)θ,A1(g1) . . . An(gn)Φ〉 can be approximated with the rela-
tion (10.52). From the other side, we have
a+(ft1) . . . a+(f
tn)θ ≈ a+(tr1) . . . a+(trn)θ
by Lemma 10.2 of Section 10.1. Using these remarks, we obtain that
relation (10.52) implies relation (10.45).
Let us note that the condition ‖D‖ = 1 can be relaxed here as
well, by assuming that either the condition (10.51) or the condition
‖Da+(f1) . . . a+(fn)θ‖ ≈ ‖a+(f1) . . . a+(fn)θ‖ (10.53)
holds. (In (10.53), we assume that the functions f1(x), . . . , fn(x) ∈ Shave distant supports; the error should not exceed
Cνn(f)d−N ,
where N is an arbitrary number, d is the minimal distance between
the supports, and C depends on N).
The following statement is used in Section 11.5.
Theorem 10.3. Let S and S be scattering matrices constructed for
energy operators H and H, the momentum operator P, and a family
of operators A (this family must be asymptotically commutative
relative to the operators H,P, as well as relative to the operators
H, P). Let us suppose that T is a unitary operator satisfying the
following conditions:
(1) TΦ = Φ,
(2) TΦ(k) = Φ(k),
(3) if A ∈ A, then T−1AT ∈ A (here, Φ and Φ are ground states
and Φ(k) and Φ(k) are single-particle states corresponding to the
energy operators H and H).
If D is a dressing operator for the operator H, satisfying the
conditions of Theorem 10.2, then the operator D = TD is a dressing
operator for the energy operator H.
The proof of this theorem consists of checking that the conditions
of Theorem 10.2 apply to the operator D. This verification can be
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234 Mathematical Foundations of Quantum Field Theory
done with the following transformations:
〈Da+(f1) . . . a+(fn)θ,A1(g1) . . . An(gn)Φ〉
= 〈Da+(f1) . . . a+(fn)θ, T−1A1(g1) . . . An(gn)TT−1Φ〉
= 〈Da+(f1) . . . a+(fn)θ,AT1 (g1) . . . ATn (gn)Φ〉
≈∑π
⟨Φ(f1), ATi1(gi1)Φ
⟩. . .⟨Φ(fn), ATin(gin)Φ
⟩=∑π
〈TΦ(f1), Ai1(gi1)TΦ〉 . . . 〈TΦ(fn), Ain(gin)TΦ〉
=∑π
〈Φ(f1), Ai1(gi1)Φ〉 . . . 〈Φ(fn), Ain(gin)Φ〉
(here, we have used the notation T−1AiT = ATi ).
The statements we have proved in this section allow us to give a
new definition of Møller matrices and the scattering matrix. Namely,
if the operator D satisfies conditions (10.52) and (10.53), then we
can define Møller matrices using equations (10.41) and (10.42). The
scattering matrix, as usual, can be defined by the relation S = S∗+S−;
in the case when the scattering matrix is unitary, this definition is
equivalent to the formula
S = S−1+ S−
= slimt→+∞,t0→−∞
exp(iHast)D−1 exp(−iH(t− t0))D exp(−iHast0).
(10.54)
Let us now consider how the the formulas (10.41), (10.42) and
(10.54) are connected to the physical picture of particle scattering.
For this discussion, we will fix an operator D, satisfying the
conditions (10.52) and (10.53).
Let us first consider the collision of two particles. A single particle
with the wave function f will be written as the vector
Φ(f) =
∫f(k)Φ(k)dk = Da+(f)θ.
Let us ask the following question: which vector in the space H should
be used to describe a state with two particles having wave functions
f1 and f2?
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Axiomatic Scattering Theory 235
Before we answer this question, note that its formulation in
such a general form is physically meaningless: we should assume
that the particles are far from each other. More formally, if two
particles have wave functions f1(k), f2(k), we should assume that
the supports of the functions f1(x), f2(x) (the wave functions of these
particles in coordinate representation) are far from each other. With
this narrowed framing of the question, we can now give a definite
answer. Namely, we should represent such a state with the vector
Da+(f1)a+(f2)θ.
Indeed, condition (10.52), imposed on the operator D, corre-
sponds to the physical representation of the state with two distant
particles Φ(f1) and Φ(f2).
Analogously, the vector Da+(f1) . . . a+(fn)θ, in the case when the
supports of the functions f1(x), . . . , fn(x) are far from each other,
describes the state of n spatially separated particles.
It should now be easy to see that the collision of two particles can
be described by the vector
x(t) = exp(−iHt)S−a+(f1)a+(f2)θ,
where f1, f2 are non-overlapping functions. Indeed, from the rela-
tion (10.41), it follows that for t→ −∞, we have
x(t) ≈ D exp(−iHast)a+(f1)a+(f2)θ = Da+(f
t1)a+(f
t2)θ,
and hence as t → −∞, x(t) describes the state of two spatially
separated particles Φ(f t1) and Φ(f t2) (technically, it is wrong to say
that the supports of the functions f t1, f t2 are far from each other;
as discussed above, it is in fact their essential supports that can be
shown to be far from each other).
In particular, the vector x(0) = S−a+(f1)a+(f2)θ depicts the
state of a system of two particles at t = 0, whose wave functions at
t → −∞ are f t1, f t2 (if each particle moves freely, then they can be
described at t = 0 by the wave functions f1 and f2). Analogously, we
can interpret other vectors of the form S−ξ and S+η.
It should now be clear that the operator S defines the transition
from the initial state to the final state in a particle collision. Indeed,
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236 Mathematical Foundations of Quantum Field Theory
if η = Sξ, then the vector
x(t) = exp(−iHt)S−ξ = exp(−iHt)S+η
depicts the collision process where we begin (at t→ −∞) in the state
x(t) ≈ D exp(−iHast)ξ,
and end (at t→ +∞) in the state
x(t) ≈ D exp(−iHast)η.
Let us now investigate the relationship between the matrix
entries of the scattering matrix (scattering amplitudes) and the
experimentally observed quantities.
For simplicity, let us consider the collision of two particles. We
assume that a beam of particles with momentum q scatters on a
resting particle. Let us assume that the beam has unit flux (the
meaning of this, in both the classical and quantum situations, is
explained in the Chapter 5, Section 5.2). In the simplest case, two
particles remain at the end of the scattering process, however we will
consider the more general situation where new particles can be born
and we can thus have more than two particles at the end. We should
predict the probability (or, more precisely, the probability density)
that at the end we observe n particles with momenta (p1, . . . ,pn).
It will be more convenient to consider the probability σG that the
vector of momenta lies in the domain G in 3n-dimensional space E3n
(if the particles are identical, we should assume that the domain G is
invariant with respect to permutations). This number describes how
many collisions per second end up with n particles, whose momentum
vector p1, . . . ,pn belongs to the domain G. This number is called the
effective collision cross-section (and called the effective differential
cross-section in the case of an infinitesimally small domain G).
Let us now show that the collision cross-section σG can be
expressed using the scattering amplitudes by the following formula:
σG =1
n!
1
|v(q)|
∫G|sn,2(p1, . . . ,pn|q, 0)|2δ(p1 + · · ·+ pn − q)
× δ(ω(p1) + · · ·+ ω(pn)− ω(q)− ω(0))dp1 . . . dpn. (10.55)
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Axiomatic Scattering Theory 237
Here, sn,k are functions that are related to the matrix elements Sn,kof the scattering matrix by the relation
Sn,k(p1, . . . ,pn|q1, . . . ,qk)
=⟨Sa+(q1) . . . a+(qk)θ, a
+(p1)a+(pn)θ⟩
= sn,k(p1, . . . ,pn|q1, . . . ,qk)δ(p1 + · · ·+ pn − q1 − · · · − qk)
× δ(ω(p1) + · · ·+ ω(pn)− ω(q1)− · · · − ω(qk)) (10.56)
(sometimes the term scattering amplitudes refers to the functions sn,k
as well). The symbol v(k), as always, denotes the function ∂ω(k)∂k ; we
assume that v(0) = 0. Formula (10.55) is proved below under the
assumption that q 6= 0 and the function sn,2 is continuous at points
(p1, . . . ,pn|q, 0), where (p1, . . . ,pn) ∈ G.
To understand the collision cross-sections in the case at hand,
we will use the construction described in Section 5.2. In general, the
following discussion is largely similar to the discussion in Section 5.2.
Namely, let us assume that the process of collision is described by
the vector D exp(−iH0t)ξα for t→ −∞, where
ξα =
(∫exp(ikα)f(k)a+(k)dk
)(∫g(k)a+(k)dk
)θ, (10.57)
f(k) and g(k) are normalized wave functions differing from zero only
in a small neighborhood of the points q and 0, respectively; α is a
vector orthogonal to the vector v(q) (as explained in Section 5.2, this
vector is analogous to the impact parameter in classical mechanics.)
The collision process is described by the vector
xα(t) = exp(−iHt)S−ξα,
and at the end of the process (with t→ +∞) we obtain the state
D exp(−iH0t)(Sξα).
The probability that for initial state (10.57) we obtain the final
state Sξα of n particles with momenta (p1, . . . ,pn), belonging to
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238 Mathematical Foundations of Quantum Field Theory
the domain G, can be written in the form
wα(G) =1
n!
∫G
∣∣∣∣∫ dk1dk2Sn,2(p1, . . . ,pn|k1,k2)
× f(k1) exp(ik1α)g(k2)
∣∣∣∣2 dp1 . . . dpn (10.58)[we have used the relation
Sξα =∑m
1
m!
∫Sm,2(p1, . . . ,pm|k1,k2)f(k1)
× exp(ik1α)g(k2)a+(p1) . . . a+(pm)θdp1 . . . dpmdk1dk2
].
The collision cross-section σG is equal to5
σG =
∫α⊥v(q)
dαwα(G). (10.59)
Substituting into (10.59) the expression (10.58) and integrating over
α, we see that
σG =1
n!
∫Gdp1 . . . dpn
∫dk1dk
′1dk2dk
′2(sn,2(p1,
. . . ,pn|k1,k2)sn,2(p1, . . . ,pn|k′1,k′2)f(k1)f(k′1)
× g(k2)g(k′2)δ(kT1 − k′T1 ))
=1
n!
∫Gdp1 . . . dpn
∫dk1dk
′1dk2dk
′2(sn,2(p1, . . . ,pn|k1,k2)
× sn,2(p1, . . . ,pn|k′1,k′2)f(k1)f(k′1)g(k2)g(k′2)
× δ(kT1 − k′T1 )δ(p1 + · · ·+ pn − k1 − k2)δ(p1 + · · ·+ pn
−k′1 − k′2)δ(ω(p1) + · · ·+ ω(pn)− ω(k1)− ω(k2))
× δ(ω(p1) + · · ·+ ω(pn)− ω(k′1)− ω(k′2)) (10.60)
5Strictly speaking, this formula contains a limit: the probabilities wα(G) shouldbe defined with the functions fν(k), gν(k), whose supports contract to the pointsq, 0, respectively, as ν → ∞.
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Axiomatic Scattering Theory 239
[here the symbols kT1 , k′T1 denote the projections of the vectors k1,
k′1 on the plane orthogonal to the vector v(q)].
In further calculations, we will use the following relations:
δ(p1 + · · ·+ pn − k1 − k2)δ(p1 + · · ·+ pn − k′1 − k′2)
= δ(p1 + · · ·+ pn − k1 − k2)δ(k1 + k2 − k′1 − k′2),
δ(ω(p1) + · · ·+ ω(pn)− ω(k1)− ω(k2))δ(ω(p1) + · · ·
+ω(pn)− ω(k′1)− ω(k′2))
= δ(ω(p1) + · · ·+ ω(pn)
−ω(k1)− ω(k2))δ(ω(k1) + ω(k2)− ω(k′1)− ω(k′2)),
δ(kT1 − k′T1 )δ(k1 + k2 − k′1 − k′2)δ(ω(k1) + ω(k2)− ω(k′1)− ω(k′2))
=|v(q)|
(v(k1)− v(k2))v(q)δ(k1 − k′1)δ(k2 − k′2)
+α(k1,k2)δ(k′1 − β(k1,k2))δ(k′2 − γ(k1,k2)),
where β(k1,k2), γ(k1,k2) are non-trivial solutions of the following
system:
ω(k1) + ω(k2)− ω(β)− ω(γ) = 0,
k1 + k2 − β − γ = 0,
kT1 − βT = 0
(i.e. solutions other than β = k1,γ = k2).
Using these formulas, we can transform expression (10.60) into
the form
σG =1
n!
∫Gdp1 . . . dpn
∫dk1dk2
(|v(q)|
(v(k1)− v(k2)) · v(q)
× |sn,2(p1, . . . ,pn|k1,k2)|2|f(k1)|2|g(k2)|2
× δ(p1 + · · ·+ pn − k1 − k2)δ(ω(p1) + · · ·+ ω(pn)
− ω(k1)− ω(k2))
)
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240 Mathematical Foundations of Quantum Field Theory
+1
n!
∫Gdp1 . . . dpn
∫dk1dk2(sn,2(p1,
. . . ,pn|k1,k2)sn,2(p1, . . . ,pn|β(k1,k2),γ(k1,k2))
× f(k1)f(β(k1,k2)g(k2)g(γ(k1,k2))α(k1,k2)
× δ(p1 + · · ·+ pn − k1 − k2)δ(ω(p1)
+ · · ·+ ω(pn)− ω(k1)− ω(k2))).
To obtain the final expression for σG, we should use the fact that
for normalized functions f(k) with support in a small neighborhood
of the point q and for functions λ(k), continuous at the point q, we
have the relations ∫λ(k)|f(k)|2dk = λ(q),∫
λ(k)f(k)f(ρ(k))dk = 0 (10.61)
(in formulas (10.61) there is a limit in which the support of the
function f(k) contracts to the point q; ρ(k) is a function that is
continuous at the point q and satisfying the condition ρ(q) 6= q).
10.5 Generalizations
In this section, we describe some generalizations of the results of
previous sections of this chapter.
1. The condition of strong convexity imposed on the dispersion
law ω(p) can be substantially relaxed. Namely, it is enough to require
that for any open set G ⊂ E3 the function ω(p) is not linear (i.e. for
any neighborhood of a point p0 ∈ E3 there is another point p1 ∈ E3
such that ∂ω(p)∂p |p=p1 6=
∂ω(p)∂p |p=p0).
In this case, the definition of the Møller matrix should be modi-
fied: in the formula (10.1), we should consider only non-overlapping
families of functions f1, . . . , fn. The main modification required in the
proofs is to use Lemma 10.5 of Section 10.3 in place of Lemma 10.2
of Section 10.1.
These modifications in the definitions and proofs allow us to
construct a scattering theory in the one- and two-dimensional cases,
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Axiomatic Scattering Theory 241
i.e. in the case when the momentum operator P has only one or
two components instead of three. (In higher dimensions, there are no
essential modifications.)
2. Up to this point, we have only considered situations with a
single one-particle state. The generalization to an arbitrary number
of one-particle states is straightforward.
A good operator B, in this case, can be defined as a smooth
operator with (1) B∗Φ = 0; (2) we can find a particle Φi(k) and a
function φB(k) for which BΦ = Φi(φB) =∫φB(k)Φi(k)dk. [Recall
that in Section 10.1 we have agreed to fix a complete orthogonal
system of particles Φ1(k), . . . ,Φs(k).]
The modifications of the statements and their proofs consist
mostly of adding indices; perhaps the most important change comes
from the need to use the word “total” instead of “dense” in the
formulation of Lemma 10.1.
It is easy to check that the Møller matrices and the scattering
matrix do not depend on the choice of complete system of particles
(i.e. they depend only on the operators H,P, and the algebra A).
3. All the constructions in this chapter can be adapted, with
the appropriate modifications, to the case when we replace the
condition of asymptotic commutatitivty by the condition of asymp-
totic anticommutatitivity of the family of operators A (asymptotic
anticommutativity means that for any two operators A,B ∈ A and
any n, we can find C and r such that
‖ [A,B(x, t)]+‖ ≤ C1 + |t|r
1 + |x|n
).
If we have an asymptotically anticommutative family A, we
impose an additional condition that any vector A1 . . . A2k+1Φ, where
Ai ∈ A, is orthogonal to the vacuum Φ (this condition becomes
necessary to construct the scattering matrix). The space H can then
be expanded into a direct sum of two subspaces Hg and Hu in such a
way that the vector A1 . . . AnΦ belongs to the space Hg, if n is even,
and to the space Hu, if n is odd. We will call vectors in the spaces
Hg and Hu even and odd, respectively.
Let us use the symbol A to denote the smallest algebra of
operators that contains the family A. The algebra A, like the linear
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242 Mathematical Foundations of Quantum Field Theory
space above, can be represented as a direct sum Ag + Au, where
every operator A ∈ Ag asymptotically commutes with every operator
B ∈ A and every operator A ∈ Au asymptotically anticommutes
with every operator B ∈ Au (this can be easily checked if we
note that the operator A1 . . . Am and the operator B1 . . . Bn, where
Ai ∈ A, Bj ∈ A, asymptotically commute in the case when one of
the numbers m,n is even and asymptotically anticommute in the case
when both numbers m and n are odd). Operators from the family Agwill be called even and operators from the family Au will be called
odd. It is easy to check that an even operator preserves the parity
of vectors (i.e. transforms the spaces Hg and Hu into themselves),
while an odd operator transforms an even vector into an odd vector
and an odd vector into an even vector.
Let us suppose that the particle system Φ1(k), . . . ,Φs(k) is chosen
in such a way that each particle has a definite parity. In other words,
the particles can be split into even, whose vectors Φi(f) ∈ Hg, and
odd, whose vectors Φi(f) ∈ Hu. Even particles should be considered
as bosons, while odd particles as fermions. This implies that the
space Has can be constructed as the Fock space representation of the
relations
[ai(k), aj(k′)]∓ = [a+
i (k), a+j (k′)]∓ = 0,
[ai(k), a+j (k′)]∓ = δijδ(k− k′),
where the anticommutator is used in the case when both particles
Φi(k),Φj(k) are odd and the commutator is used in the other cases.
One can say that
Has = F s(L2(E3 ×N1))⊗ F a(L2(E3 ×N2)),
where N1 is the set of even particles and N2 is the set of odd particles.
In the case considered in this subsection, the construction of the
scattering matrices is performed in the same way as when the family
A is asymptotically commutative.
In the situation at hand, the elementary particles are fermions
while bosons arise only as composite particles. If the elementary
particles can be either fermions or bosons, then the family A must
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Axiomatic Scattering Theory 243
be a union of an asymptotically commutative family A1 and an
asymptotically anticommutative family A2, where each operator
from A1 must asymptotically commute with any operator from A2.
All the results of Chapter 10, as well as the results of the following
chapters, can be easily extended to the case when we have both
fermions and bosons as elementary particles.
4. We have previously assumed that the single-particle spectrum
does not intersect the multi-particle spectrum (in other words, that
the laws of conservation of energy and momentum prohibit the decay
of particles). This condition is not always satisfied. However, in the
theory of elementary particles, if a particle is stable, then its decay in
all known cases is prohibited either by the conservation of energy and
momentum or by some other conservation law. In this case, one can
prove all the results established above by the same methods. More
precisely, in axiomatic scattering theory, it is enough to assume that
the space H expands into a direct sum of the spaces Ni, that are
invariant with respect to the operators H,P, in such a way that for
every i, the spectrum of the operators H,P in the subspace H1 ∩Nidoes not intersect the spectrum of these operators in the subspace
M∩Ni (here, M is the multi-particle subspace). In this case, however,
it is necessary to tighten the cyclicity requirement by assuming that
vectors of the form AΦ, where A ∈ A, are dense in each space Ni.5. Let us describe a generalization of the concept of an asymp-
totically commutative algebra that allows us to study asymptotic
commutative algebras containing unbounded operators.
Let us suppose that the energy operator H and the momentum
operator P = (P1, P2, P3) act on the Hilbert spaceH. We will assume,
as always, that H, P1, P2, P3 are commuting self-adjoint operators,
whose spectrum satisfies the conditions outlined in Section 10.1.
Let us fix a linear subspace D that is dense in H, contains the
ground state Φ of the energy operatorH, and is invariant with respect
to operators of the form exp(iHt− iPx). (In this subsection, we will
assume that these conditions always apply to the operators H,P and
the set D.) A family A, consisting of operators defined on the set D
that transform the set into itself, will be called asymptotically Abelian
algebra, if
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244 Mathematical Foundations of Quantum Field Theory
C1. For every operator A ∈ A, the operators A+ and A(x, t) also
belong to A; for every pair of operators A,B ∈ A, their linear
combination λA+ µB and product AB also belong to A.
C2. For any operators A1, . . . , Ar ∈ A and arbitrary function f ∈S(E4r), the operator∫
f(x1, t1, . . . ,xr, tr)A1(x1, t1) . . . Ar(xr, tr)drxdrt (10.62)
belongs to the algebra A and the number⟨∫f(x1, t1, . . . ,xr, tr)A1(x1, t1) . . . Ar(xr, tr)d
rxdrtΦ,Φ
⟩(10.63)
continuously depends on the function f ∈ S (the integral
in (10.62) is understood in a weak sense).6
C3. For any operators A,B,A1, . . . , Ar ∈ A and any number n, we
can find numbers C and s such that
| 〈[A(x, t), B]A1(ξ1, τ1) . . . Aj(ξj , τj)Φ,
Aj+1(ξj+1, τj+1) . . . Ar(ξr, τr)Φ〉 |
≤ C(1 + |t|s)1 + |x|n
(1 +
r∑i=1
|ξi|+r∑i=1
|τi|
)k, (10.64)
where k is a number not depending on n.
We will assume that the vector Φ is a cyclic vector of the
asymptotic commutative algebra A. Then, using the earlier consid-
erations of this chapter, we can define Møller matrices and scattering
matrices, corresponding to the operators H,P, and the algebra A,
prove their existence, and transfer to this case the results of earlier
considerations. (The definitions, theorems, and proofs do not require
any significant modifications.)
6The second part of the condition C2 implies that the function 〈A1(x1, t1) . . .Ar(xr, tr)Φ,Φ〉 is a locally summable function of moderate growth.
March 27, 2020 15:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch10 page 245
Axiomatic Scattering Theory 245
Let us add to these results the following statement which will be
used in Section 11.4.
Suppose thatB(k|t) is an operator function, which is a generalized
function in the variable k, that satisfies the following conditions:
(1) exp(iHτ − iPa)B(k|t) exp(−iHτ + iPa) = exp(−ika)B(k|t+ τ);
(2) B+(k|t)Φ = 0;
(3) vectors of the form∫f(k)B(k|t)dkΦ belong to the single-particle
subspace;
(4) the functions ρm,n defined by the equation
ρm,n(k1, t1, . . . ,km, tm|k′1, t′1, . . . ,k′n−1, t′n−1)
× δ(k1 + · · ·+ km − ε1k′1 − · · · − εnk′n)
=⟨A1(k1|t1) . . . Am(km|tm)B(k′1, ε1|t′1) . . . B(k′n, εn|0)
⟩T,
where Ai(k|t) = (2π)−3∫Ai(x, t) exp(ikx)dx; Ai ∈ A; B(k,
1|t) = B+(k|t); B(k,−1|t) = B(k|t) belongs to the space
S(E3(m+n−1)) over the variables k1, . . . ,km,k′1, . . . ,k
′n−1 for
fixed t1, . . . , tm, t′1, . . . , t
′n−1. [We will call such generalized func-
tions “good”; this name is motivated by the fact that for every
good operator B, we can construct a good operator gener-
alized function B(k|t) = (2π)−3∫B(x, t) exp(ikx)dx.] Then,
for any system of smooth functions with compact support
f1(k), . . . , fm(k), we have
limt→±∞
B(f t1|t) . . . B(f tn|t)Φ = S±a+(φf1) . . . a+(φfn)θ,
where
f ti (k) = fi(k) exp(−iω(k)t),
B(f ti |t) =
∫f ti (k)B(k|t)dk,
B(k|0)Φ = φ(k)Φ(k).
The proof of this statement follows the reasoning in Section 10.1.
Let us now define the notion of asymptotically commutative
families of operator generalized functions.
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246 Mathematical Foundations of Quantum Field Theory
A family B of operator generalized functions A(x, t) is called
asymptotically commutative if there exists an asymptotically com-
mutative algebra A that contains all operators of the form
A(f) =∫f(x, t)A(x, t)dxdt, where A(x, t) belongs to the family
B, f ∈ S(E4). (We will assume that the vector Φ is a cyclic vector of
the family of operators A(f). The operators A(f) should have a com-
mon domain of definition D and transform it into itself; the domain
of definition D′ of operators belonging to the algebra A may coincide
with the set D, but in general must at least contain it. When we say
that the algebra A contains the operator A(f), we mean that in this
algebra, we can find an operator coinciding with A(f) on the set D.)
The scattering matrix corresponding to the operators H,P and
the asymptotically commutative family of operator generalized func-
tions B can be defined as the scattering matrix constructed for H,P,
and the asymptotic algebra A, containing all operators A(f). (It is
easy to see that this scattering matrix does not depend on the choice
of algebra A.) We can define Møller matrices analogously.
The following theorem gives sufficient conditions for the asymp-
totic commutativity of a family of operator generalized functions.
Let us suppose that B is a family of operator generalized functions
satisfying the following conditions:
D1. For any operator generalized function A(x, t) ∈ B, the operators
A(f) =∫f(x, t)A(x, t)dxdt, where f ∈ S(E4), are defined on a
set D and transform it into itself.
D2. The functional 〈A(f)ψ1, ψ2〉, where A ∈ B, ψ1, ψ2 ∈ D, f ∈S(E4), continuously depends on the function f ∈ S(E4) (i.e. the
real-valued generalized function 〈A(x, t)ψ1, ψ2〉 is a generalized
function of moderate growth).
D3. If A ∈ B, then
exp(iHτ − iPa)A(x, t) exp(−iHτ + iPa) = A(x + a, t+ τ).
D4. For every function A(x, t) in family B, the family also contains
the adjoint operator generalized function A+(x, t).
D5. For any operator generalized functions A1(x, t), . . . , Ar(x, t),
A(x, t), B(x, t) in B, we can find a δ > 0 such that the real-valued
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Axiomatic Scattering Theory 247
generalized function
T (x, t, ξ1, τ1, . . . , ξr, τr)
= 〈[A(x + a, t+ α), B(a, α)]A1(ξ1, τ1)
. . . Aj(ξj , τj)Φ, Aj+1(ξj+1, τj+1) . . . Ar(ξr, τr)Φ〉
in the domain satisfying |t| < δ|x| can be represented in the
form
T (x, t, ξ1, τ1, . . . , ξr, τr)
=∑
|α|≤N(n)
D(α)
(1 +∑i
|ξi|2 +∑i
|τi|2)k
× (1 + |x|2)−n σα,n(x, t, ξ1, τ1, . . . , ξn, τn)
(here, n is an arbitrary natural number; σα,n is a bounded
continuous function; D(α) is a differential operator of order |α|with constant coefficients; k is a number not depending on n).
D6. The vector Φ is a cyclic vector of the family of operators A(f),
where A(x, t) ∈ B, f ∈ S(E4).
Then, the family B is asymptotically Abelian.
To prove this theorem, let us consider the family A consisting of
linear combinations of operators of the form∫f(x1, t1, . . . ,xn, tn)A1(x1, t1) . . . An(xn, tn)dnxdnt, (10.65)
where f ∈ S(E4n), Ai(x, t) ∈ B. As it is shown in Appendix A.7,
expressions of the form (10.65) can be seen as operators defined on
some linear subspace D′ and transform D′ into itself (it follows that
we can take D′ to be the set of all vectors of the form AΨ, where
A ∈ A; Ψ ∈ D; expression of the form (10.65) defines an operator on
the set D by the operator analog of the kernel theorem). It is easy to
check that the set D′ contains the set D and is invariant with respect
to the operators exp(iHt− iPa).
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248 Mathematical Foundations of Quantum Field Theory
It is easy to check that the family A, when viewed as a family
of operators defined on the set D′, is asymptotically commutative.
Conditions C1 and C2 can be easily checked with the help of the
statements proven in Appendix A.7. For example, if A ∈ A is an
operator of the form (10.65), then the operators
A+ =
∫f(x1, t1, . . . ,xn, tn)A+
n (x1, t1) . . . A+1 (xn, tn)dnxdnt,
A(x, t) =
∫f(x1 − x, t1 − t, . . . ,xn − x, tn − t)
×A1(x1, t1) . . . An(xn, tn)dnxdnt
are also contained in A. Inequality (10.64) follows from condition D5.
Let us now give a simple example of objects satisfying the
conditions of axiomatic scattering theory.
Let us consider the Fock space H = F (L2(E3)) with the free
Hamiltonian H0 =∫ε(k)a+(k)a(k)dk and the momentum operator
P =∫
ka+(k)a(k)dk. If ε(k) is a smooth, strongly convex function,
satisfying the condition ε(k1)+ε(k2) > ε(k1 +k2), then the spectrum
of these operators satisfies the conditions outlined in Section 10.1.
[The ground state of this system is the Fock vacuum θ and the single-
particle state is defined by the formula Φ(k) = a+(k)θ.] The space
Has can be identified with the spaceH; the Hamiltonian Has becomes
the Hamiltonian H0 in this identification.
The operator generalized functions
a+(x, t) = (2π)−3/2
∫exp(−ikx)a+(k, t)dk,
a(x, t) = (2π)−3/2
∫exp(ikx)a(k, t)dk,
where
a+(k, t) = exp(iH0t)a+(k) exp(−iH0t)
= exp(iω(k)t)a+(k),
a(k, t) = exp(iH0t)a(k) exp(−iH0t) exp(−iω(k)t)a(k),
constitute an asymptotically commutative family B0 of generalized
functions (as always, we assume that these operator generalized
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Axiomatic Scattering Theory 249
functions act on the linear manifold S∞ consisting of vectors of the
form ∑n
∫φn(k1, . . . ,kn)a+(k1) . . . a+(kn)θdnk,
where the functions φn belong to the space S and only a finite number
of these functions are non-zero). The family B0 can be shown to
be asymptotically commutative with the aid of the theorem proven
above. A different proof of the same fact can be obtained if we note
that operators of the form∫f(x, t)a+(x, t)dxdt and
∫f(x, t)a(x, t)dxdt,
where f ∈ S(E4) are contained in the asymptotically commutative
algebra A0 consisting of operators of the form∑m,n
∫fm,n(k1, . . . ,km|p1, . . . ,pn)a+(k1)
. . . a+(km)a(p1) . . . a(pn)dmkdnp, (10.66)
where fm,n ∈ S(E3(m+n)), the sum in (10.66) is finite, and operators
of the form (10.66) are viewed as operators, defined on the set S∞. It
is easy to check directly that A0 is an asymptotically commutative
algebra; this can also be shown by results we formulate below. It is
easy to verify that the Møller matrix S± and the scattering matrix
S, constructed for the operators H0,P and the family A0, are trivial
(i.e. they are all the identity operator).
It is sometimes convenient to assume that an asymptotically
commutative algebra is endowed with a topology (this will be useful
in Sections 10.6 and 11.5). We will now give some conditions that
when applied to a topological operator algebra guarantee that it will
be asymptotically commutative.
Let us assume that on the Hilbert space H we have, as before, an
energy operator H, a momentum operator P, and a linear subspace
D, satisfying the conditions outlined above, and let us consider a set
of operators A obeying the following conditions:
T1. The set A consists of operators that are defined on the set D
and transform D into itself.
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250 Mathematical Foundations of Quantum Field Theory
T2. For any A,B in A, the following operators are also in A: the
linear combination λA + µB (i.e. the set can be viewed as a
linear space), the product AB (this implies that A is an algebra
of operators), and the adjoints A+, B+.
T3. The linear space A is equipped with a locally convex topology;
the space A must be complete, and the product and conjugation
operations must be continuous in the topology A.7
T4. The vector AΦ ∈ H continuously depends on A ∈ A in the
topology of the algebra A [i.e. we can find a seminorm p(A) on
A such that
‖AΦ‖ ≤ p(A)].
T5. If A ∈ A, then the operator A(x, t) is in A and it continuously
depends on x, t in the topology on A; for any seminorm p(A) on
A and any compact set F ⊂ A, we can find another seminorm
q(A) and a number k, such that for A ∈ F , x ∈ E3,−∞ < t <∞we have
p(A(x, t)) ≤ (1 + |x|k + |t|k)q(A).
T6. For any seminorm p(A) on A, any compact set F ⊂ A, and any
natural number n, there exists a seminorm q(A) such that
p([A(x), B]) ≤ q(A)q(B)
1 + |x|n
for all A,B ∈ F , x ∈ E3.
(If the locally convex topology on A is specified by a system
of seminorms ‖A‖α, then conditions T5 and T6 should be
reformulated in the following way: for all seminorms ‖A‖α,
compact sets F ⊂ A, and natural numbers n, there must exist
a seminorm ‖A‖λ in the system and numbers k,C such that
‖A(x, t)‖α ≤ C(1 + |x|k + |t|k)‖A‖λ,
‖[A(x), B]‖α ≤C‖A‖λ‖B‖λ
1 + |x|n
for any A,B ∈ F .)
7Conditions T1–T3 mean that the set A is a complete locally convex topologicalalgebra with involution.
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Axiomatic Scattering Theory 251
T7. The vector Φ is a cyclic vector with respect to the family of
operators A.
We can now state the following proposition:
A family of operators A satisfying conditions T1–T6 is an asymp-
totically commutative algebra.
Indeed, condition C1 follows from conditions T1 and T2. To
check C2, note that for any A1, . . . , An ∈ A the operators
A1(x1, t1) . . . An(xn, tn) continuously depend on x1, t1, . . . ,xn, tn in
the topology on A and for any seminorm p(A) on A we can find a
number k such that
p(A1(x1, t1) . . . An(xn, tn)) ≤ 1 + |x|k + |t|k (10.67)
(this follows from conditions T3 and T5). These remarks and the
completeness of A imply that the integral (10.62) converges in the
topology A and defines an element of the algebra A. It follows from
condition T4 and inequality (10.67) that
〈A1(x1, t1) . . . An(xn, tn)Φ,Φ〉
is a continuous function of moderate growth in the variables
x1, t1, . . . ,xn, tn. This proves the second part of condition C2. Finally,
we can easily check C3 by using T3–T6.
The statement we have just proved allows us to say that conditions
T1–T7 are sufficient to construct a scattering theory starting with
the operators H,P and the algebra A.
Let us now return to the operators H0 =∫ε(k)a+(k)a(k)dk,P =∫
ka+(k)a(k)dk and the algebra A0, consisting of operators of the
form (10.66). We will show that the algebra can be endowed with a
topology in a way that satisfies conditions T1–T7. The algebra A0
can be represented as the union of linear spaces A(m,n) consisting of
operators of the form
A =
∫fm,n(k1, . . . ,km|p1, . . . ,pn)a+(k1) . . . a+(km)a(p1)
. . . a(pn)dmkdnp.
Elements of the space A(m+n) are in one-to-one correspondence with
functions in the space S(E3(m+n)); this allows us to transfer the
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252 Mathematical Foundations of Quantum Field Theory
topology of the space S(E3(m+n)) to the space A(m+n). Hence, we
can consider A(m+n) to be a complete locally convex space.
Let us consider a system of seminorms on algebra A0 that are
continuous functions on each subspace A(m,n) ⊂ A0. This system of
seminorms defines a topology on A0 which is called the inductive
limit topology ; the algebra A0 is complete in this topology (see, for
example, Robertson et al. (1980)).
It is easy to check that the algebra A0, with the topology
introduced above, satisfies conditions T1–T7. To verify conditions
T5 and T6, it helps to use the following lemmas.
Lemma 10.8. For any norm ‖f‖α,β on the space S(E3(m+n)) we
can find a norm ‖f‖γ,δ on S and numbers C, k such that for any
function f ∈ S(E3(m+n)) we have
‖UtVxf‖α,β ≤ C(1 + |x|k + |t|k)‖f‖γ,δ[here, Vx and Ut denote operators in S(E3(m+n)) transforming the
function f(k1, . . . ,km|p1, . . . ,pn) to the functions
exp
i m∑j=1
kj − tn∑j=1
pj
x
f(k1, . . . ,km|p1, . . . ,pn),
exp
−i m∑j=1
ε(kj)−n∑j=1
ε(pj)
t
f(k1, . . . ,km|p1, . . . ,pn),
respectively ].
Lemma 10.9. Foranynorm‖f‖α,β onthespaceS(E3(m+m′+n+n′−2r))
and any number N , we can find norms ‖f‖γ,δ and ‖f‖γ′,δ′ on the
spaces S(E3(m+n)) and S(E3(m′+n′)) and a number C so that
‖λr(Vxf, g)‖α,β ≤C
1 + |x|N‖f‖γ,δ‖g‖γ′,δ′ .
[Here, r > 0 and λr(f, g) denotes the function∫f(k1, . . . ,km|p1, . . . ,pn−r,q1, . . . ,qr)
×g(q1, . . . ,qr,k′1, . . . ,km′−r|p′1, . . . ,p′n)dq1 . . . dqr.]
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Axiomatic Scattering Theory 253
It follows from Lemma 10.8 that for any seminorm p(A) on the
space A(m+n) we can find a seminorm q(A) on A(m+n) and a number
k so that for any operator A ∈ Am,n we have
p(A(x, t)) ≤ (1 + |x|k + |t|k)q(A).
It follows from Lemma 10.9 that for any seminorm p(A) on
the space A0 and for any number N we can find seminorms q(A)
and q′(A) on the spaces A(m,n) and A(m′,n′) so that for any A ∈A(m,n), B ∈ A(m′,n′), we have
p([A(x), B]) ≤ q(A)q′(B)
1 + |x|N.
To finish checking conditions T5 and T6, we should note that every
compact subset of the algebra A0 is contained in a direct sum of a
finite number of subspaces A(m,n) (see Robertson et al. (1980)).
Therefore, the algebra A0 satisfies conditions T1–T7; as we have
remarked above, this implies that A0 is asymptotically commutative.
10.6 Adiabatic theorem in axiomatic scattering
theory
We will now show how in axiomatic scattering theory one can express
Møller matrices and scattering matrices in terms of their adiabatic
analogs.
First, we will define the adiabatic Møller matrix and the adiabatic
scattering matrix in a slightly more general situation than was
considered in Section 4.1.
Let the self-adjoint operator H(g) act on the Hilbert space H,
with the parameter g taking values in the interval [0, 1]. We will
fix a continuous function h(τ) of a real variable τ which rapidly
decays at infinity and equals 1 at τ = 0. The symbol Uα(t, t0)
will denote the evolution operator, constructed with the time-
dependent Hamiltonian H(h(ατ)), and the symbol Sα(t, t0) will
denote the operator exp(iH(0)t)Uα(t, t0) exp(−iH(0)t0). Otherwise,
we can define the operator Sα(t, t0) as the solution to
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254 Mathematical Foundations of Quantum Field Theory
the equation
idSα(t, t0)
dt= exp(iH(0)t)(H(h(αt))−H(0)) exp(−iH(0)t)Sα(t, t0)
with the initial condition Sα(t0, t0) = 1.
We will similarly use U(t, t0|g(τ)) to denote the evolution operator
corresponding to the time-dependent Hamiltonian H(g(τ)), where
g(τ) is a function taking values in [0, 1]; using this notation, we can
write Uα(t, t0) = U(t, t0|h(ατ)).
We will call the operators Sα(0,±∞) = slimt→±∞ Sα(0, t) the
adiabatic Møller matrices and the the operator Sα = Sα(∞,−∞) =
slim t→∞t0→−∞
Sα(t, t0) will be called the adiabatic S-matrix. The con-
struction of the adiabatic S-matrix and the adiabatic Møller matrices
provided in Section 4.1 for the pair of operators H,H0 corresponds
to H(g) = H0 + g(H −H0) and h(τ) = exp(−|τ |).In what follows, for the sake of simplifying the proofs, we will
assume that the function h(τ) satisfies a few somewhat stronger
conditions than we have assumed above. Namely, we will assume
that this function is smooth, even, has compact support, and satisfies
h(0) = 1 in a neighborhood of the point τ = 0. The radius of this
neighborhood will be denoted by the symbol δ and the radius of the
support of the function h(τ) will be denoted by ∆ (i.e. h(τ) = 1 for
|τ | < δ and h(τ) = 0 for |τ | > ∆).
Let us now consider the situation when the energy operator in
an axiomatic scattering theory depends on a parameter g. More
precisely, suppose that we have an energy operator H(g) that acts on
the Hilbert space H, where the parameter g is in the interval [0, 1], a
momentum operator P = (P1, P2, P3), and a family of operators A,
satisfying the following conditions:
(a) The self-adjoint operators H(g), P1, P2, P3 commute for every
g. In the space H, there exists a vector Φ (the ground state of the
energy operator H(g), not depending on the parameter g) and a
vector function Φ(k|g), generalized with respect to k and continuous
in the parameter g [single-particle state of the operator H(g)], for
which we have
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Axiomatic Scattering Theory 255
(1) H(g)Φ = PΦ = 0;
(2) H(g)Φ(k|g) = ω(k|g)Φ(k|g),
where ω(k|g) is a positive function, infinitely differentiable in k, g
and strongly convex in k;
(3) PΦ(k|g) = kΦ(k|g);
(4) 〈Φ(k|g),Φ(k′|g)〉 = δ(k− k′);
(5) for any k0 ∈ E3, 0 ≤ g0 ≤ 1, we can find an operator A ∈ A,
such that 〈Φ(k0|g0), AΦ〉 6= 0;
(6) for any k0 ∈ E3, 0 ≤ g0 ≤ 1, there exist ε > 0, δ > 0, so that for
any point (ω,k) belonging to the multi-particle spectrum of the
operators (H(g),P) and satisfying |k− k0| < δ, |g − g0| < δ, we
have the inequality ω > ω(k0|g0) + ε.
(b) Let us denote by D the set of vectors of the form AΦ where
A ∈ A. We assume that D is dense in H and invariant with respect
to the operators exp(iPx) and U(t, t0|g(τ)), where g(τ) is a smooth
function taking values in [0, 1]. For any two operators A,B ∈ A,
the linear combination λA + µB, the product AB, and the adjoint
operators A+, B+ all belong to A (in other words, the family A is
an operator algebra with an involution). We will assume that A is
equipped with a locally convex topology, in which A is complete and
the product and involution operations are continuous.
The topology on A should satisfy the following conditions:
(1) 〈AΦ,Φ〉 continuously depends on A ∈ A;
(2) if A ∈ A, g(τ) is a smooth function with values in [0, 1], x ∈ E2,
and −∞ < t0, t1 <∞, then the operator
U(t1, t0|g(τ))A(x)U(t0, t1|g(τ)) ∈ A;
for every seminorm p(A) on A and every compact F ⊂ A we can
find a seminorm q(A) on A and a number k such that for A ∈ F ,
we have
p(U(t1, t0|g(τ))A(x)U(t0, t1|g(τ))
≤ (1 + |t1 − t0|k + |x|k)q(A). (10.68)
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256 Mathematical Foundations of Quantum Field Theory
In particular, A(x, t|g) ∈ A; we will suppose that this operator
is a smooth function in g with respect to the topology A, the
operator ∂m
∂gmA(x, t|g) continuously depends on x, t, g and
p
(∂m
∂gmA(x, t|g)
)≤ (1 + |x|k + |t|k)q(A), (10.69)
where p(A) is any seminorm on A; m is an arbitrary whole
number; the seminorm q(A) on A and the number k depend
on the seminorm p, the number m, and the compact set F . [Here
and further, A(x) denotes the operator exp(−iPx)A exp(iPx)
and A(x, t|g) denotes exp(iH(g)t)A(x) exp(−iH(g)t)];
(3) for any seminorm p(A) on A, any number n, and any compact
F ⊂ A, we can find a seminorm q(A) on A such that for all
A,B ∈ A,x ∈ E3, A ∈ F we have
p([A(x), B]) ≤ q(A)q(B)
1 + |x|n. (10.70)
These conditions are sufficient to guarantee that starting from the
operators H(g),P, and the algebra A, we can construct the Møller
matrices S±(g) and the scattering matrix S(g) = S∗+(g)S−(g). (Not
very precisely, one can say that we require the operators H(g),P, and
the algebra A to satisfy the conditions T1–T7 of the Section 10.5,
which guarantee the existence of Møller matrices, but we additionally
require these conditions to be satisfied uniformly in g).
We will now show that we can derive the following relations from
the above conditions:
S−(1) = slimα→0
Sα(0,−∞) exp
(i
α
∫ρ(k)a+
in(k)ain(k)dk
),
(10.71)
S+(1) = slimα→0
Sα(0,+∞) exp
(i
α
∫ρ(k)a+
out(k)aout(k)dk
),
(10.72)
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Axiomatic Scattering Theory 257
where
ρ(k) =
∫ 0
−∞(ω(k|h(σ))− ω(k|0))dσ,
ain(k) = S−(0)a(k)S−1− (0),
aout(k) = S+(0)a(k)S−1+ (0)
are in- and out-operators, constructed for the energy operator H(0).
The proof of these relations (the adiabatic theorem) is the main result
of this section.
Of course, one can easily state the analog of relations (10.71) and
(10.72) for the operators S±(g), where 0 ≤ g ≤ 1; we will not stop
to do this.
The relations (10.71) and (10.72) are clearly equivalent to the
relations
S−(1) = slimα→0
slimt→−∞
Uα(0, t)S−(0)Wα(t), (10.73)
S+(1) = slimα→0
slimt→+∞
Uα(0, t)S+(0)Wα(t), (10.74)
where Wα(t) is an operator in the space Has, defined by the formula
Wα(t) = exp
(i
∫rα(k|t)a+(k)a(k)dk
),
rα(k|t) =
∫ 0
tω(k|h(ατ))dτ
(to verify this equivalence, it is enough to note that
exp(−iH(0)t)S−(0) = S−(0) exp
(−i∫ω(k|0)ta+(k)a(k)dk
);
1
αρ(k) = lim
t→−∞(rα(k|t)− tω(k|0))).
We will prove the adiabatic theorem in the form (10.73) and
(10.74). The proof will use the following series of lemmas.
Lemma 10.10. For any seminorm p(A) on A, any compact F ⊂ A,
and any number n, we can find a seminorm q(A) on A and a number
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258 Mathematical Foundations of Quantum Field Theory
k, such that
p([U(t1, t0|g(τ))A(x)U(t0, t1|g(τ)), B])
≤ 1 + |t1 − t0|k
1 + |x|nq(A)q(B)
(here, A,B ∈ F , g(τ) is a smooth function with values in [0, 1],
−∞ < t1, t0 <∞,x ∈ E3).
The proof of this lemma quickly follows from inequalities (10.68)
and (10.70) and the remark that the operator U(t1, t0|g(τ)) com-
mutes with exp(iPx).
Lemma 10.11. If φ(x, t) is a piecewise-continuous function, decay-
ing faster than a power function, then the operator∫φ(x, t)A(x, t|g)dxdt,
where A ∈ A, 0 ≤ g ≤ 1, belongs to the algebra A and is infinitely
differentiable with respect to g in the topology of A, and furthermore
the operator
dk
dgk
∫φ(x, t)A(x, t|g)dxdt
continuously depends on φ ∈ T , A ∈ A, g ∈ [0, 1] in the topology
on A.
[We will assume that the space T of piecewise-continuous func-
tions with decay faster than a power function is endowed with a
topology by means of the norm family
‖φ‖λ = supx∈E3,−∞<t<∞
(1 + |x|λ + |t|λ)φ(x, t).
]The proof of this lemma is based on the completeness of the algebra
A and condition b(2).
Lemma 10.12. Suppose that Ag, Bg ∈ A are two families of
operators, infinitely differentiable in g in the topology on A. Let us
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Axiomatic Scattering Theory 259
assume that B∗gΦ = 0 for all g (this condition is satisfied if Bg is a
good operator). Then the function
ρ(k|g) =
∫〈Ag(x)Φ, BgΦ〉 exp(ikx)dx
is infinitely differentiable with respect to k, g.
Indeed, let us consider the function
ρ(x|g) = 〈Ag(x)Φ, BgΦ〉 =⟨B∗gAg(x)Φ,Φ
⟩.
By Lemma 10.10, the function
∂m
∂gmρ(x|g) =
m∑k=0
Ckm
⟨∂kB∗g∂gk
∂m−kAg(x)
∂gm−kΦ,Φ
⟩
=
m∑k=0
Ckm
⟨[∂kB∗g∂gk
,∂m−kAg(x)
∂gm−k
]Φ,Φ
⟩
obeys the inequality ∣∣∣∣ ∂m∂gm ρ(x|g)
∣∣∣∣ ≤ C
1 + |x|n
(here, m and n are arbitrary natural numbers). It follows from this
inequality that the function ρ(k|g) =∫ρ(x|g) exp(ikx)dx is infinitely
differentiable.
Before we move on to the next lemma, let us note that the vector
generalized function Φ(k|g) defined by conditions a(2), a(3), a(4)
is not uniquely specified: a vector generalized function Φ′(k|g) =
exp(iλ(k|g))Φ(k|g), where λ(k|g) is a real measurable function, will
satisfy the same conditions.
Lemma 10.13. For any operator A ∈ A, we can find a real
measurable function λ(k|g) so that⟨AΦ,Φ′(k|g)
⟩= 〈AΦ, exp(iλ(k|g))Φ(k|g)〉
is infinitely differentiable in k and g.
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260 Mathematical Foundations of Quantum Field Theory
Let us suppose that B1g , B
2g are two families of good operators that
are infinitely differentiable in g in the topology of A; the functions
r1(k|g) and r2(k|g) will be defined by the relation
BigΦ =
∫ri(k|g)Φ(k|g)dk.
Let us consider the function
ν(x|g) =⟨B1g(x)Φ, B2
gΦ⟩
and note that
ν(x|g) =
⟨exp(iPx)
∫r1(k|g)Φ(k|g)dk,
×∫r2(k|g)Φ(k|g)dk
⟩=
∫exp(ikx)r1(k|g)r2(k|g)dk. (10.75)
It follows from relation (10.75) and Lemma 10.12 that the function
r1(k|g)r2(k|g) is infinitely differentiable in k and g.
Let us now show that for all points (k0, g0) we can find an
infinitely differentiable family of good operators Bg, such that
〈Bg0Φ,Φ(k0|g0)〉 6= 0.
We will use the fact that for every operator A ∈ A and every
smooth function with compact support σ(k, ω|g), which is equal to
0 for ω ≤ 0 and for any point (ω,k) belonging to the multi-particle
spectrum of the operators H(g),P, we can construct a family of good
operators Bσ,Ag by means of the formula
Bσ,Ag =
∫σ(x, t|g)A(x, t|g)dxdt,
where
σ(x, t|g) =1
(2π)4
∫exp(ikx− iωt)σ(k, ω|g)dkdω
(see Lemma 10.1 in Section 10.1). By Lemma 10.11, this family is
infinitely differentiable in g. It is easy to check that⟨Bσ,Ag Φ,Φ(k|g)
⟩= σ(k, ω(k|g)|g) 〈AΦ,Φ(k|g)〉 . (10.76)
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Axiomatic Scattering Theory 261
For every point k0, g0, we can find a function σ such that
σ(k0, ω(k0|g0)|g0) 6= 0. This observation, combined with condi-
tion a(5) shows that for every point k0, g0 we can find an operator
A ∈ A such that ⟨Bσ,Ag0 Φ,Φ(k0|g0)
⟩6= 0.
We can now use the following statement.
Let F = φγ(k) be a family of measurable complex functions
on the convex subset M of the Euclidean space En, satisfying (a)
for any two functions φγ ∈ F , φδ ∈ F , the function φγ(k)φδ(k)
is infinitely differentiable; (b) for any point k ∈ M , we can find a
function φγ ∈ F , not equalling zero at the point k. Then we can find
a measurable real function λ(k) with the property that the product
of any function φγ ∈ F with exp(iλ(k)) is a smooth function.
(The proof of this statement, based on several theorems from the
topology of fiber bundles, is given in Fateev and Shvarts (1973).)
Applying this statement, we can find a real function λ(k|g) such
that for any family of good operators Bg ∈ A, which are infinitely
differentiable in g in the topology of A, the function
exp(iλ(k|g))r(k|g) = exp(iλ(k|g)) 〈BgΦ,Φ(k|g)〉
is infinitely differentiable.
It follows from relation (10.76) that the same choice of the
function λ(k|g) implies the differentiability of all functions of the
form exp(iλ(k|g)) 〈AΦ,Φ(k|g)〉, where A ∈ A; this finishes the proof
of the lemma.
Remark 10.4. Using the same arguments, one can prove, without
requiring condition a(5), that for any operator A ∈ A, the function
|〈Φ(k|g), AΦ〉|2 is infinitely differentiable. This allows us to consider
the value of the function |〈Φ(k|g), AΦ〉| at individual points. There-
fore, condition a(5) has a precise meaning.
In what follows, we will assume that the single-particle state
Φ(k|g) is chosen in such a way that for any operator A ∈ A, the
function 〈AΦ,Φ(k|g)〉 is infinitely differentiable in k, g [Lemma 10.13
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262 Mathematical Foundations of Quantum Field Theory
implies that this can always be done by replacing Φ(k|g) with
Φ′(k|g) = exp(iλ(k|g))Φ(k|g) if necessary].
Lemma 10.14. For any smooth function with compact support
f(k|g), we can find a family of good operators Bfg , infinitely dif-
ferentiable in g in the topology of A, such that
BfgΦ =
∫f(k|g)Φ(k|g)dk.
To prove this lemma, it is convenient to use the relation
Bψφg =
∫φ(x|g)Bψ
g (x)dx (10.77)
[more precisely, if a family of good operators Bψ
g ∈ A is infinitely
differentiable in g and
Bψg Φ =
∫ψ(k|g)Φ(k|g)dk,
φ(k|g) is smooth function with compact support, and
φ(x|g) =1
(2π)3
∫φ(k|g) exp(ikx)dk,
then formula (10.77) specifies a family of good operators, infinitely
differentiable in g and satisfying the condition
Bψφg Φ =
∫ψ(k|g)φ(k|g)Φ(k|g)dk
].
Let us note that for every point (k0, g0) there exists a neighbor-
hood U , an operator A ∈ A, and a function σ, such that the function
〈Bσ,Ag ,Φ,Φ(k|g)〉 is non-zero in the neighborhood U (here, Bσ,A
g is a
family of good operators, constructed in the proof of Lemma 10.12).
If the support of the function f(k|g) is contained in the neighborhood
U , then the necessary family Bfg can be obtained with the help of
relation (10.77), using Bψg = Bσ,A
g , Bψφg = Bf
g in the relation. In other
words, we should choose function (10.76) as the function ψ(k|g) and
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Axiomatic Scattering Theory 263
the function
f(k|g)
〈Bσ,Ag Φ,Φ(k|g)〉
as the function φ(k|g) (clearly, the latter function is smooth and has
compact support if we define it to be zero at the values where both
the numerator and the denominator are zero).
An arbitrary function f can be represented as a finite sum of
functions fi, for which the family Bfig can be obtained by the
construction above. It is clear that the operators
Bfg =
∑Bfig
are infinitely differentiable in g and satisfy the conditions
BfgΦ =
∫f(k|g)Φ(k|g)dk,
(Bfg )∗Φ = 0.
However, these functions do not, a priori, satisfy the condition of
smoothness entering the definition of a good operator. To satisfy
this condition, we can replace the operators Bfg by the operators
Bfg =
∫µ(x, t|g)B(x, t|g)dxdt,
where
µ(x, t|g) =1
(2π)4
∫µ(k, ω|g) exp(ikx− iωt)dkdω,
µ(k, ω|g) is a smooth function with compact support, equal to 1 if
(k, g) ∈ supp f , ω = ω(k|g).
Lemma 10.15. The function ν(k|g) defined by the formula⟨dΦ(f |g)
dg,Φ(f ′|g)
⟩=
∫ν(k|g)f(k)f ′(k)dk, (10.78)
where f(k), f ′(k) are smooth functions with compact support, is
smooth in k and g.
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264 Mathematical Foundations of Quantum Field Theory
Let us first note that, by Lemma 10.14, the vector Φ(f |g) can
be represented in the form BfgΦ, where the operator Bf
g is infinitely
differentiable in g; hence, the derivative dΦ(f |g)dg exists. Furthermore,
the bilinear form
A(f, f ′) =
⟨dΦ(f |g)
dg,Φ(f ′|g)
⟩is translation-invariant [i.e. A(f, f ′) = A(fx, f
′x), where fx(k) =
exp(−ikx)f(k)] and, therefore, can be written in the form (10.78).
Let us consider the function
ζ(x|g) =
⟨dΦ(f |g)
dg,Φ(f ′x|g)
⟩=
⟨dΦ(f |g)
dg, exp(−iPx)Φ(f ′|g)
⟩
=
⟨dBf
g
dgΦ, Bf ′
g (x)Φ
⟩.
It is clear that
ζ(x|g) =
∫exp(ikx)ν(k|g)f(k)f ′(k)dk. (10.79)
It follows from relation (10.79) and Lemma 10.12 that the function
ν(k|g)f(k)f ′(k) is smooth. Since f(k), f ′(k) are arbitrary smooth
functions, the proof of the lemma follows.
It follows from Lemma 10.15 that the single-particle state Φ(k|g)
can be chosen in such a way that for every f, f ′ we have⟨dΦ(f |g)
dg,Φ(f ′|g)
⟩= 0. (10.80)
[If equality (10.80) is not satisfied, then we can satisfy it if we replace
Φ(k|g) with the generalized vector function
Φ(k|g) = exp(iτ(k|g))Φ(k|g),
where τ(k|g) is a solution of the equation
i∂τ(k|g)
∂g= −ν(k|g).
]
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Axiomatic Scattering Theory 265
Lemma 10.15 implies that τ(k|g) is a smooth function and hence the
function 〈AΦ, Φ(k|g)〉 is smooth for all A ∈ A.
In what follows, we will assume that the single-particle state
Φ(k|g) is chosen such that it satisfies condition (10.80).
Let us denote by t0 a moment in time, satisfying the relation
t0 ≤ −∆α (recall that the symbol ∆ denotes the support radius of the
fixed function h(τ), so that h(τ) = 0 for τ ≤ αt0).
Lemma 10.16. For any smooth function with compact support f(k),
we have
Uα(t, t0)Φ(f |0)
= exp(irα(P|t)− irα(P|t0))(Φ(f |h(αt)) + αξ1(f |αt)
+ · · ·+ αnξn(f |αt) + αn+1ηn+1(f, α, t, t0)),
where ξi(f |σ) is defined by the equations
idξi−1(f |σ)
dσ= (H(h(σ))− ω(P|h(σ)))ξi(f |σ), (10.81)⟨
dξi(f |σ)
dσ,Φ(f ′|h(σ))
⟩= 0, (10.82)
ξ0(f |σ) = Φ(f |h(σ)), (10.83)
ξ0(f |σ0) = 0 for j > 0 (10.84)
and ‖ηn+1(f, α, t, t0)‖ ≤ C (where C does not depend on α, t, t0, but
may depend on f and n).
The proof of this lemma is based on arguments similar to the
arguments in Section 4.3. The vector Ψ(t) = Uα(t, t0)Φ(f |0) satisfies
the equation
idΨ
dt= H(h(αt))Ψ(t).
By using the change of variables σ = αt, this equation can be written
in the form
iαdΨ
dσ= H(h(σ))Ψ(σ). (10.85)
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266 Mathematical Foundations of Quantum Field Theory
We will look for solutions of (10.85) of the form
Ψ(σ) = exp
(− iα
∫ σ
σ0
ω(P|h(σ′))dσ′)
(ξ0(f |σ) + αξ1(f |σ)
+ · · ·+ αnξn(f |σ) + αn+1ηn+1(f, α, σ)).
Let us note that equation (10.81) specifies the vector ξi(f |σ) up
to a constant belonging to the kernel of the operator H(h(σ)) −ω(P|h(σ)) (i.e. the vector is specified up to a vector from the
single-particle subspace). Equations (10.82)–(10.84) fix this constant
vector uniquely. Further on (see Lemma 10.17), the vector ξi(f |σ) is
infinitely differentiable in σ.
The function ηn+1(f, α|σ) satisfies the equation
iαdηn+1(f, α|σ)
dσ= (H(h(σ))− ω(P|h(σ)))ηn+1(f, α|σ)
− idξn(f |σ)
dσ(10.86)
with initial condition ηn+1(f, α|σ0) = 0. From this equation and
the remark that∥∥dξn(f |σ)
dσ
∥∥ is bounded above by a constant, not
depending on σ, it is easy to obtain the inequality
‖ηn+1(f, α|σ)‖ ≤ C
α
(the constant C depends on f). Noting the fact that
ηn(f, α|σ) = ξn(f |σ) + αηn+1(f, α|σ),
we see that
‖ηn(f, α|σ)‖ ≤ const.
This finishes the proof of the lemma.
Lemma 10.17. The vector ξi(f |σ), entering the formulation of
Lemma 10.16, can be written in the form
ξi(f |σ) = Df,σi Φ,
where Df,σi is a family of operators, infinitely differentiable in σ in
the topology on A, hence, this vector is infinitely differentiable in σ.
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Axiomatic Scattering Theory 267
We will prove this lemma by induction. Suppose the operatorDf,σi−1
is given. Let us consider the vector ζi(f |σ), defined by the relations
idξi−1(f |σ)
dσ= (H(h(σ))− ω(P|h(σ)))ζi(f |σ),⟨
ζi(f |σ),Φ(f ′|h(σ))⟩
= 0,
where f , f ′ are arbitrary smooth functions with compact support.
This vector can be written in the form
ζi(f |σ) = γ(H(h(σ)),P|h(σ))dξi−1(f |σ)
dσ. (10.87)
Here,
γ(ω,p|g) = i(ω − ω(p|g))−1,
if p ∈ supp f , and (ω,p) belongs to the spectrum of the operators
H(g),P, but ω 6= ω(p|g); if ω = ω(p|g), then γ(ω,p|g) should be
equal to zero. The function γ(ω,p|g) can be chosen to be smooth
and equal to zero for large |p|.Using relation (10.87), we can write
ζi(f |σ) =
∫γ(t,x|h(σ))Ef,σi−1(x, t|σ)Φdxdt, (10.88)
where γ(t,x|g) is the Fourier transform of γ(ω,p|g) in the variables
ω,p and Ef,σi−1 = (d/dσ)Df,σi−1. Equation (10.88) implies that ζi(f |σ) =
F f,σi Φ, where F f,σi =∫γ(t,x|h(σ))Ef,σi−1(x, t|σ)dxdt (this operator is
infinitely differentiable in σ, by Lemma 10.11). The vector ξi(f |g)
can be expressed in terms of the vector ζi(f |σ) in the following way:
ξi(f |σ) = ζi(f |σ) + Φ(λσi f |h(σ)), (10.89)
where λσi (k) is a function satisfying the equation∫∂λσi (k)
∂σf(k)f ′(k)dk = −
⟨dζi(f |σ)
dσ,Φ(f ′|h(σ))
⟩(10.90)
with the initial condition λσ0i (k) = 0 for σ0 < −∆. By introduc-
ing the vector generalized functions ξi(k|σ) and ζi(k|σ) with the
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268 Mathematical Foundations of Quantum Field Theory
relations ξi(f |σ) =∫f(k)ξi(k|σ)dk, ζi(f |σ) =
∫f(k)ζi(k|σ)dk, the
formulas (10.89) and (10.90) can be written in the form
ξi(k|σ) = ζi(k|σ) + λσi (k)Φ(k|h(σ)),
∂λσi (k)
∂σδ(k− k′) = −
⟨∂ζi(k|σ)
∂σ,Φ(k′|h(σ))
⟩.
The function λσi (k) is infinitely differentiable in σ; this can be shown
using the ideas in the proof of Lemma 10.15. We can therefore select
an operator Df,σi of the form
Df,σi = F f,σi +B
λσi f
h(σ).
It is easy to see that the operator Df,σi is infinitely differentiable
in σ and, therefore, satisfies the conditions of the lemma (the
differentiability of the operator F f,σi has already been shown, while
operator Bλσi f
h(σ) can be written in the form
Bλσi f
h(σ) = Bµσi f
h(σ) =
∫µσi (x)Bf
h(σ)(x)dx,
where µσi (k) = β(k)λσi (k) and the symbol β(k) denotes a smooth
function with compact support, equal to 1 when k ∈ suppf).
Remark 10.5. For all σ in the interval [−δ, δ], where the function
h equals 1, we can show the following equations by induction:
dξi(f |σ)
dσ= 0; ζi(f |σ) = 0; F f,σi = 0;
∂λσi (k)
∂σ= 0; Df,σ
i = Bλσi f1 .
Let us define the functions φt,α(x), φt,α(k) by the formulas
φt,α(x) = (2π)−3
∫φt,α(k) exp(ikx)dk,
φt,α(k) = φ(k) exp(irα(k|t)).
Lemma 10.18. Let φ(k) be a smooth function with compact support.
For any whole number n and any ε > 0 we can find a number Cn,ε,
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Axiomatic Scattering Theory 269
not depending on x, t, and α, so that
|φt,α(x)| ≤ Cn,ε1 + (ρ(x, V t
α(φ))− ε|t|)n
(here, V tα(φ) is a set of points which can be written in the form
∇rα(k|t), where k ∈ suppφ, and ρ(x, A) is the distance between the
point x and the set A). There exists a constant C, not depending on
x, t, α, so that
|φt,α(x)| ≤ C|t|−3/2,∫|φt,α(x)|dx ≤ C(1 + |t|3/2). (10.91)
The proof is analogous to the proof of Lemma 10.2 in Section 10.1.
Lemma 10.19. If f(k), φ(k) are smooth functions with compact
support, then the vector
Γ(t) = Uα(t, t0)S−(0)Wα(t0)a+(f φ)θ,
where t0 ≤ −∆α , can be written in the form
Γ(t) = QtsΦ +Rs(t),
where
Qts = Qts(f, φ, α) =
∫φt,α(x)N t,α
s (x)dx, (10.92)
N t,αs =
s∑i=0
Df,αti ,
‖Rs(t)‖ ≤ Cs(f, φ)αs+1(1 + |t|3/2). (10.93)
Indeed,
Wα(i0)a+(f φ)θ = a+(f φt0,α)θ.
Therefore,
Γ(t) = Uα(t, t0)Φ(fφt0,α|0)
= Uα(t, t0)φt0,α(P)Φ(f |0)
= φt0,α(P)Uα(t, t0)Φ(f |0).
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270 Mathematical Foundations of Quantum Field Theory
Hence, we can conclude from Lemmas 10.16–10.18 that
Γ(t) = φt0,α(P) exp(irα(P|t)− irα(P|t0))
×
(s∑i=0
αiξi(f |αt) + αs+1ηs+1(f, α, t, t0)
)
= φt,α(P)
(s∑i=0
Df,αti Φ
)+Rs(t)
= φt,α(P)N t,αs Φ +Rs(t)
=
∫φt,α(x)N t,α
s (x)Φdx +Rs(t),
which proves the necessary statement [to obtain an estimate for Rs,
we should use an estimate for ‖ηs+1‖, used in Lemma 10.16, and
inequality (10.91)].
Remark 10.6. The relation
Uα(t′, t)Uα(t, t0) = Uα(t′, t0)
implies that
Uα(t′, t)Γ(t) = Γ(t′). (10.94)
Using equation (10.94) and Lemma 10.18, we can conclude that
Uα(t′, t)QtsΦ = Qt′s Φ +Rs(t, t
′), (10.95)
where ‖Rs(t, t′)‖ ≤ Cαs+1 and the constant C depends on f and φ
(but does not depend t, t′, α). We will use relation (10.95) later.
Lemma 10.20. Let φ1(k), φ2(k) be non-overlapping smooth func-
tions with compact support. Then there exists a number c > 0 such
that the distance between V tα(φ1) and V t
α(φ2) is larger than c|t|.
It follows from the strong convexity of ω(k|g) that
d2ω(k|g) ≥ λdk2 (10.96)
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Axiomatic Scattering Theory 271
(here, λ is positive and does not depend on k, g, if k runs over a
bounded set; the second differential is taken over only the variable k).
Inequality (10.96) implies that d2rα(k|t) ≥ λ|t|dk2, from which we
can conclude that the second derivative of rα(k|t), in any direction,
cannot be less than λ|t|. Indeed, we can set
e =k1 − k2
|k1 − k2|, ν(ζ) = rα(k2 + ζe|t)
and conclude that ν ′′(ζ) ≥ λ|t|. Hence, it is clear that
|∇rα(k1|t)−∇rα(k2|t)|
≥ |v′(|k1 − k2|)− v′(0)| ≥ λ|t||k1 − k2|.
Therefore, if the distance between the sets suppφ1 and suppφ2 is
larger than ε, then the distance between the sets V tα(φ1) and V t
α(φ2)
is larger than λ|t|ε. This finishes the proof of the lemma.
Lemma 10.21. Let f1(k), f2(k), φ1(k), φ2(k) be smooth functions
with compact support and suppose that the supports of the functions
φ1, φ2 do not intersect. Then
p([Uα(t+ τ, t)Q1,ts Uα(t, t+ τ), Q2,t
s ]) ≤ C|τ |r
1 + |t|n
[here, p(A) is an arbitrary seminorm on the algebra A; n is an
arbitrary number; −∞ < τ , t <∞, |τ | ≤ |t|; C and r do not depend
on t, τ , and α; the symbol Qi,ts denotes the operator
Qi,ts = Qts(fi, φi, α),
defined by the relation (10.92)].
The proof of this lemma follows directly from Lemmas 10.10,
10.18, and 10.20.
Lemma 10.22. Let φ1(k), . . . , φn(k) be smooth functions with
compact and non-intersecting supports, let f1(k), . . . , fn(k) be smooth
functions with compact support, and let t ≤ −aα−ε, t0 ≤ −∆α , a > 0,
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272 Mathematical Foundations of Quantum Field Theory
ε > 0, s ≥ 32n+ 1. Then the vector
Ψn(t) = Uα(t, t0)S−(0)Wα(t0)b+(f1φ1) . . . b+(fnφn)θ
can be written in the form
Ψn(t) = Q1,ts . . . Qn,ts Φ + π(t), (10.97)
where
Qi,ts = Qts(fi, φi, α),
‖π(t)‖ ≤ C|t− t0|α2,
and the number C does not depend on t, t0, and α.
It is easy to see that the vector Ψn(t) does not depend on t0, if t0is less than −∆/α, hence we may assume that t0 is the largest integer
obeying t0 < −∆/α. We will assume that t is an integer and we will
prove the relation (10.97) by induction on t; the generalization to
real-valued t does not present any obstacles.
Let us suppose that
Ψn(t) ≈ Q1,ts . . . Qn,ts Φ. (10.98)
Using equation (10.95), we can conclude that
Ψn(t+ 1) = Uα(t+ 1, t)Ψn(t)
≈ Uα(t+ 1, t)Q1,ts . . . Qn,ts Φ
= Uα(t+ 1, t)Q1,ts . . . Qn−1,t
s Uα(t, t+ 1)Uα(t+ 1, t)Qn,ts Φ
≈ Uα(t+ 1, t)Q1,ts . . . Qn−2,t
s Uα(t, t+ 1)Uα(t+ 1, t)
×Qn−1,ts Uα(t, t+ 1)Qn,t+1
s Φ (10.99)
An additional error, arising in the transition from approxima-
tion (10.98) to approximation (10.99) by (10.95), can be written in
the form
Uα(t+ 1, t)Q1,ts . . . Qn−1,t
s Uα(t, t+ 1)Rs(t, t+ 1),
March 27, 2020 15:20 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch10 page 273
Axiomatic Scattering Theory 273
where ‖Rs(t, t + 1)‖ ≤ Cαs+1(1 + |t|3/2) and any seminorm of
the operator Qi,ts on the A does not exceed C(1 + |t|3/2) [the
last statement follows from (10.91)]. Using these remarks and the
inequalities |t| ≤ |t0|, s ≥ 32n + 1, we can show that as we go
from (10.98) to (10.99) the error does not increase by more than
Cα2.
By Lemma 10.21, we can see that
Ψn(t+ 1) ≈ Uα(t+ 1, t)Q1,ts . . . Qn−2,t
s
×Uα(t, t+ 1)Qn,t+1s Uα(t+ 1, t)Qn−1,t
s Φ. (10.100)
Similarly, the additional error obtained from transferring to (10.100)
from (10.99) does not exceed Cα2 (moreover, this error does not
exceed Cαr, where r is arbitrary). To prove this, we should use, in
addition to Lemma 10.21, the inequality t < −aα−ε and the estimate
for Qi,ts shown above.
Using equation (10.95) again, we obtain
Ψn(t+ 1) ≈ Uα(t+ 1, t)Q1,ts
. . . Qn−2,ts Uα(t, t+ 1)Qn,t+1
s Qn−1,t+1s Φ.
Applying equation (10.95) several times and then applying
Lemma 10.21, we obtain
Ψn(t+ 1) ≈ Qn,t+1s . . . Q1,t+1
s Φ,
≈ Q1,t+1s . . . Qn,t+1
s Φ. (10.101)
The error arising from each transformation does not exceed Cα2.
Hence, we have derived (10.101) from (10.98) and have proven that
the error in (10.101) also does not exceed Cα2. The inductive step
is shown. Since the case t = t0 for equation (10.97) is obvious,
Lemma 10.22 is proven.
We can now finally easily show the main result of this section.
In the formulation of Lemma 10.22, let us take t = − δα , where δ
is such that h(τ) ≡ 1 for −δ ≤ τ ≤ δ. Then we can show that
φt,αi (x) = φi(x, t)
= (2π)−3
∫φi(k) exp(ikx− iω(k|1)t)dk, (10.102)
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274 Mathematical Foundations of Quantum Field Theory
Ψn(t) ≈n∏i=1
∫ φi(x, t)
s∑j=0
αiDfi,−δj (x)
dx
Φ,
Dfi,−δ0 = Bfi
1 ,
Dfi,−δj = B
λ−δj fi1 . (10.103)
The error in (10.102) does not exceed Cα2(− δα + ∆
α ) = const α and
inequality (10.103) follows from the remark to Lemma 10.17.
By the definition of the Møller matrix S− and the rela-
tions (10.102) and (10.103) we obtain
S−(1)b+(f1φ1) . . . b+(fnφn)θ
= limt→−∞
n∏i=1
(∫φi(x, t)B
fi1 (x, t|1)dx
)Φ
= limt→−∞
exp(iH(1)t)n∏i=1
(∫φi(x, t)B
fi1 (x)dx
)Φ
= limt→−∞
exp(−iH(1)t)
n∏i=1
(∫φi(x, t)D
fi,−δ0 (x)dx
)Φ
= limα→0
exp
(−iH(1)
δ
α
) n∏i=1
(∫φi
(x,− δ
α
)
×
s∑j=0
αjDfi,−δj (x)
dx
Φ
= limα→0
exp
(−iH(1)
δ
α
)Ψn
(− δα
)= lim
α→0Uα(0, t0)S−(0)Wα(t0)b+(f1φ1) . . . b+(fnφn)θ. (10.104)
To derive equality (10.73) from relation (10.104), it is enough to
recall that if limAnx = Ax for all x in a dense set and the
norms of the operators An are uniformly bounded, then slimAn =A.
Relation (10.74) follows analogously.
March 27, 2020 16:3 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch11 page 275
Chapter 11
Translation-Invariant Hamiltonians(Further Investigations)
11.1 Connections between the axiomatic theory
and the Hamiltonian formalism
Let H be a formal translation-invariant Hamiltonian of the
form (8.3). Let us suppose that there exists an operator realization
(H, H, P, a(k, ε, t),Φ) of the Hamiltonian H (recall that H is a
Hilbert space; H, P are the operators of energy and momentum;
a(k, ε, t) are operator generalized functions; Φ is a ground state).
Let us consider a family A consisting of two operator generalized
functions
a(x, 1, t) = (2π)−3/2
∫exp(ikx)a(k, 1, t)dk,
a(x,−1, t) = (2π)−3/2
∫exp(−ikx)a(k,−1, t)dk.
We will use the term scattering matrix of the formal Hamiltonian
H to refer to the scattering matrix, constructed with the operators
H, P, and the family A (here, we assume that the family A and the
operators H, P satisfy the conditions that guarantee the existence of
the scattering matrix; see Section 10.5).
The relation between this definition and the definition of the
scattering matrix provided in Chapter 9 will be addressed later.
Currently, we will note that in the case when the Hamiltonian H
satisfies the conditions in Section 9.2, then it follows from the results
of Section 10.3 that the scattering matrix constructed in Section 9.2
coincides with the above definition.
275
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276 Mathematical Foundations of Quantum Field Theory
Indeed, let λ(k) ∈ S(E3), f(k), and µ(τ) be smooth functions
with compact support and B =∫λ(k)µ(τ)a+(k, τ)dkdτ . It is clear
that ∫f(x|t)B(x, t)dx
=
∫f(x|t) exp(−ikx)λ(k)µ(τ)a+(k, t+ τ)dkdxdτ
=
∫f(k)λ(k)µ(τ)a+(k, t+ τ)dkdτ,
and, therefore, in the conditions of Section 9.2, we have
wlimt→−∞
∫f(x|t)B(x, t)dx =
∫f(k)λ(k)µ(τ)Λ
−1(k)a+
in(k, τ)dkdτ
=
∫f(k)λ(k)µ(ω(k))Λ
−1(k)a+
in(k)dk,
(11.1)
where a+in(k, τ), ain(k, τ) are in-operators, defined in Section 9.2;
µ(ω(k)) =∫µ(τ) exp(iω(k)τ)dτ .
Relations (9.18) and (9.21) imply that⟨BΦ, a+
in(p)Φ⟩
=
∫λ(k)µ(τ)
⟨a+(k, τ)Φ, a+
in(p)Φ⟩dkdτ
=
∫λ(k)µ(ω(k))
⟨a+(k)Φ, a+
in(p)Φ⟩dk
= λ(p)µ(ω(p))ρ(1,p) = λ(p)µ(ω(p))Λ−1
(p).
(11.2)
Comparing equations (11.1) and (11.2) with relation (10.37),
we see that the in-operators, defined in Section 9.2, coincide with
the in-operators defined in Section 10.3; an analogous proof can be
carried out for out-operators and for scattering matrices.
We should note, however, that the present definition of scattering
matrix can also be used in the presence of bound states. (We define
bound state as a single-particle state Φ(k) satisfying the condition
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Translation-Invariant Hamiltonians (Further Investigations) 277
〈Φ(k), a+(k′, t)Φ〉 = 0. Other single-particle states are called ele-
mentary. The construction of Section 9.2 works only for elementary
particles.)
The questions of the existence of an operator realization of the
formal Hamiltonian H and whether the operators H, P, and the
operator A satisfy the requirements in Section 10.5 turn out to be
non-trivial.
In this section, we will show that these questions can be answered
in the framework of perturbation theory; more precisely, we will
construct the relevant objects as formal power series in the coupling
constant.1
Let us consider the Hermitian and translation-invariant formal
expression
H = H0 + V
=
∫ε(k)a+(k)a(k)dk +
∑r≥1
gr∑m,n
∫v(r)m,n(k1, . . . ,km|p1, . . . ,pn)
× δ(k1 + · · ·+ km − p1 − · · · − pn)a+(k1) · · · a+(km)a(p1)
· · · a(pn)dmkdnp. (11.3)
a+(k), a(k) are, as usual, symbols satisfying CCR. We will assume
that ε(k) is a smooth function whose derivatives are of moderate
growth, the functions v(r)m,n belong to the space S and for every r, only
a finite number of the functions v(r)m,n are non-zero. The number g will
be called the coupling constant. If the sum in (11.3) is infinite, then
we will consider it simply as a formal power series without assuming
1In other words, we will consider formal expressions∑
nAngn, where An are
numbers, vectors, or operators; if An are operators, we will assume that theyare all defined on the same domain. The sums and products of two (numberor operator) formal series
∑nAng
n and∑
nBngn are understood as the series∑
n(An + Bn)gn and∑
n(∑
k+l=nAkBl)gn. The rest of this section will be
considered in the framework of perturbation theory (without always stating so).In order to give precise mathematical meaning to the results in this section andtheir proofs, all the relations should be understood in terms of formal power series.
March 27, 2020 16:3 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch11 page 278
278 Mathematical Foundations of Quantum Field Theory
anything about its convergence. The set of all formal expressions
having these properties will be denoted by M.
An expression H ∈ M can be seen as a formal translation-
invariant Hamiltonian (the most interesting are the Hamiltonians
in the set M that can be written in the form H = H0 + gW , where
W does not already contain the coupling constant g; however, it will
be convenient to consider the general case).
An operator realization of a Hamiltonian H ∈ M will be defined
to be a Hilbert space H with the action of the energy operator H =∑Hrg
r and the momentum operator P, the operator generalized
functions a(k, ε, t) =∑grar(k, ε, t) and the vector Φ, satisfying the
operator realization conditions in every order with respect to g (the
series∑Hrg
r and∑grar(k, ε, t) are to be understood as formal
power series in g without convergence assumptions). For the operator
H =∑grHr, we can construct the operators exp(iHt) =
∑grUr(t)
which constitute a group of unitary operators.2 The operators Ur(t)
and the operators∫f(k)ar(k, ε, t)dk must be defined on the same
domain D and must transform it to itself; the set D must contain
the vector Φ.
In Section 11.3, we will show that for every Hamiltonian H ∈M,
we can construct an operator realization in the above sense (for
Hamiltonians that generate vacuum polarization, we will assume
that the function ε(k), entering in the definition of the Hamiltonian
H0, satisfies the condition ε(k1 + k2) < ε(k1) + ε(k2), however this
requirement is not necessary).
Any translation-invariant Hamiltonian H = H0 + W generates
a family of Hamiltonians H0 + gW , depending on the parameter g.
If the Hamiltonian H0 +W satisfies the conditions which guarantee
that the Heisenberg equations (8.5) make sense (these conditions
are described in Section 8.1), then H0 + gW ∈ M; therefore, for
2The operators∑grUr(t) constitute a group of unitary operators if∑
k+l=r
Uk(t)Ul(t′) = Ur(t+ t′),
Uk(0) = 0 for k > 0,
U0(t) = 1.
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Translation-Invariant Hamiltonians (Further Investigations) 279
these Hamiltonians, an operator realization can be constructed in
the framework of perturbation theory.
In Section 11.3, we will show that the operators H =∑grHr, P
and the family of operator generalized functions consisting of two
functions a(x, ε, t) =∑grar(x, ε, t) =
∑gr∫
exp(iεkx)ar(k, ε, t)dk,
ε = ±1, satisfy the conditions of Section 10.5 in every order.
The proof uses the additional assumption that the Hamiltonian in
consideration belongs to the subset M of M, whose Hamiltonians
obey the condition that their functions ε(k) satisfy ε(k1 + k2) <
ε(k1) + ε(k2) and are not linear on any open subset.
It follows that the Møller matrices and the scattering matrices
exist in the framework of perturbation theory (more precisely, the
Møller matrices S± and the scattering matrix S can be written as
formal power series S± =∑grS
(r)± , S =
∑grS(r)). To derive this, it
is necessary to analyze the construction of Møller matrices, given in
Chapter 10, and check that the statements proven in this chapter
hold for formal power series in g as well. To see this, note that
we must modify the Møller matrix construction by assuming that
the functions f1(k), . . . , fn(k), entering the formula (10.1), form a
non-overlapping family of functions3 (for details, see Section 11.5).
11.2 Heisenberg equations and canonical
transformations
Let us consider a formal translation-invariant Hamiltonian H,
belonging to the class M described in Section 11.1. As noted
in Section 8.1, if v(r)m,0 6≡ 0 (the Hamiltonian generates vacuum
polarization), then the expression H does not define a self-adjoint
operator on Fock space. Furthermore, the operator exp(iHt) cannot
be constructed in this case even in the framework of perturbation
theory (in terms of a formal power series in g).
This can be easily shown by calculating the operator S(t, t0) =
exp(iH0t) exp(−iH(t − t0)) exp(−iH0t0) by means of the diagram
techniques developed in Section 6.4. For example, in the expression
3The same modification of the Møller matrix definition is used for a differentpurpose in Section 10.5, part 5.
March 27, 2020 16:3 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch11 page 280
280 Mathematical Foundations of Quantum Field Theory
Fig. 11.1 Divergent diagram.
corresponding to the diagram depicted in Fig. 11.1, we run into the
meaningless expression
δ(k1 + · · ·+ km)δ(k1 − km+1) · · · δ(km − k2m)δ(km+1 + · · ·+ k2m)
= δ2(k1 + · · ·+ km)δ(k1 − km+1) · · · δ(km − k2m).
A similar situation arises for all diagrams without outer vertices (for
vacuum loops): in the expression corresponding to the diagram, the
integrand contains a product of δ-functions whose arguments are
linearly dependent, rendering the expression meaningless.
Therefore, we cannot write a solution to the Schrodinger equation
idψdt = Hψ even in perturbation theory.
However, using the expression (11.3), we can formally write
the Heisenberg equations which remain meaningful in perturbation
theory.
Indeed, the Heisenberg equations,
∂a(k, t)
∂t= i[H, a(k, t)], (11.4)
∂a+(k, t)
∂t= i[H, a+(k, t)], (11.5)
can be written in the form
i∂a(k, t)
∂t− ε(k)a(k, t) = I(k, t), (11.6)
−i∂a+(k, t)
∂t− ε(k)a+(k, t) = I+(k, t), (11.7)
March 27, 2020 16:3 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch11 page 281
Translation-Invariant Hamiltonians (Further Investigations) 281
where
I(k, t) =∑r≥1
gr∑m,n
m
∫v(r)m,n(k,k1, . . . ,km−1|p1, . . . ,pn) (11.8)
× δ
k +
m−1∑i=1
ki −n∑j=1
pj
a+(k1, t) (11.9)
· · · a+(km−1, t)a(p1, t) · · · a(pn, t)dm−1kdnp. (11.10)
(These equations can be obtained by calculating the commutator
on the right-hand side of the equations (11.4) and (11.5) by using
CCR, distributivity, and the relation [AB,C] = [A,C]B +A[B,C].)
By using the initial conditions a+(k, 0) = a+(k), a(k, 0) = a(k),
equations (11.6) and (11.7) can be written in the form
a(k, t) = exp(−iε(k)t)a(k)
+
∫ t
0exp(−iε(k)(t− τ))I(k, τ)dτ, (11.11)
a+(k, t) = exp(iε(k)t)a+(k)
+
∫ t
0exp(iε(k)(t− τ))I+(k, τ)dτ. (11.12)
We then look for solutions to equations (11.11) and (11.12) in the
form
a(k, t) = exp(−iε(k)t)a(k)
+∑r≥1
gr∑m,n
∫f r,tm,n(k1, . . . ,km|p1, . . . ,pn)
× δ
k +
m∑i=1
ki −n∑j=1
pj
a+(k1) · · · a+(km)a(p1)
· · · a(pn)dmkdnp, (11.13)
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282 Mathematical Foundations of Quantum Field Theory
a+(k, t) = exp(iε(k)t)a+(k)
+∑r≥1
gr∑m,n
∫f r,tm,n(k1, . . . ,km|p1, . . . ,pn)
× δ
k +m∑i=1
ki −n∑j=1
pj
a+(p1) · · · a+(pn)a(k1)
· · · a(kn)dmkdnp. (11.14)
Substituting these expressions for a(k, t), a+(k, t) into (11.11) and
(11.12) and equating terms having the same order in g, we obtain a
recurrence relation for f r,tm,n, from which it is clear that the functions
f r,sm,n belong to the space S and for every order of g (i.e. for every r),
only a finite number of functions f r,tm,n are non-zero.
(Here, we view a(k, t) as formal power series consisting of the
symbols a(k), a+(k), obeying CCR; all expressions arising in the
solution can be converted to normal form using CCR. The series
in g are viewed formally.)
It is easy to check that ‖f r,tm,n‖α,β does not grow faster than a
power of t, i.e.
‖f r,tm,n‖α,β ≤ C(1 + |t|k) (11.15)
(here, ‖ · ‖α,β is a norm on the space S; C and k depend on r, α, β).
Inequality (11.15) follows from the following statement.
Theorem 11.1. Let σ(k1, . . . , kn) be a smooth real function whose
derivatives do not grow faster than a power. Then for any norm
‖ · ‖α,β on the space S, we can find a norm ‖ · ‖γ,δ on the space Sand numbers C, r such that for any function φ ∈ S
‖ exp(iσ(k1, . . . , kn)t)φ(k1, . . . , kn)‖α,β≤ C(1 + |t|r)‖φ(k1, . . . , kn)‖γ,δ. (11.16)
Indeed, using inequality (11.16), we see that the norm of the
function
gt(k1, . . . , kn) =
∫ t
0exp(iσ(k1, . . . , kn)(t− τ))hτ (k1, . . . , kn)dτ
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Translation-Invariant Hamiltonians (Further Investigations) 283
satisfies the inequality
‖gt‖α,β ≤ C|t|(1 + |t|r) sup0≤τ≤t
‖hτ‖γ,δ,
which implies inequality (11.15) by induction. It is easy to prove that
for fixed t, the expression a+(k, t), a(k, t) satisfies CCR:
[a(k, t), a(k′, t)] = [a+(k, t), a+(k′, t)] = 0,
[a(k, t), a+(k′, t)] = δ(k− k′) (11.17)
(it is important to note that the calculations of the commutators
in (11.17) do not require limits: all the involved series depend
on a finite number of convergent integrals). In order to check
relation (11.17), we can differentiate them in t, for example,
∂
∂t[a(k, t), a+(k′, t)] = i[[H, a(k, t)], a+(k′, t)]
+ i[a(k, t), [H, a+(k′, t)]]
= i[H, [a(k, t), a+(k′, t)]],
which, with the initial condition [a(k, 0), a+(k′, 0)] = [a(k), a+(k′)] =
δ(k− k′), allows us to conclude that [a(k, t), a+(k′, t)] = δ(k− k′).
We will say that the formulas
b(k) = α(k)a(k) +∑r≥1
gr∑m,n
∫h(r)m,n(k1, . . . ,km|p1, . . . ,pn)
× δ
k +m∑i=1
ki −n∑j=1
pj
a+(k1)
· · · a+(km)a(p1) · · · a(pn)dmkdnp, (11.18)
b+(k) = α(k)a+(k) (11.19)
+∑r≥1
gr∑m,n
∫h
(r)m,n(k1, . . . ,km|p1, . . . ,pn)
× δ(k +
m∑i=1
ki −n∑j=1
pj)a+(p1) · · · a+(pn)
× a(k1) · · · a(km)dmkdnp, (11.20)
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284 Mathematical Foundations of Quantum Field Theory
where α(k) is a smooth function whose derivatives do not grow faster
than a power of |k|, h(r)m,n ∈ S, and for every r, all but a finite number
of the functions h(r)m,n are zero, specify a canonical transformation if
the following relations are satisfied:
[b(k), b(k′)] = [b+(k), b+(k′)] = 0,
[b(k), b+(k′)] = δ(k− k′).
Roughly speaking, the canonical transformations turn the sym-
bols a+(k), a(k), satisfying CCR, into new symbols b+(k), b(k), also
satisfying CCR.
It is easy to prove that the set of canonical transformations
constitutes a group. This means that the composition of two canonical
transformations λ and µ produces a new canonical transformation λµ
and that the inverse of a canonical transformation is also a canonical
transformation.
It was shown above that the transformation from a+(k), a(k) to
the Heisenberg operators a+(k, t), a(k, t) is a canonical transforma-
tion; it therefore follows that any expression, H ∈ M defines a one-
parameter family of canonical transformations; we will denote these
transformations by σtH .
It is useful to note that the canonical transformation σtH can be
obtained as a limit of canonical transformations in finite volume.
Namely, using the HamiltonianHΩ, obtained as a finite-volume cutoff
of H, we can introduce the operators ak(t), a+k (t), where k ∈ TΩ, by
the formulas
ak(t) = exp(iHΩt)ak exp(−iHΩt),
a+k (t) = exp(iHΩt)a
+k exp(−iHΩt).
The operators a+k (t), ak(t), for fixed t, clearly satisfy CCR; the trans-
formation from the operators a+k , ak to the operators a+
k (t), ak(t)
specifies a canonical transformation in the volume Ω. Let us write
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Translation-Invariant Hamiltonians (Further Investigations) 285
the operators a+k (t), ak(t) in normal form:
ak(t) = exp(−iε(k)t)ak
+∑r≥1
gr∑
ki∈TΩpj∈TΩ
Ωf r,tm,n(k1, . . . ,km|p1, . . . ,pn)
× δk1+···+kmp1+···+pn
(2π
L
) 32
(m+n−2)
a+k1· · · a+
kmap1 · · · apn ,
a+k (t) = exp(iε(k)t)a+
k
+∑r≥1
gr∑
ki∈TΩpj∈TΩ
Ωfr,tm,n(k1, . . . ,km|p1, . . . ,pn)
× δk1+···+kmp1+···+pn
(2π
L
) 32
(m+n−2)
a+p1· · · a+
pnak1 · · · akm .
For the functions Ωf r,tm,n, just as for the functions f r,tm,n, we can derive
recurrent formulas from the Heisenberg equations. It is easy to check
that the recurrent formulas admit the limit Ω → ∞, and therefore,
it follows that
limΩ→∞
Ωf r,tm,n = f r,tm,n.
This equality is what we meant by the statement that the canonical
transformation σtH can be obtained as a limit of canonical transfor-
mations in finite volume.
Let the Hamiltonian H ∈ M be given by the formula (11.3) and
the canonical transformation λ by the relations (11.18) and (11.20).
Let us further replace in H the symbols a+(k), a(k) by the
expressions b+(k), b(k), defined by the formulas (11.18) and (11.20),
and bring the resulting formal expression to normal form using CCR,
while discarding infinite constants.
It is easy to check that we will obtain an expression H ∈ M; we
will say that the Hamiltonian H is obtained from the Hamiltonian
H by the canonical transformation λ, and write H = λ(H).
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286 Mathematical Foundations of Quantum Field Theory
Let us note that if the canonical transformation λ can be written
in the form λ = σtA, where A ∈M, then
H = λ(H) = limΩ→∞
λΩ(H),
where
λΩ(H) = exp(itAΩ)HΩ exp(−itAΩ)− cΩ
(the limit should be understood in the sense of the limits of the
coefficients of the functions in normal form of the operator λΩ(H) to
the coefficients of the functions in the expression λ(H)). The proof
of this follows from the fact that, as stated above, the canonical
transformation σtA can be obtained as a limit of canonical trans-
formations in finite volume; the constant cΩ arises from discarding
infinite constants after the canonical transformation; we can assume
that
cΩ = 〈exp(itAΩ)HΩ exp(−itAΩ)θ, θ〉 .
If we formally calculate the commutator [A,B] of two expressions
A,B ∈ M and multiply it by i, then we obtain a new expression
belonging to M. It is easy to check that the set M, equipped with
the operation i[A,B] and the obvious structure of vector space, is a
Lie algebra. The group of canonical transformations G can be seen
as an infinite-dimensional Lie group. It is possible to show that Mis a Lie algebra of the group G. The canonical transformations σtHclearly constitute a one-parameter subgroup of G, corresponding to
the element H of the Lie algebra M; the map taking the element
H ∈ M to the element σ1H is the exponential map from M to G.
Each element of the Lie group defines an automorphism of the
corresponding Lie algebra; in our case of interest, the canonical
transformation λ corresponds to the automorphism of the Lie algebra
M that maps the formal expression H to the formal expression λ(H).
11.3 Construction of an operator realization
Let us consider a Hamiltonian H =∑Hrg
r ∈ M that does not
generate vacuum polarization. The formal expressions Hr therefore
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Translation-Invariant Hamiltonians (Further Investigations) 287
define Hermitian operators Hr on Fock space with domain S∞ (see
Section 3.2, where it is shown that in the case of vacuum polarization,
this statement ceases to be true). In the framework of perturbation
theory, we can construct the operator exp(iHt) =∑grUr(t),
where the operators Ur(t) are defined on S∞ and transform S∞into itself. This statement can be proved by means of diagram
techniques described in Section 6.4; roughly speaking, the proof
depends on the remark that in the case at hand, all the diagrams
represent mathematically sensible expressions. (Before constructing
a perturbation theory series for exp(iHt), it is necessary to express
the Hamiltonian H in the form H ′0 + V ′, where
H ′0 = H0 +∑r≥1
gr∫v
(r)1,1(k)a+(k)a(k)dk,
and use the diagram techniques for the operator S(0, t) =
exp(−iH ′0t) exp(iHt).) Another proof is laid out in Section 11.5.
The operator generalized functions a(k, ε, t) =∑grar(k, ε, t) are
defined by the formula
a(k, ε, t) = exp(iHt)a(k, ε) exp(−iHt),
where a(k, 1) = a+(k), a(k,−1) = a(k) are the operators of
annihilation and creation in Fock space. Since the operators Ur(t)
and∫f(k)a(k, ε)dk transform the set S∞ into itself, the operators∫f(k)ar(k, ε, t)dk =
r∑s=0
Us(t)
(∫f(k)a(k, ε)dk
)Ur−s(−t)
inherit the property for any function f ∈ S.
It is clear that the energy operator H =∑Hrg
r, the momentum
operator P =∫ka+(k)a(k)dk, the operator generalized functions
a(k, ε, t) =∑grar(k, ε, t), and the vector Φ = θ constitute an
operator realization of the Hamiltonian H ∈M.
To construct an operator realization for a Hamiltonian generating
vacuum polarization, we will use the following statement.
Theorem 11.2. Let λ be a canonical transformation and let H =
λ(H) ∈ M be the Hamiltonian obtained from the Hamiltonian H
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288 Mathematical Foundations of Quantum Field Theory
by the canonical transformation λ. Then the operator realization of
the Hamiltonian H can be obtained from the operator realization
(H, H, P, a(k, ε, t),Φ) of the Hamiltonian H, by keeping the space
H, the operators of energy and momentum H, P, and the vector Φ
the same and replacing the operator generalized functions a(k, ε, t)
by the operator generalized functions
a(k,−1, t) = a(k, t) = α(k)a(k, t)
+∑r≥1
gr∑m,n
∫h(r)m,n(k1, . . . ,km|p1, . . . ,pn)
× δ
m∑i=1
ki −n∑j=1
pj − k
a+(k1, t)
· · · a+(km, t)a(p1, t) · · · a(pn, t)dmkdnp, (11.21)
a(k, 1, t) = a+(k, t) = α(k)a(k, t)
+∑r≥1
gr∑m,n
∫h
(r)m,n(k1, . . . ,km|p1, . . . ,pn)
× δ
m∑i=1
ki −n∑j=1
pj − k
a+(p1, t)
· · · a+(pn, t)a(k1, t) · · · a(kn, t)dmkdnp (11.22)
[we assume that the canonical transformation λ is given by the
formulas (11.18) and (11.20)].
To prove the formulated statement, let us note that the
only dependence on the concrete form of the translation-invariant
Hamiltonian in the definition of an operator realization is in the
condition that the operator generalized function a(k, ε, t) satisfies
the Heisenberg equations.
In order to check that this condition is satisfied, it is necessary to
differentiate equation (11.21) with respect to t, using the fact that
∂a(k, ε, t)
∂t= i[H, a(k, ε, t)]
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Translation-Invariant Hamiltonians (Further Investigations) 289
by the definition of the operator realization of the Hamiltonian H.
Expressing a in terms of a by the canonical transformation λ−1, we
obtain an equation for a(k, ε, t); simple but cumbersome calculations
allow us to bring the resulting equations to the form
∂a(k, ε, t)
∂t= i[H, a(k, ε, t)].
We will further consider the Hamiltonian H = H0 + V ∈ M,
assuming that the function ε(k), entering the definition of the
Hamiltonian H0, satisfies the inequality ε(k1 + k2) < ε(k1) + ε(k2).
With the help of the statement proven above, we can easily convert
the construction of the operator realization of this Hamiltonian to
the already-solved problem of construction of the operator realization
of a Hamiltonian that does not generate vacuum polarization. We
should use the following statement proven (in different terms) in
Section 9.4.
Theorem 11.3. Let the Hamiltonian H = H0 + V ∈ M be given
and suppose the function ε(k) satisfies the inequality ε(k1 + k2) <
ε(k1) + ε(k2). Then there exists a canonical transformation λ such
that the Hamiltonian H = λ(H) ∈M can be represented in the form
H = H0 + V , (11.23)
where
V =∑r
gr∑m,n
∫v(r)m,n(k1, . . . ,km|p1, . . . ,pn)
× δ
m∑i=1
ki −n∑j=1
pj
a+(k1) · · · a+(km)a(p1)
· · · a(pn)dmkdnp,
V does not contain summands of type (m, 0) or summands of type
(m′, 1) with m′ 6= 1(
i.e. v(r)m,0 = 0, v
(r)m′,1 = 0 for m′ 6= 1
).
The transformation λ was previously called a Faddeev transfor-
mation; let us recall that it was constructed in the form σ1A, where
A ∈M.
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290 Mathematical Foundations of Quantum Field Theory
One can say that the Faddeev canonical transformation transforms
the Hamiltonian H into the Hamiltonian H, for which the bare
vacuum coincides with the dressed vacuum and the bare single-
particle state coincides with the dressed single-particle state.
Let us now show that any two operator realizations of a
Hamiltonian H ∈ M are unitarily equivalent. With the help of
Faddeev transformation, we can reduce the problem to the case
when the Hamiltonian does not generate vacuum polarization [recall
that above we also assumed the condition ε(k1 + k2) < ε(k1) +
ε(k2)]. Indeed, from two unitarily equivalent operator realizations
of the Hamiltonian H by means of the Faddeev transformation,
we can obtain two unitarily equivalent operator realizations of the
Hamiltonian not generating vacuum polarization.
In order to prove the uniqueness of the operator realization in
the case when the Hamiltonian H ∈ M does not generate vacuum
polarization, it is enough to check that for any operator realiza-
tion (H′, H ′,P′, a′(k, ε, t),Φ′) of the Hamiltonian H, the relation
a′(k,−1, t)Φ′ = 0 is satisfied (then the unitary operator U estab-
lishing the unitary equivalence of this operator realization and the
operator realization constructed above can be defined as the operator
satisfying the condition Ua(k, ε, 0) = a′(k, ε, 0)U,UΦ = Φ′).4 Let
us now check the relation a′(k,−1, t)Φ′ = (∑grar(k,−1, t))Φ′ = 0.
Consider the equations for a′r(k,−1, t) following from the Heisenberg
equations. It is easy to check that they have the form
i∂a′r(k,−1, t)
∂t− ε(k)a′r(k,−1, t) = Fr, (11.24)
where Fr is an expression composed of a′s(k, 1, t) and a′s(k,−1, t)
with s < r. In order to prove the necessary relation, we should show
that a′r(k,−1, t)Φ′ = 0; we will perform the proof by induction in r.
4The existence of the operator U satisfying these conditions follows from theresults of Section 6.1 (more precisely, from their analogs concerning the unitaryequivalence of the Fock representations of CCR, written in the form of a formalexpression in powers of g). Using the Heisenberg equations, we can show thatUa(k, ε, t) = a′(k, ε, t)U . From the condition (b) in the definition of operatorrealization, we obtain that UH = H ′U,UP = P′U .
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Translation-Invariant Hamiltonians (Further Investigations) 291
Suppose that a′s(k,−1, t)Φ′ = 0 for s < r. By the absence of vacuum
polarization, every summand in the expression Fr will contain at
least one operator of the form as(k,−1, t) and therefore FrΦ′ = 0.
Using equation (11.24) on the vector Φ, we can see that
i∂
∂ta′r(k,−1, t)Φ′ = ε(k)a′r(k,−1, t)Φ′,
from which it follows that
a′r(k,−1, t)Φ′ = exp(−iε(k)t)a′r(k,−1, 0)Φ′.
Let us now calculate the expectation value of the energy in the state∫f(k)a′(k,−1, t)dkΦ′:⟨H ′∫f(k)a′(k,−1, t)dkΦ′,
∫f(k)a′(k,−1, t)dkΦ′
⟩=
⟨1
i
∫f(k)
∂
∂ta′(k,−1, t)dkΦ′,
∫f(k)a′(k,−1, t)dkΦ′
⟩=∑s≥r
1
i
∫f(k)f(k′)
⟨∂
∂ta′s(k,−1, t)Φ′, a′s′(k
′,−1, t)Φ′⟩dkdk′
= C2rg2r + C2r+1g
2r+1 + · · · ,
where
C2r =1
i
∫f(k)f(k′)
⟨∂
∂ta′r(k, 1, t)Φ
′, a′r(k′,−1, t)Φ′
⟩dkdk′
= −∫ε(k)f(k)f(k′) exp(−i(ε(k)− ε(k′))t)
×⟨a′r(k,−1, 0)Φ′, a′r(k
′,−1, 0)Φ′⟩dkdk′.
Since ⟨a′r(k,−1, 0)Φ′, a′r(k
′,−1, 0)Φ′⟩
= σ(k)δ(k− k′),
where σ(k) ≥ 0, we see that
C2r = −∫ε(k)|f(k)|2σ(k)dk ≤ 0.
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292 Mathematical Foundations of Quantum Field Theory
If a′r(k,−1, t)Φ′ 6≡ 0, then σ(k) 6≡ 0, and therefore, we can find a
function f(k) such that C2n < 0. This contradicts the non-negativity
assumption of the operator H ′.
Let us now check the applicability conditions of axiomatic field
theory in each power of g. Let us first consider the spectrum of the
operators H, P in the operator realization (H, H, P, a(k, ε, t),Φ) of
a Hamiltonian of the form (11.23). For this Hamiltonian, the (bare)
Fock vacuum θ and the bare single-particle state a+(k)θ satisfy the
conditions
Hθ = Pθ = 0,
Ha+(k)θ =(ε(k) +
∑grv
(r)11 (k)
)a+(k)θ,
Pa+(k)θ = ka+(k)θ.
Therefore, θ can be viewed as the ground state of the operator H
(dressed vacuum) and a+(k)θ can be viewed as a single-particle state
of operators H, P (dressed single-particle state). The joint spectrum
of the operators H, P consists of the point 0, corresponding to the
ground state, the set of points (ω(k),k) = (ε(k) +∑grvr1,1(k),k)
(single-particle spectrum), and the multi-particle spectrum, whose
points have the form (ω+∑
r≥1 grωr,k+
∑r≥1 g
rkr), where the point
(ω,k) belongs to the multi-particle spectrum of the operators H0, P
(i.e. can be presented in the form (ε(k1)+ · · ·+ε(km),k1 + · · ·+km)).
If the Hamiltonian H belongs to the class M [i.e. the function ε(k)
is not linear on any open set and satisfies the condition ε(k1 +k2) <
ε(k1) + ε(k2)], then the spectrum of the operators H, P satisfies the
conditions formulated in Section 10.5.
In order to finish the exposition of the case at hand, it remains
to be checked that the family B of operator generalized functions,
consisting of two functions
a(x, ε, t) = (2π)−3/2
∫exp(iεkx)a(k, ε, t)dk, ε = ±1,
is asymptotically commutative, in the sense of Section 10.5, in every
order of perturbation theory with respect to g. In order to show that
B is asymptotically commutative, let us note that the operators of
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Translation-Invariant Hamiltonians (Further Investigations) 293
the form∫f(x)a(x, ε, t)dx that we have considered belong to the
algebra A, consisting of operators of the form∑gr∑m,n
∫f (r)m,n(k1, . . . ,km|p1, . . . ,pn)a+(k1)
· · · a+(km)a(p1) · · · a(pn)dmkdnp,
where the functions f(r)m,n ∈ S and for every r, only finitely many
functions f(r)m,n are non-zero. It is easy to check (using the results
of Section 11.2, specifically inequality (11.15)) that the algebra Ais asymptotically commutative in every order in g. (This statement
also follows from the results of Section 11.5.)
Therefore, the family B is asymptotically commutative.
Let us return to considering an arbitrary Hamiltonian H =
H0 + V ∈ M. Using the operator realization of this Hamiltonian
constructed by means of Faddeev transformation, we can obtain
an operator realization of the Hamiltonian H of the form (11.23).
In these constructions, the energy and momentum operators in the
operator realization of the Hamiltonian H coincide with the energy
and momentum operators in the operator realization of the Hamilto-
nian H; this remark implies that they satisfy the necessary spectrum
conditions. Operator generalized functions a(k, ε, t), entering in
the operator realization of the Hamiltonian H, are related to the
operator generalized functions a(k, ε, t) in the operator realization
of the Hamiltonian H through the canonical transformation (see
the relation (11.21)). It therefore follows that the family B of two
operator generalized functions a(x, ε, t), ε = ±1, with which we
may construct the scattering matrix for the Hamiltonian H, is an
asymptotically commutative algebra in every order of g, and there-
fore the operators∫f(x)a(x, ε, t)dx belong to the asymptotically
commutative family A.
These statements imply that the scattering matrix of the Hamil-
tonian H ∈ M can be constructed via axiomatic scattering theory in
terms of formal series in g.
It is now possible to apply axiomatic scattering theory to study
(in the framework of perturbation theory) scattering matrices in the
Hamiltonian formalism.
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294 Mathematical Foundations of Quantum Field Theory
First, let us note that the discussions imply that the scattering
matrix of the Hamiltonian H coincides with the scattering matrix of
the Hamiltonian H, obtained from H by means of Faddeev canonical
transformation.5 Indeed, these two scattering matrices are built on
the asymptotically commutative families B and B, which are both
contained in the same asymptotically commutative algebra A; by
the results of Chapter 10, the Møller and scattering matrices based
on these families are identical.
The statement we have just proved implies the following equiva-
lence theorem.
If the Hamiltonians H1 ∈ M and H2 ∈ M are related by a
canonical transformation, then their Møller and scattering matrices
coincide.
Indeed, let α be a canonical transformation transforming the
Hamiltonian H1 to the Hamiltonian H2; let β be the Faddeev
canonical transformation transforming the Hamiltonian H2 to the
Hamiltonian H of the form (11.23). Then the canonical transforma-
tion αβ transforms the Hamiltonian H1 to the Hamiltonian H.
The scattering matrices of the Hamiltonians H1 and H2 above
coincide with the scattering matrices of the Hamiltonian H and,
therefore, coincide with each other. The same reasoning applies to
Møller matrices.
11.4 Dressing operators for translation-invariant
Hamiltonians
In this section, we will consider translation-invariant Hamiltonians
from class M.
Let us first examine the case where the Hamiltonian H ∈ Mdoes not generate vacuum polarization. The dressing operator for
5It is important to note that in this statement in place of the Faddeevtransformation, we can select any canonical transformation transforming theHamiltonian H into a Hamiltonian of the form (11.23), not necessarily theconcrete transformation constructed in Section 9.4.
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Translation-Invariant Hamiltonians (Further Investigations) 295
this Hamiltonian is defined as an operator satisfying the condition
S± = slimt→±∞
exp(iHt)D exp(−iHast). (11.25)
(The Hamiltonian in consideration defines the operator H on Fock
space H = F (L2(E3))), the asymptotic space Has = F (L2(E3)) is
naturally identified with the space H, therefore, the Møller matrices
S± act on the space H. The symbol Has denotes the asymptotic
Hamiltonian which we also view as an operator on H:
Has =
∫ω(k)a+(k)a(k)dk,
where ω(k) is the energy of the single-particle state of the Hamilto-
nian H. All operators in the formula (11.25), just as for the rest of
this section, are viewed as formal series in g. The limit in (11.25) is
understood as a strong limit in each perturbation order on the set of
linear combinations of vectors of the form a+(f1) · · · a+(fn)θ, where
f1(k), . . . , fn(k) are non-overlapping smooth functions with compact
support.)
The following theorem describes a broad class of dressing
operators for Hamiltonians H ∈ M that do not generate vacuum
polarization.
Theorem 11.4. Let D be an operator on Fock space H = F (L2(E3)),
satisfying the following conditions:
(1) Dθ = θ; (11.26)
(2) the vector generalized function Da+(k)θ is δ-normalized and is
an eigenvector for the operator H;
(3) the operator D can be represented as a product of operators of
the form expA or N(expA), where A = A1 + iA2, A1 ∈ M,
A2 ∈M.
Then the operator D is a dressing operator for the Hamiltonian H.
(The symbol N(expA) denotes the normal exponent; see Section 6.3.)
This theorem can be proved with the help of description of
dressing operators in axiomatic scattering theory (see Section 10.4).
In the following section, we present a proof of this theorem in the
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296 Mathematical Foundations of Quantum Field Theory
case when the operator D is represented in the form of a product of
operators of the form expA, where A ∈ M. In the section at hand,
however, we will provide a different proof of the formulated theorem.
To begin, let us consider the case where D = N(expA), where
A =∑m
∫Am,1(k1, . . . ,km)δ(k1 + · · ·+ km − p)
× a+(k1) · · · a+(km)a(p)dk1 · · · dkmdp.
Let us first note that in this case, we have the relation
[D, a+(k)] = B(k)D, (11.27)
where B(k) is a generalized operator function defined by the formula
B(k) =∑m
∫Am,1(k1, . . . ,km)δ(k− k1 − · · · − km)
× a+(k1) · · · a+(km)dmk
(the relation (11.27) follows immediately from the equality
[N(An), a+(k)] = B(k) · nN(An−1),
which we obtained using the formula
[R1 · · ·Rm, R] =∑
1≤i≤mR1 · · ·Ri−1[Ri, R]Ri+1 · · ·Rm).
Let us rewrite relation (11.27) in the form
Da+(k) = E(k)D, (11.28)
where E(k) = a+(k) +B(k). Using formulas (11.28) and (11.26), we
see that
Da+(k1) · · · a+(kn)θ = E(k1) · · ·E(kn)θ,
in particular
Da+(k)θ = E(k)θ
= a+(k) +∑m
∫Am,1(k1, . . . ,km)
× δ(k− k1 − · · · − km)a+(k1) · · · a+(km)dmk. (11.29)
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Translation-Invariant Hamiltonians (Further Investigations) 297
It follows from relation (11.29) that
exp(iHt)D exp(−iH0t)a+(f1) · · · a+(fn)θ
= exp(iHt)Da+(f t1) · · · a+(f tn)θ
= exp(iHt)
(∫f t1(k1)E(k1)dk1
)· · ·(∫
f tn(kn)E(kn)dkn
)θ
=
(∫f t1(k1)E(k1, t)dk1
)· · ·(∫
f tn(kn)E(kn, t)dkn
)θ
(here, f t(k) = exp(−itω(k))f(k)).
To check that D is a dressing operator, it is sufficient to establish
that E is a good operator generalized function in the sense of
Section 10.5; this can be easily done following the analysis of
Heisenberg operators a(k, ε, t) provided in Section 11.2.
Let us remark that knowing the single-particle state Φ(k) =
Da+(k)θ and using the formula (11.29), we can find the function
Am,1 and the rest of the dressing operator of the considered type.
From this remark, it is easy to derive that the condition that the
operator A belongs to the class M + iM, for the case at hand, is
unnecessary (it is necessary to use the following expression of the
single-particle state in terms of perturbation theory
Φ(k) = Φ(k|g)
= exp(iσ(k|g))
(a+(k) +
∑m
∫φm(k1, . . . ,km)
× δ(k− k1 − · · · − km)a+(k1) · · · a+(km)dmk
)θ,
where exp(iσ(k|g)) is an arbitrary phase factor; φm(k1, . . . ,km) =∑m φ
(r)m (k1, . . . ,km)gr; the functions φ
(r)m belong to the space S, and
for every r only finitely many of these functions are non-zero).
The proven statement allows us to conclude that the definition of
scattering matrix given in Section 9.1 (formula (9.10)) is equivalent
to the other definitions.
Let us now return to the slightly more general case where D =
N(expA), where A ∈M+iM (i.e. A = A1+iA2, A1 ∈M, A2 ∈M).
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298 Mathematical Foundations of Quantum Field Theory
We will write A in the form A =∑Am,n, where Am,n =
∑r A
(r)m,ngr,
A(r)m,n =
∫A(r)m,n(k1, . . . ,km|p1, . . . ,pn)
× δ(k1 + · · ·+ km − p1 − · · · − pn)a+(k1)
· · · a+(km)a(p1) · · · a(pn)dmkdnp,
and define the operator A′ in the same way as the operator A except
only keeping terms of type (m, 1) (i.e. we set A′ =∑
mAm,1).
The operator D′ = N(expA′) satisfies the conditions
D′θ = θ,
D′a+(k)θ = Da+(k)θ
(terms containing no less than two annihilation operators go to zero
when acting on a+(k)θ).
As was shown above, the operator D′ = N(expA′) is a dressing
operator, i.e.
slimt→±∞
exp(iHt)D′ exp(−iHast) = S±. (11.30)
In order to derive from formula (11.30), the necessary relation
slimt→±∞
exp(iHt)D exp(−iHast) = S±,
it is enough to show that
slimt→±∞
exp(iHt)(D −D′) exp(−iHast) = 0. (11.31)
The proof of equation (11.31) in each order of perturbation theory
immediately follows from the relation
limt→±∞
N(A(r1)m1n1
· · ·A(rs)msns
)a+(f t1)· · · a+
(f ts)θ = 0,
where at least one of the operators A(ri)mini satisfies the condition
ni ≥ 2 (i.e. contains at least two annihilation operators), and the
functions f1(k), . . . , fs(k) constitute a non-overlapping family of
smooth functions with compact support. The above completes the
proof of the theorem in the case when the operator D takes the
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Translation-Invariant Hamiltonians (Further Investigations) 299
form N(expA). The proof of the theorem in the general case can
be reduced to the cases already considered using the following two
lemmas.
Lemma 11.1. Any operator of the form exp(A1 + iA2), where
A1, A2 ∈ M, can be written in the form N(exp(A′1 + iA′2)), where
A′1, A′2 ∈M.
Lemma 11.2. Any operator of the form exp(A1 + iA2)× exp(A′1 +
iA′2), where A1, A2, A′1, A
′2 ∈ M, can be written in the form
N(exp(A′′1 + iA′′2)), where A′′1, A′′2 ∈M.
We will not provide proofs of these lemmas.
Let us now consider the case of an arbitrary Hamiltonian H ∈M.
The symbol HΩ, as usual, denotes the operator H with finite volume
cutoff (see Section 8.1); the operator HΩ acts on the Fock space
FΩ = F (L2(Ω)). In perturbation theory, we can calculate the
operator exp(iHΩt) (in other words, HΩ can be viewed as a self-
adjoint operator in the framework of perturbation theory).
The family of operators DΩ will be called a family of dressing
operators for the Hamiltonian H if the scattering matrix S of the
Hamiltonian H can be written in the form
S = limt→∞t0→−∞
limΩ→∞
exp(i(cΩ +HasΩ)t)(DΩ)−1
× exp(−iHΩ(t− t0))DΩ exp(−i(cΩ +HasΩ)t0). (11.32)
(The operators DΩ act on the space FΩ, the symbols HasΩ denote the
asymptotic Hamiltonian Has with volume cutoff, and cΩ is a constant
depending on the volume. The limit in (11.32) can be understood in
the sense of the convergence of matrix elements (see Section 9.3) or
via the relation
S = slimt→∞t0→−∞
slimΩ→∞
iΩ exp(i(cΩ +HasΩ)t)(DΩ)−1
× exp(iHΩ(t− t0))DΩ exp(−i(cΩ +HasΩ)t0)i∗Ω,
where iΩ is a natural embedding of the Fock space FΩ = F (L2(Ω))
into the Fock space F (L2(E3)).)
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300 Mathematical Foundations of Quantum Field Theory
Let us sketch the proof of the following theorem.
Theorem 11.5. Let the family of operators DΩ satisfy the following
conditions:
(1) The operator DΩ transforms the Fock vector θ and the bare
single-particle state a+k θ into normalized eigenvectors of the
operator HΩ.
(2) The operator DΩ can be represented as the product of opera-
tors of the form N(expAΩ) and expAΩ, where AΩ are oper-
ators obtained from a volume cutoff of operators of the form
A = A1 + iA2 and A1, A2 are formal expressions from the class
M, not depending on Ω.
Then DΩ is a family of dressing operators. (More precisely, it
can be shown that formula (11.32) holds, where cΩ is defined by the
formula HΩDΩθ = cΩD
Ωθ.)
Let us begin with the remark that with the help of analogs
of Lemmas 11.1 and 11.2 for operators on the space FΩ, we can
reduce the proof to the case when the operator DΩ has the form
N(expAΩ), where A = A1 + iA2, A1 ∈ M, A2 ∈ M. Let us
first suppose that the Hamiltonian H does not generate vacuum
polarization. For such a Hamiltonian, the operators exp(iHΩt), DΩ =
N(expAΩ), exp(iHasΩt) have limits as Ω → ∞; they converge,
respectively, to the operators exp(iHt), N(expA) = D, exp(iHast).
This can be derived from the relations Hθ = 0, Hasθ = 0, Aθ = 0
(the last of these relations follows from the equation DΩθ = θ). It
is easy to check that on the right-hand side of equation (11.32), we
can transform the limit of products into a product of limits. Noting
that the constant cΩ in the case at hand is equal to 0, we obtain that
equation (11.32) is equivalent to the relation
S = slimt→∞t0→−∞
exp(iHast)D−1 exp(−iH(t− t0))
×D exp(−iHast0),
which follows from the already-proven relations (11.25) and the
formula S = S∗+S−. Thus, the necessary statement is proven for
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Translation-Invariant Hamiltonians (Further Investigations) 301
Hamiltonians not generating vacuum polarization. The case of an
arbitrary Hamiltonian H ∈ M reduces to the considered case by
Faddeev transformation. Indeed, by the results of Section 9.4, there
exists a formal expression W ∈M, such that as Ω→∞, we have
exp(iWΩ)HΩ exp(−iWΩ) ≈ H ′Ω + cΩ,
and, therefore,
exp(iWΩ)eiHΩt exp(−iWΩ) ≈ ei(H′Ω+cΩ)t (11.33)
where H ′Ω is obtained from a Hamiltonian H ′ that does not generate
vacuum polarization by means of volume cutoff; cΩ is the energy
of the ground state of the operator HΩ. Using formula (11.33), we
conclude that
limΩ→∞
exp(i(cΩ +HasΩ)t)(DΩ)−1 exp(−iHΩ(t− t0))
×DΩ exp(−i(cΩ +HasΩ)t0)
= limΩ→∞
exp(iH ′asΩt′)(D′Ω)−1 exp(−iH ′Ω(t− t0))
×D′Ω exp(−iH ′asΩt0), (11.34)
where
D′Ω = exp(iWΩ)DΩ
(to derive equation (11.34), one needs to use the coincidence of
the asymptotic Hamiltonians Has and H ′as, corresponding to the
Hamiltonians H and H ′). Applying the statement of the theorem
to a Hamiltonian H ′ not generating vacuum polarization, we see
that the operators D′Ω constitute a family of dressing operators
for the Hamiltonian H ′. This implies that the right-hand side of
equation (11.34), in the limit as t → ∞, t0 → −∞, equals
the scattering matrix of the Hamiltonian H ′. Since the scattering
matrices of the Hamiltonians H and H ′ coincide, equation (11.34)
proves the theorem in the general case.
11.5 Perturbation theory via the axiomatic approach
Let us suppose that on the Hilbert space H, we have defined
an energy operator H0, a momentum operator P, a linear dense
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302 Mathematical Foundations of Quantum Field Theory
subspace D of H, and a complete locally convex topological algebra
with involution A, consisting of operators acting on D. Let us
further assume that these objects satisfy conditions T1–T7 from
Section 10.5.
Suppose that W is an operator from the algebra A, satisfying the
condition WΦ = 0. Let us now prove that the family of operators
H(g) = H0 + gV = H0 + g
∫W (x)dx
together with momentum operator P and algebra A, satisfy conditions
T1–T7 from Section 10.5 in each power of the parameter g (the
integral is understood in the strong sense). Imposing some conditions
on the joint spectrum of the operators H0,P, we will prove that
starting with the operators H(g),P, and the algebra A, one can
construct the Møller matrices S±(g) as well as the scattering matrix
S(g), at least as formal series in powers of g.6
Let us first note that the operator V ∈∫W (x)dx is defined on
the set AΦ (i.e. on the set of vectors of the form AΦ, where A ∈ A)
and transforms this set into itself. Indeed,(∫W (x)dx
)AΦ =
∫[W (x), A]Φdx.
By condition T6, it follows that for any seminorm p(A) on A and
any n, we have
p([W (x), A]) ≤ C
1 + |x|n. (11.35)
By the completeness of the algebra A and equation (11.35), the
integral∫
[W (x), A]dx converges in the topology of A and defines
an element of the algebra A. This completes the proof.
Later in the chapter, we will prove that when calculating the
operator exp(−iH(g)t) = exp(−i(H0+gV )t) in terms of perturbation
6This statement, just as the rest of the statements in this section, applies tothe more general class of Hamiltonians of the form H0 +
∑r≥1 g
r∫Wr(x)dx, and
Wr ∈ A,WrΦ = 0, and the series in g is understood as a formal series.
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Translation-Invariant Hamiltonians (Further Investigations) 303
theory in each order of g, we obtain an operator, defined on the set
AΦ, that transforms this set into itself (this implies that the operator
H(g) can be seen as a self-adjoint operator, at least in perturbation
theory). Indeed, from the results of Section 4.1, it follows that
exp(−iH(g)t) =∑(g
i
)rUr(t),
where
Ur(t) =1
r!exp(−iH0t)
∫ t
0dτ1
∫ τ1
0dτ2 · · ·
∫ τr−1
0dτrV (τ1) · · ·V (τr)
=1
r!exp(−iH0t)
∫ t
0dτ1
∫ τ1
0dτ2 · · ·
∫ τr−1
0dτr
∫drxW (x1, τ1)
· · · W (xr, τr),
V (τ) = exp(iH0τ)V exp(−iH0τ) =
∫W (x, τ)dx,
W (x, τ) = exp(i(H0τ −Px))W exp(−i(H0τ −Px)).
Let us define the operator Rr(x1, t1, . . . ,xr, tr|A), where A ∈ A,
by induction and the relations
R0(A) = A,
Rr(x1, t1, . . . ,xr, tr|A)
= [W (x1, t1), Rr−1(x2, t2, . . . ,xr, tr|A)].
It is easy to check that the operator Rr continuously depends on
x1, t1, . . . ,xr, tr in the topology of the algebra A and
p(Rr(x1, t1, . . . ,xr, tr|A)) ≤ C(1 + |t1|s + · · ·+ |tr|s)1 + |x1|n + · · ·+ |xr|n
(11.36)
(here, p(A) is any seminorm on A, n is an arbitrary number and the
numbers s and C depend on the seminorm p and the number n).
Inequality (11.36) can be easily derived by induction and by means
of conditions T5 and T6.
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304 Mathematical Foundations of Quantum Field Theory
Let us now note that
W (x1, t1) · · ·W (xr, tr)AΦ = Rr(x1, t1, . . . ,xr, tr|A)Φ
(this can also be easily checked by induction). Thus,
Ur(t)AΦ = Br(t)Φ,
where
Br(t) =1
r!
∫ t
0dτ1
∫ τ1
0dτ2 · · ·
∫ τr−1
0dτr
∫drx exp(−iH0t)
×Rr(x1, τ1, . . . ,xr, τr|A) exp(iH0t).
From the inequality (11.36) and the completeness of algebra A,
it follows that for any A ∈ A, the integral defining the operator
Br(t) converges in the topology of A and, therefore, Br(t) ∈ A.
Thus, the operator Ur(t) is defined on the set AΦ and transforms
the set into itself. It therefore follows that the operator A(x, t|g) =
exp(iH(g)t − iPx)A exp(−iH(g)t + iPx), in each order of the
perturbation series, is defined on the set AΦ and then transforms
the set into itself:
A(x, t|g) =∑
grAr(x, t),
where the operator
Ar(x, t) =r∑
α=0
Uα(−t)A(x)Ur−α(t)
(1
i
)racts on the set AΦ).
Let us now show that the operator Ar(x, t) ∈ A is continuous
in x, t in the topology on A and that for any seminorm p(A) on A,
we have
p(Ar(x, t)) ≤ (1 + |x|s + |t|s)q(A), (11.37)
where the number s and the seminorm q(A) on A depend on the
seminorm p and the number r; A runs over a compact set F ⊂ A(in other words, we will prove that condition T5 from Section 10.5
holds for operators H(g),P, and the algebra A in each order of the
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Translation-Invariant Hamiltonians (Further Investigations) 305
perturbation theory in g). To do this, let us note that from the
Heisenberg equation,
1
i
∂A(x, t|g)
∂t= [H0 + gV,A(x, t|g)],
follows the relation
1
i
∂Ar(x, t)
∂t= [H0, Ar(x, t)] + [V,Ar−1(x, t)]. (11.38)
With the relation (11.38) and the initial condition Ar(x, 0) = 0 for
r ≥ 1, we can express Ar(x, t) through Ar−1(x, t):
Ar(x, t) = exp(iH0t)Ar(x, t) exp(−iH0t),
Ar(x, t) = i
∫ t
0exp(−iHτ)[V,Ar−1(x, τ)] exp(iHτ)dτ
= i
∫ t
0dτ
∫dξ exp(−iHτ)[W (ξ), Ar−1(x, τ)] exp(iHτ).
(11.39)
Using this expression, the necessary statement follows by induction.
We obtain
p(exp(−iHτ)[W (ξ), Ar−1(x, τ)] exp(iHτ)) ≤ p1(A)(1 + |x|2 + |t|s)1 + |ξ|n
,
(11.40)
where A ∈ F , p is an arbitrary seminorm, n is an arbitrary number,
and the seminorm p1 and the number s depend on p and n.
Inequality (11.40) implies that the integral in (11.39) converges
uniformly in x, t in the topology on A if x, t belongs to a bounded
set. From this remark, it follows that Ar(x, t) ∈ A and that it con-
tinuously depends on x, t; the inequality (11.37) follows immediately
from the inequality (11.40).
It is now easy to prove that the operators H(g),P, and the algebra
A, whose elements take the form D′ = AΦ, satisfy conditions T1–T7
in Section 10.5 in each order of the perturbation series in g. Indeed,
we have already shown that the set D′ is invariant in each order of
perturbation expansion of the operators exp(iH(g)t). Its invariance
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306 Mathematical Foundations of Quantum Field Theory
with respect to the operators exp(iPx) follows from the relation
exp(−iPx)AΦ = A(x)Φ. Conditions T1–T4, T6, and T7 do not
depend on the choice of the energy operator H(g) and, therefore,
are satisfied in the case at hand. We have already checked condition
T5 by perturbation theory.
Let us now show that the joint spectrum of the operators H(g),P
satisfies the conditions on the spectrum of energy and momentum
operators specified in Section 9.5 (at least in perturbation theory).
Let us suppose that the necessary conditions7 are satisfied for
g = 0 and let us further assume that for any point k ∈ E3, we can find
an operator A ∈ A, such that | 〈Φ0(k), AΦ〉 | 6= 0 (the symbol Φ0(k)
denotes a single-particle state of the operator H0). The arguments
applied in the slightly more complicated situation in the proof of
Lemmas 10.12–10.14 in Section 10.6, allow us to show that the single-
particle state Φ(k) can be chosen such that the functions 〈Φ(k), AΦ〉for all A ∈ A will be infinitely differentiable. If the single-particle
state Φ(k) is chosen this way, then for any smooth function with
compact support, we can find a good operator Bf such that BfΦ =∫f(k)Φ(k)dk.
It follows from WΦ = 0 that the ground state Φ of the operator
H0 is a stationary state of the operator H(g) for any g; this implies
that when we calculate the ground state of the operator H(g) in
perturbation theory, we obtain the vector Φ. We can find single-
particle states of the operator H(g) in perturbation theory in the
form
Φ(k|g) =∞∑r=0
grΦr(k),
using the relations
(H0 + gV )(∑
grΦr(k))
=(∑
grωr(k))(∑
grΦr(k)), (11.41)
7In order to prove the existence of Møller matrices and scattering matricesconstructed with the operators H(g),P, and the algebra A, it is enough to assumethat the joint spectrum of the operators H0,P satisfies conditions 4 and 5 fromSection 10.5.
March 27, 2020 16:3 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch11 page 307
Translation-Invariant Hamiltonians (Further Investigations) 307
P(∑
grΦr(k))
= k(∑
grΦr(k)), (11.42)⟨
∂
∂g
∑grΦr(k),
∑grΦr(k
′)
⟩= 0. (11.43)
By the relations (11.41) and (11.42), we obtain that∫ωr(k)f(k)f ′(k)dk
=⟨V Φr−1(f),Φ0(f ′)
⟩−⟨Φr−1(ω1f),Φ0(f ′)
⟩− · · ·
−⟨Φ1(ωr−1f),Φ0(f ′)
⟩, (11.44)
Φr(f) = γ(H0,P)(−V Φr−1(f) + Φr−1(ω1f) + · · ·+ Φ0(ωrf))
+ Φ0(λrf), (11.45)
where f(k) is a smooth function with compact support; Φr(f) =∫f(k)Φr(k); γ(ω,k) is a smooth function with compact support
equal to (ω−ω0(k))−1, if k ∈ supp f and (ω,k) belongs to the multi-
particle spectrum of the operators H,P and also if k = 0, ω = 0. It
should be equal to zero if ω = ω0(k). Using relations (11.43)–(11.45),
we can find functions λr(k), ωr(k), and vectors Φr(k); simultaneously
by induction on r, we can prove that ωr(k) and λr(k) are smooth
functions and the vector Φr(f) can be written in the form
Φr(f) = B(r)f Φ, (11.46)
where B(r)f ∈ A. (The representation (11.46) follows from equa-
tion (11.45) with the help of the reasoning employed in the proof
of Lemma 10.17 in Section 10.6; the infinite differentiability of ωr(k)
and λr(k) can be shown by a slight modification of the proof of
Lemma 10.15 in Section 10.6.)
Let us now show that the Møller matrices S±(g), corresponding
to the operators H(g),P, and the algebra A, can be constructed at
least in the framework of perturbation theory. More precisely, we will
construct the Møller matrix as a formal series
S±(g) =∑
grS(r)± ,
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308 Mathematical Foundations of Quantum Field Theory
where S(r)± are operators defined on the set D ⊂ Has, consisting of
linear combinations of vectors of the form a+(f1) · · · a+(fn)θ (here,
f1(k), . . . , fn(k) is a non-overlapping family of smooth functions with
compact support. (Recall that this condition on a family of functions
means that for any k ∈ supp fi,k′ ∈ supp fj , i 6= j, the gradients of
the functions ω0(k) at the points k and k′ do not coincide.)
To construct Møller matrices, we repeat the construction in
Section 10.1, considering all objects, entering into the formula (10.1),
as formal series in powers of g. The proof of the correctness of
this definition is nearly identical to the proof in Chapter 10. It is
important to note that the estimate (10.7) cannot be proven in every
order in g, in other words, if f(k) is a smooth function with compact
support with
f(x|t) = (2π)−3
∫exp
−it∑r≥0
grωr(k) + ikx
f(k)dk
=∑r≥0
grfr(x|t),
then we cannot prove the inequality
|fr(x|t) ≤ C|t|−3/2].
Therefore, we also cannot prove that the Lemma 10.4 of Section 10.1
holds in each order of perturbation theory. However, in the proof
of the correctness of the definition of the Møller matrix relying
on formula (10.1), we can substitute this lemma with Lemma 10.5
from Section 10.3, which holds in each order in g (the family of
functions f1, . . . , fn in formula (10.1) should be assumed to be a
non-overlapping family of smooth functions with compact support).
The scattering matrix S is defined, as always, by the relation
S = S∗+S−. (From what we have proved, it does not follow that S
has the form S =∑grS(r), where S(r) are operators with a common
domain, however, the matrix elements⟨Sa+(k1) · · · a+(km)θ, a+(p1) · · · a+(pn)θ
⟩=⟨S−a
+(k1) · · · a+(km)θ, S+a+(p1) · · · a+(pn)θ
⟩(11.47)
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Translation-Invariant Hamiltonians (Further Investigations) 309
for ki 6=kj ,pi 6=pj can be viewed as formal series in g. More
precisely, these matrix elements can be written in the form∑
r≥0 gr
S(r)(k1, . . . ,km |p1, . . . ,pn), where S(r) are generalized functions,
defined on the class of test functions φ(k1, . . . ,km |p1, . . . ,pn) ∈S, having support belonging to the set where ki 6= kj ,pi 6=pj . Hence, by scattering matrix, we will mean the set of matrix
elements (11.47).)
We can now use the results of Chapter 10 to investigate the Møller
matrices S±(g) and the scattering matrix S(g), corresponding to the
operators H(g),P, and the algebra A. Let us note that, in particular,
with the help of the statements proven above, we can easily prove
the assumptions in the adiabatic theorem from Section 10.6 in each
order of g. Unfortunately, this is not enough to prove, in each order of
perturbation theory, the theorem that gives an expression of Møller
matrix S±(g) in terms of the adiabatic Møller matrix Sα(0,±∞|g).
Inspection of the proof of the adiabatic theorem shows that the proof
depends on relations of the form∣∣∣∣exp
(i
αν(k|g)
)∣∣∣∣ ≤ const,
where ν(k|g) =∑
r≥0 grνr(k) is a real function; these relations
clearly do not hold in each order of g. In particular, to prove
Lemma 10.25, from Section 10.6, it is essential that
p(Qt(fi, φi, α)) = p(Qti) ≤ C(1 + |t|n),
where C is a constant not depending on α; this inequality with n = 32
follows from the inequality∫| ˜φt,αi (x)|dx ≤ C(1 + |t|3/2),
which does not hold in the framework of perturbation theory.
However, it is easy to verify that all the proofs in Section 10.6 hold in
the framework of perturbation theory if we allow the use of partial
summation of perturbation series. This allows us to say that the
adiabatic theorem can be proven in the framework of perturbation
theory with partial summation.
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310 Mathematical Foundations of Quantum Field Theory
Let us now come back to Hamiltonians that do not generate
vacuum polarization using the results we have proved.
Let us take as our Hilbert space H the space F (L2(E3)) and let
the energy operator H0 and the momentum operator P be defined
by the formulas
H0 =
∫ε(k)a+(k)dk,
P =
∫ka+(k)a(k)dk.
The function ε(k) is presumed to be a smooth function whose
derivatives grow not faster than a power. We also assume that the
function ε(k) satisfies the condition ε(k1 + k2) < ε(k1) + ε(k2) and
does not coincide with a linear function on any open set.) Let us
suppose that the algebra A is the algebra of operators of the form
∞∑m,n
∫fm,n(k1, . . . ,km|p1, . . . ,pn)
× a+(k1) · · · a+(km)a(p1) · · · a(pn)dmkdnp, (11.48)
where fm,n ∈ S and the summation in (11.48) is presumed to be finite
(operators of the form (11.48) can be viewed as operators defined on
the set S∞). In Section 10.5, it was shown that the operators H0,P,
and the algebra A satisfy conditions T1–T7. This allows us to apply
to them all the statements proven in this chapter.
Let us now note that the operator
H0 +∑r≥1
∫Wr(x)dx, (11.49)
where Wr ∈ A,Wrθ = 0, can be written in the form
H0 +∑r≥1
gr∑m,n
∫w(r)m,n(k1, . . . ,km|p1, . . . ,pn)
× δ(k1 + · · ·+ km − p1 − · · · − pn)
× a+(k1) · · · a+(km)a(p1) · · · a(pn)dmkdnp,
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Translation-Invariant Hamiltonians (Further Investigations) 311
i.e. it is a translation-invariant Hamiltonian not generating vacuum
polarization. (It is easy to check that the Hamiltonian (11.49) belongs
to the class M described in Section 11.1.)
As was shown in this chapter, we can associate Møller matrices
and scattering matrix written as formal series in powers of g to the
Hamiltonian (11.49) (we have already discussed this in Section 11.3,
where we have provided a different proof of this statement).
We can now apply the results of Chapter 10 to gain insight
about the scattering matrix of a translation-invariant Hamiltonian
not generating vacuum polarization.
For example, the following statement follows immediately from
Theorem 10.3, Section 10.4 if we note that for any free Hamiltonian
H0, the identity operator is a dressing operator and satisfies the
conditions of Theorem 10.2 in Section 10.4.
Let H ∈ M, B1, . . . , Bn ∈ M be Hamiltonians not generating
vacuum polarization. Suppose that the operator D = exp(iB1) · · ·exp(iBn) transforms the bare single-particle state a+(k)θ into the
single-particle state of the Hamiltonian H. Then the operator D is a
dressing operator for the Hamiltonian H.
Furthermore, let H = H0 + V ∈ M be a Hamiltonian not
generating vacuum polarization and let the function ε(k) be strictly
convex. Then from the results of Section 10.6, we can conclude that
the definition of the Møller matrix in terms of the adiabatic Møller
matrix given in Section 9.1 is equivalent to the other definitions.
It therefore follows that in the case at hand, Definition 9.2 of the
scattering matrix in Section 9.3 is equivalent to the other definitions
(both of these statements, as noted above, hold in the framework of
perturbation theory with partial summation).
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Chapter 12
Axiomatic Lorentz-InvariantQuantum Field Theory
12.1 Axioms describing Lorentz-invariant scattering
matrices
Let H be a Hilbert space of states. To discuss the Lorentz invariance
of a theory, we should first of all assume that we have a unitary rep-
resentation of an inhomogeneous Lorentz group P (Poincare group)
on the space H. We will denote the unitary operator corresponding
to the element g ∈ P by U(g). The group P contains the group of
space–time translations T as a subgroup; we will define the energy
operator H and the momentum operator P by the relation
exp(−i(Ha0 −Pa)) = U(1, a), (12.1)
where the symbol (1, a) denotes the translation x′ = x+a (in general,
(Λ, a) denotes the element of P defined by the transformation
x′ = Λx+ a).
However, if we only have a representation of the Poincare group,
still the corresponding scattering matrix cannot be defined. We will
assume, as in the axiomatic approach to scattering theory described
in Chapter 10, that we have an asymptotically commutative algebra
A on the space H; we impose conditions guaranteeing the Lorentz
invariance of the scattering matrix.
To begin, we will assume that the presentation U(g) and the
algebra A satisfy the following axioms:
A1. Spectrality : The energy operator H has a single ground state
Φ; the vector Φ is Lorentz-invariant (i.e. U(g)Φ = Φ for any
transformation g ∈ P).
313
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314 Mathematical Foundations of Quantum Field Theory
The vector Φ is called the physical vacuum.
A2. Cyclicity : The vector Φ is a cyclic vector of the algebra A.
A3. Lorentz invariance of the algebra A: If the operator A ∈ A, then
for any transformation g ∈ P , the operator U(g)AU−1(g) also
belongs to the algebra A.
In what follows, axiom A3 can be replaced with the following
weaker condition.
A3′. For any transformation g ∈ P , the algebra A asymptotically
commutes with the algebra U(g)AU−1(g) (i.e. with the alge-
bra consisting of operators of the form U(g)AU−1(g), where
A ∈ A).
Let us analyze the consequences of axiom A1 on the joint
spectrum of the operators H,P.
Lorentz invariance of the vector Φ implies, in particular, that
exp(−iHt)Φ = Φ and exp(iPa)Φ = Φ; hence,
HΦ = PΦ = 0. (12.2)
It is easy to check that the converse holds as well (i.e. from
equation (12.2) and the uniqueness of the ground state, the Lorentz
invariance of Φ follows).
Since the ground state Φ of the energy operator H has zero energy,
by equation (12.2), the operator H is non-negative. This implies that
the joint spectrum of the operators H,P is contained in the half-
space consisting of points (ω,p) where ω ≥ 0. However, the joint
spectrum of the operators H,P must be invariant with respect to
homogenous Lorentz transformations (see Appendix A.9). Therefore,
the spectrum of the operators H,P lies in the cone W , consisting
of points (ω,p), that satisfy the inequality ω2 ≥ p2 (any point
with ω2 < p2 can be transformed to a point (ω′,p′) with ω′ < 0
by a homogenous Lorentz transformation). The physical vacuum Φ
corresponds to the point 0 = (0, 0, 0, 0) ∈W .
Let us consider an irreducible invariant subspace K of the spaceH[i.e. a subspace invariant with respect to the operators U(g) and not
containing non-trivial subspaces that are also invariant with respect
to the operators U(g)]. For example, the subspace A, consisting of
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Axiomatic Lorentz-Invariant Quantum Field Theory 315
vectors of the form λΦ (the vacuum subspace), is irreducible and
invariant. All other irreducible invariant subspaces cannot contain
the vector Φ (otherwise, they would contain the subspace A and not
be irreducible). We will therefore assume that the subspace K does
not contain the vector Φ.
A representation of the Poincare group P by the operators U(g),
viewed as operators on the subspace K, is irreducible. The energy
operator H on the space K cannot vanish, otherwise every vector
x ∈ K is a ground state of the operator H, which contradicts the
uniqueness of the ground state (axiom A1). On the other hand, the
same axiom implies that the operator H on the subspace K (and on
the whole space) is non-negative.
As follows from known facts about irreducible representations of
the group P (see Appendix A.9), the joint spectrum of the operators
H and P on the space K constitutes the set Uµ, consisting of points
(ω,p) ∈ E4 with ω2 − p2 = µ2, ω ≥ 0. Here, µ is an arbitrary non-
negative number, the set Uµ with µ > 0 forms the top-half of a
hyperboloid. Furthermore, one can prove that there exists a unitary
map (an isomorphism) Ψ between the space L2(E3 × N), where N
is a finite set, and the space K, having the property
U(g)Ψ(f) = Ψ(V (g)f), (12.3)
where V (g) is an irreducible presentation of type (µ, n), described in
Appendix A.9. It follows from the relation (12.3) that
HΨ(f) = Ψ(hf),
PΨ(f) = Ψ(pf),(12.4)
where h and p are operators in the space L2(E3×N) defined by the
formulas
(pf)(k, α) = kf(k, α),
(hf)(k, α) =√
k2 + µ2f(k, α).(12.5)
The mapping Ψ can be viewed as n generalized vector functions
Ψα(k) related to Ψ by the formula
Ψ(f) =∑α∈N
∫f(k, α)Ψα(k)dk
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316 Mathematical Foundations of Quantum Field Theory
(here, n is the number of elements in the set N and α runs over
the set N). From the unitarity of the mapping Ψ, it follows that the
generalized vector functions Ψα(k) satisfy the relations⟨Ψα(k),Ψα′(k′)
⟩= δα
′α δ(k− k′)
(orthogonality and normalization by a δ-function); from equa-
tions (12.4) and (12.5), it follows that
PΨα(k) = kΨα(k),
HΨα(k) =√
k2 + µ2Ψα(k).
This implies that the functions Ψα(k) can be viewed as single-particle
states in the sense of the definitions previously used in this book. In
ordinary terminology, the collection of functions Ψ1(k), . . . ,Ψn(k)
(i.e. the mapping Ψ) is identified with a particle having mass µ and
spin s = n−12 . A particle having spin 0 is called scalar.
The smallest subspace, containing all irreducible invariant sub-
spaces differing from A, will be called the single-particle subspace
and will be denoted by B.
It is easy to check that the space B can be represented as a direct
sum of mutually orthogonal irreducible invariant subspaces Ki; using
the isomorphisms Ψi, described above, between the spaces L2(E3 ×Ni) and Ki, we can construct an isomorphism Ψ =
∑i Ψi between
the space L2(E3 ×N) =∑
i L2(E3 ×Ni) and the space B (N is the
union of the sets Ni).
It is clear from these remarks that the single-particle subspace B
defined above coincides with the single-particle subspace in the sense
of the definition in Section 10.1.
The multi-particle subspace M will be defined as the orthogonal
completion of the direct sum A+B (the space A+B can be written
as the sum of all irreducible invariant subspaces).
The space Has will be defined as the Fock space F (B).
Using the isomorphism between the space B and the space
L2(E3 × N), constructed above, we can introduce the operator
generalized functions a+(k, s), a(k, s) on Has, where k ∈ L3, s ∈ N ,
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Axiomatic Lorentz-Invariant Quantum Field Theory 317
satisfying the relations
[a(k, s), a(k′, s′)] = [a+(k, s), a+(k′, s′)] = 0,
[a(k, s), a+(k′, s′)] = δss′δ(k− k′).
The space B is invariant with respect to the operators U(g). The
representation of P by the operators U(g) on the space B clearly
induces a representation on Has; we will denote the operators of
this representation by Uas(g). The asymptotic Hamiltonian Has and
the asymptotic momentum operator Pas are naturally defined by the
relation
exp(−i(Hasa0 −Pasa)) = Uas(1, a),
where (1, a) is a translation.
It is easy to see that the asymptotic Hamiltonian and the
asymptotic momentum operator can be expressed in terms of the
operators a+(k, s), a(k, s) by the formulas
Has =∑i
∑s∈Ni
∫ √k2 + µ2
i a+(k, s)a(k, s)dk,
Pas =∑s∈N
∫ka+(k, s)a(k, s)dk.
We will now formulate an additional axiom which will allow us
to construct a scattering theory in this case (the strong spectral
condition).
A4. The joint spectrum of the operators H,P on the subspace C of
the space H, containing all the irreducible invariant subspaces
of the operators U(g), does not intersect the joint spectrum of
these operators on the orthogonal complement of the space C.
The spectrum of the energy operator H contains a gap (i.e. there
exists an ε > 0, such that the single point of the spectrum of the
operator H on the ray (−∞, ε) is the point 0, corresponding to
the physical vacuum).
It is clear that the space C equals the direct sum of the
subspace A and the single-particle subspace B, therefore, the joint
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318 Mathematical Foundations of Quantum Field Theory
spectrum of the operators H,P on the space C is identical to the
union of the vacuum spectrum (i.e. the spectrum of the operators
H,P on the space A) and the single-particle spectrum (i.e. the
spectrum of the operators H,P on the space B). It follows from the
statements above that the vacuum spectrum consists of the single
point 0 and the single-particle spectrum can be written as the union
of several sets Uµi , specified by the relation ω2 − k2 = µ2i , ω ≥ 0.
The orthogonal complement to the space C = A+B, referred to
as the multi-particle space and denoted by M , the joint spectrum of
H,P on M will be called the multi-particle spectrum.
By using these definitions, we can reformulate axiom A4.
A4′. The vacuum spectrum, single-particle spectrum, and the multi-
particle spectrum are pairwise non-intersecting.
Thus, the spectrum of the operators H,P on the space H is
structured as follows. It contains the point 0, corresponding to the
vector Φ, and several sets Uµi with µi > 0, constituting the single-
particle spectrum, and the multi-particle spectrum, which does not
contain the point 0 and does not intersect any of sets Uµi .
Let us now formulate the following statement. If the axioms 1–4
are satisfied, then the Møller matrices S± and the scattering matrix
S can be defined as in Section 10.1; using the ideas of Section 10.1,
one can prove the existence of the Møller matrices and the scattering
matrix. The Møller matrices and scattering matrix defined in this
manner are Lorentz-invariant (i.e. they satisfy the equations
U(g)S± = S±Uas(g), (12.6)
Uas(g)S = SUas(g) (12.7)
for any transformation g ∈ P).
Indeed, the definitions and discussions in Section 10.1 can be
applied to the case at hand. (The conditions in Section 10.1 are not
necessarily filled, since the definition of the single-particle space in
Section 10.1 is different from the definition in this section; similarly,
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Axiomatic Lorentz-Invariant Quantum Field Theory 319
the definitions of the space Has differ. However, in light of the fact
that the single-particle space B defined in this section is contained in
the single-particle space in the sense of Section 10.1, the discussions
in Chapter 10 do not need to be modified in this case.) The Lorentz
invariance of the Møller matrices and the scattering matrix can be
proved by the discussions in Section 6, Chapter 6 in Jost (1965); we
will sketch another proof in Chapter 13.
Let us note that the results of this section hold if the asymp-
totically commutative family of operators is replaced by an asymp-
totically commutative family of operator generalized functions (in
the sense of Section 10.5). The domain of definition of these
operator generalized functions should be invariant with respect to
the operators U(g), where g ∈ P. Conditions A2 and A3 require
corresponding modifications (we ought to require that the vector Φ
is cyclic vector for this family and that for every generalized operator
function A(x, t), the corresponding generalized operator functions
Ag(x, t) = UgA(x, t)U−1g , where g is an arbitrary element of the
group P, should also belong to the family).
12.2 Axiomatics of local quantum field theory
To formulate the axioms of Lorentz-invariant theory, we added
extra conditions to the axiomatic scattering theory constructed in
Chapter 10. These extra conditions guarantee the Lorentz invariance
of the scattering matrix.
Here, we will provide a different set of axiomatics introduced by
Haag and Araki.
Let us fix a representation Ug of the Poincare group P on the
Hilbert space H, and, as in the previous section, let us require the
spectral condition (axiom A1).
Let us assume that for every bounded set O ⊂ E4, there is a
collection R(O) of bounded operators on the space H. We assume
that this collection is closed under addition and multiplication of
operators, multiplication by a scalar, involution A → A∗, and weak
limits (such a collection of operators is called a W ∗-algebra).
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320 Mathematical Foundations of Quantum Field Theory
Let us assume that the algebra R(O) has the following properties:
R1. If O1 ⊂ O2, then R(O1) ⊂ R(O2).
R2. Covariance with respect to the Poincare group:
UgR(O)U−1g = R(gO).
R3. Local commutativity: If any point of the set O1 is separated by
a space-like interval from any point of the set O2, then R(O1)
and R(O2) commute with each other.
R4. Cyclicity: The physical vacuum Φ is cyclic with respect to the
union R of the algebras R(O).
It is easy to see that the union R of the algebras R(O) is an
asymptotically commutative algebra. Indeed, let A ∈ R(O1) ⊂ R,
B ∈ R(O2) ⊂ R. Then, by condition R2, we have
A(x, t) = exp(iHt− iPx)A exp(−iHt+ iPx) ∈ R(O1 − x),
where x = (x, t). Since the sets O1 and O2 are bounded, there exists
an a such that for |x|2 − |t|2 ≥ a2, any point of the set O1 − x is
separated from any point of the set O2 by a space-like interval. Using
condition R3, we obtain that for |x|2 − |t|2 ≥ a2,
[A(x, t), B] = 0.
On the other hand, ‖A(x, t)‖ = ‖A‖, and therefore,
‖[A(x, t), B]‖ ≤ 2‖A‖ · ‖B‖.
Combining these relations, we see that
‖[A(x, t), B]‖ ≤ C 1 + |t|n
1 + |x|n
for any n (to show this, it is enough to note that
1 + |t|n
1 + |x|n> c > 0 for |x|2 − |t|2 ≥ a2
).
Furthermore, the algebra R is clearly invariant with respect to
Lorentz transformations.
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Axiomatic Lorentz-Invariant Quantum Field Theory 321
Therefore, the algebra R and the representation Ug satisfy all
conditions of Section 12.1, except for axiom A4.
If we require that axiom A4 holds, then the considerations
provided in Section 12.1 allow us to prove that there exists a Lorentz-
invariant scattering matrix.
Another important axiomatics is due to Wightman. Again, we
require that the Poincare group is represented in Hilbert space Hand the spectral condition A1 is satisfied.
We assume that an operator generalized function φ(x) is defined
on the spaceH (here, x ∈ E4). More precisely, we assume that for any
function f ∈ S(E4), there is an operator φ(f); all the operators φ(f)
are defined on the same linear set D and transform D into itself;
the operator φ(f) must depend linearly on f and for any vectors
ξ, η ∈ D, the functional 〈φ(f)ξ, η〉 must continuously depend on f in
the topology of the space S (i.e. it must define a generalized function
from the space S ′(E4)).
As before, we will write φ(f) =∫φ(x)f(x)dx; the operator
generalized function φ(x) will be referred to as a quantum field.
Let us assume that the quantum field φ(x) is Hermitian (i.e. the
operators φ(f) and φ(f) are Hermitian conjugate).
We say that the representation U(g) of the Poincare group P and
the operator generalized function φ(x) (the quantum field) satisfy
the Wightman axioms,1 if the following conditions are satisfied:
W1. Transformation law of quantum fields: The operators Ug keep
the set D fixed: UgD = D. The quantum field φ(x) transforms
according to the law
U(g)φ(x)U−1(g) = φ(gx). (12.8)
More precisely, equation (12.8) implies that
U(g)φ(f)U−1(g) = φ(fg),
where
fg(x) = f(g−1x).
1Here, we have formulated the Wightman axioms in the case of a scalar field;for the general case, see, for example, Bogolyubov et al. (1969).
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322 Mathematical Foundations of Quantum Field Theory
W2. Locality (local commutativity): If the points x, y ∈ E4 are sepa-
rated by a space-like interval, then φ(x) commutes with φ(y).
More precisely, we require that the operators φ(f) and φ(g)
commute in the case when the support of the functions f and g
are separated by a space-like interval.
W3. Cyclicity : The vector Φ (the ground state of the energy
operator) is a cyclic vector of the family of operators φ(f).
The Wightman functions wm(x1, . . . , xm) will be defined by the
relation
wm(x1, . . . , xm) = 〈φ(x1) · · ·φ(xm)Φ,Φ〉 .
By the kernel theorem (see Appendix A.7), the function wn can be
viewed as a generalized function from S′(E4m).
Using the considerations of Section 8.2, one can prove the
reconstruction theorem, which states that knowing the Wightman
functions wm, we can construct a Hilbert space H, a Lorentz group
representation, and a quantum field φ(x). It is easy to work out
the conditions on the functions wm(x1, . . . , xm) which guarantee
that these functions are Wightman functions for a quantum field,
satisfying the Wightman axioms (these conditions are analogous to
conditions 1–6 from Section 8.2).2
Let us now show that by adding axiom A4 from Section 12.1 to
the Wightman axioms, we can provide a definition of Møller matrices
(hence of the scattering matrix) and prove their existence and Lorentz
invariance. To do this, it is sufficient to consider a family of operator
generalized functions φ(gx), where g runs over a Poincare group P,
and note that this family is asymptotically commutative in the sense
of Section 10.5 and it satisfies conditions A1–A4 from Section 12.1.
(Axiom A3 follows immediately from axioms W1, axiom A2 from
axiom W3, condition D5 from Section 10.5 for δ < 1 follows from
local commutativity.)
2For more details, see Streater and Wightman (2016), Jost (1965), andBogolyubov et al. (1969).
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Axiomatic Lorentz-Invariant Quantum Field Theory 323
Thus, we can see that all the statements which can be obtained
from the axioms in Section 12.1 can be derived from Wightman’s
axioms. However, there are theorems which can be proved in
Wightman’s axiomatics that have not been proven (and, apparently,
cannot be proven) in the axiomatic formulation in Section 12.1.
(This is not surprising; Wightman axiomatized local quantum field
theory; the axioms of Section 12.1 are based on weaker assumption
of asymptotic commutativity).
The most important statement of this type are dispersion rela-
tions obtained first in Bogolyubov et al. (1956) and derived from
Wightman axioms in Hepp and Epstein (1971).
Let us also mention the CPT theorem, stating that the Wightman
axioms imply CPT invariance (i.e. the existence of additional
symmetries).
Let us formulate Borchers’ theorem:
If two quantum fields φ1(x), φ2(y) on a Hilbert space satisfy the
Wightman axioms and are mutually local (i.e. [φ1(x), φ2(y)] = 0 for
any points x, y separated by a space-like interval), then they define
the same scattering matrix.
The CPT theorem can be used in the proof of this theorem.
However, this theorem has another proof: it follows from the results of
Section 10.1 (from the theorem that two asymptotically commutative
algebras define the same scattering matrix).
Finally, let us analyze the connection between the Wightman
axioms and the Haag–Araki axioms. Let us assume that the operators
φ(f) =∫f(x)φ(x)dx, entering in the Wightman axioms, are not
only Hermitian but also essentially self-adjoint for any real smooth
function f with compact support. For every such function f , one can
construct a family Tf of bounded operators, arising as functions of
the operator φ(f) (i.e. having the form α(φ(f)), where α is a bounded
function). Let us use the symbol R(O) for the smallest W ∗-algebra,
containing all families Tf , where f is a smooth real function with
support in the set O. It is easy to check that the algebras R(O)
satisfy the Haag–Araki axioms.
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324 Mathematical Foundations of Quantum Field Theory
12.3 The problem of constructing a non-trivial
example
It is natural to try to construct a Lorentz-invariant quantum theory
by means of the quantization of Lorentz-invariant classical theory.
Let us recall that in Section 8.3, we considered the question of
quantizing a classical theory corresponding to the Hamiltonian
H(π, φ) =1
2
∫π2(x)dx + V (φ).
It is easy to see that such a classical system is Lorentz invariant if
H(π, φ) =1
2
∫π2(x)dx−1
2
∫φ(x)∆φ(x)dx+
∫U(φ(x))dx. (12.9)
Indeed, the Hamiltonian equations for the system, described by the
functional Hamiltonian (12.9), have the form
∂φ(x, t)
∂t= π(x, t),
∂π(x, t)
∂t= ∆φ(x, t)− F (φ(x, t)),
where
F (φ) =∂U(φ)
∂φ.
Thus, the function φ(x) = φ(x, t) satisfies the equation
φ(x) + F (φ(x)) = 0,
which is clearly Lorentz invariant (the equation does not change form
when replacing φ(x) by φ(Λx), where Λ is a Lorentz transformation).
We can also prove the Lorentz invariance of classical systems,
described by the Hamiltonian (12.9), by considering the action
functional of such a system. This functional can be presented in the
form
S =
∫ ((∂φ
∂t
)2
− (∇φ)2
)dx−
∫U(φ(x))dx,
which is invariant under Lorentz transformations.
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Axiomatic Lorentz-Invariant Quantum Field Theory 325
Let us now consider the simplest system with the Hamiltonian of
the form (12.9), a system with
U(φ(x)) =1
2m2φ2(x). (12.10)
After quantizing this system, we obtain a quantum Hamiltonian
H =1
2
∫π2(x)dx− 1
2
∫φ(x)∆φ(x)dx +
1
2m2
∫φ2(x)dx, (12.11)
where π(x), φ(x) are symbols satisfying the relations
π+(x) = π(x), φ+(x) = φ(x),
[π(x), π(y)] = [φ(x), φ(y)] = 0,
[π(x), φ(y)] =1
iδ(x− y).
The Hamiltonian (12.11) can be written in the form (8.26), where
ν(x) = −∆δ(x)+m2δ(x). An operator realization of Hamiltonians of
the form (8.26) (free Hamiltonians) is described in Section 8.3. Using
the results of Section 8.3, we can prove that the operator realization
of the Hamiltonian (12.11) can be constructed on the Fock spaceH =
F (L2(E3)); as the energy operators and the momentum operators,
we can select the operators
H =
∫ω(k)a+(k)a(k)dk,
P =
∫ka+(k)a(k)dk,
where ω(k) =√
k2 +m2; the operators (the operator generalized
functions) φ(x, t) are defined by the formula
φ(x, t) = (2π)−3/2
∫(a+(k) exp(iω(k)t− ikx)
+ a(k) exp(−iω(k)t+ ikx))dk√2ω(k)
. (12.12)
The ground state Φ coincides with the Fock vacuum θ.
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326 Mathematical Foundations of Quantum Field Theory
The operators φ(x) = φ(x, t) satisfy the Lorentz-invariant
equation
φ(x) +m2φ(x) = 0.
It is therefore natural to expect that the quantum system is Lorentz
invariant. Actually, the representation of the translation group on
the space F (L2(E3)), given rise by the operators H and P, can be
extended to a unitary representation of Poincare group in such a way
that
U(g)φ(x)U−1(g) = φ(gx)
(here, g is a Lorentz transformation and U(g) is the corresponding
unitary operator). The necessary Lorentz group representation can
be obtained by considering on the space L2(E3) the representation
of the type (m, 1), described in Appendix A.9. (We should note that
the representation on the space L2(E3) specifies a representation on
the space F (L2(E3)); we have already used this fact in Section 12.1).
It is easy to check that the operator generalized function φ(x, t)
defined by the formula (12.12) and the representation of the Poincare
group satisfy all the Wightman axioms.
The given quantum system is called a free scalar field with
mass m. Corresponding particles are non-interacting bosons with
spin 0. The scattering matrix is trivial as it should be for non-
interacting particles.
To obtain a non-trivial Lorentz-invariant scattering matrix, we
can try to quantize a classical theory with the Hamiltonian of the
form (12.9), where the function U(φ) is not quadratic. Unfortunately,
this approach leads to problems related to the fact that in every step
one encounters singular expressions which cannot be easily given a
precise meaning. For example, the expressions of the form φn(x),
where n ≥ 2, and similarly more general expressions ν(φ(x)), where
ν(φ) is a nonlinear function, are not well defined; therefore, the
equation
φ(x) + F (φ(x)) = 0,
for the field operators φ(x) does not have precise meaning.
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Axiomatic Lorentz-Invariant Quantum Field Theory 327
Let us suppose that the Hamiltonian (12.9) can be presented
in the form H = H0 + V , where H0 is a Hamiltonian of the
form (12.10), V =∫ν(φ(x))dx. After quantization, the resulting
formal Hamiltonian can be written in the form
H = H0 + V =1
2
∫π2(x)dx− 1
2
∫φ(x)∆φ(x)dx
+1
2m2
∫φ2(x)dx +
∫ν(φ(x))dx. (12.13)
If we try to express Hamiltonian in terms of the symbols a+(k), a(k),
as in Section 8.3, and bring the resulting expression to normal form
by means of CCR, then except for the standard infinite constants,
which we have previously thrown away, we will obtain other infinite
summands. Therefore, one often considers the formal Hamiltonian
H = H0 +N
∫ν(φ(x))dx, (12.14)
where N is the normal product symbol (i.e. we express∫ν(φ(x))dx
in terms of a+(x), a(k) by the formulas (8.27), and then reorder the
symbols in normal order, assuming they commute). Replacing the
formal Hamiltonian (12.13) by the formal Hamiltonian (12.14), we
can get rid of some infinities, but other infinities remain. In particu-
lar, calculating the Green functions of the Hamiltonian (12.14) in the
framework of perturbation theory, we encounter diverging integrals.3
Unfortunately, even now, a consistent method of overcoming these
challenges in the case of arbitrary function ν(φ) or even in the case
when the function ν(φ) is a polynomial does not exist. However, if
the function ν(φ) is a polynomial of order ≤ 4 (i.e. ν(φ) = aφ3+bφ4),
one can overcome these problems in the framework of perturbation
theory. There exists a method that allows us to construct a family of
Lorentz-invariant scattering matrices that correspond to the family
of Hamiltonians of the form H0 + N∫ν(φ(x))dx. This method was
introduced by Feynman; it is called covariant renormalization.
3We should note that in the one-dimensional case (i.e. in the case when in theHamiltonian (12.14) integration is performed over a single variable) calculations ofGreen functions of the Hamiltonian (12.14) do not give rise to diverging integrals.
March 27, 2020 16:33 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch12 page 328
328 Mathematical Foundations of Quantum Field Theory
Let us consider for the sake of concreteness the case when ν(φ) =
bφ4. We consider the family of Hamiltonians of the form
H = H0(w) + bVΛ
=1
2
∫π2(x)dx +
1
2
∫w(x− y)φ(x)φ(y)dxdy
+ b
∫fΛ(x1 − x4,x2 − x4,x3 − x4)
+ φ(x1)φ(x2)φ(x3)φ(x4)d4x.
Here,w(x) is an arbitrary function, fΛ(x1,x2,x3) = Λ3f(Λx1,Λx2,Λx3),
where the function f ∈ S and satisfies the condition∫f(x1,x2,x3)
d3x = 1. It is easy to see that limΛ→∞ fΛ(x1,x2,x3) = δ(x1)
δ(x2)δ(x3) and therefore in the limit as Λ → ∞, the interaction
VΛ becomes V =∫φ4(x)dx; the replacement of V by VΛ is called
ultraviolet cutoff. (Defining VΛ, we can use the normal product
in place of the regular product; the class of Hamiltonians under
consideration does not change in this case.) Let us select the
function w(x) from the condition that energy of the particles, defined
by the Hamiltonian H0(w) + bVΛ, equals m for k = 0; we will
denote the resulting Hamiltonian H(m, b,Λ) (in the framework of
perturbation theory, the function w(x) is easy to find). The transition
to the Hamiltonian H(m, b,Λ) is called mass renormalization (the
number m corresponds to the mass of the particles defined by the
Hamiltonian). Let us denote the scattering matrix corresponding
to the Hamiltonian H(m, b,Λ) by S(m, b,Λ). In order to obtain a
Lorentz-invariant theory, we must take the limit Λ → ∞ (remove
the ultraviolet cutoff). However, taking the limit Λ→∞, we obtain
divergent expressions in the perturbation series.
In order to obtain a non-trivial Lorentz-invariant scattering
matrix in the framework of perturbation theory, we should assume
that the quantity b (bare charge) changes in the process of
removing the ultraviolet cutoff. In the perturbation theory, one can
prove the following statement: there exists a function b(m,Λ, g) such
that the scattering matrix S(m, b(m,Λ, g),Λ) has a finite, non-trivial,
Lorentz-invariant limit S(m, g) as Λ → ∞. (In order to find the
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Axiomatic Lorentz-Invariant Quantum Field Theory 329
function b(m,Λ, g), we suppose that in the process of removing the
cutoff we leave a physical quantity g fixed and express the bare charge
in terms of the cutoff parameter and this quantity. We can take as
g any matrix element of the scattering matrix; for some particular
choice, it is called physical charge or dressed charge.) The result
(which we have described in a non-standard form) is the theory of
renormalization. The Green functions Gn(x1, . . . , xn|m,Λ, g) of the
Hamiltonian H(m, b(m,Λ, g),Λ) do not have a finite limit as Λ→∞,
however, we can find a function Z(m,Λ, g) such that the functions
Z−n/2(m,Λ, g)Gn(x1, . . . , xn|m,Λ, g) have a finite Lorentz-invariant
limit as Λ→∞.
The statements above suggest the following way of constructing
objects satisfying the Wightman axioms and giving rise to non-trivial
scattering matrices. First, we must find the functions b(m,Λ, g) and
Z(m,Λ, g), such that a non-trivial finite limit
limΛ→∞
Z−n/2(m,Λ, g)wn(x1, . . . , xn|m,Λ, g) = wn(x1, . . . , xn|m, g)
exists. Here, wn(x1, . . . , xn|m, g) is a Wightman function of the
Hamiltonian H(m, b(m,Λ, g),Λ). One can hope that the functions
w(x1, . . . , xn|m, g) defined this way satisfy the conditions necessary
for constructing objects obeying the Wightman axioms and that
the corresponding scattering matrix is non-trivial. Unfortunately, at
present, this hypothesis is unproven. Moreover, in four-dimensional
space–time, an example of objects satisfying the Wightman functions
and giving rise to a non-trivial scattering matrix is not known.
Clearly, if we can construct a non-trivial scattering matrix in
Wightman’s axioms, then we will obtain a non-trivial example of
a scattering matrix in the axioms of Section 12.1. However, the other
direction does not hold — a non-trivial example in the axioms of
Section 12.1 can be easier to construct.
The situation in two-dimensional and three-dimensional space–
time is much better. A new branch of mathematical physics —
constructive field theory — was born from the attempts to find non-
trivial examples and in two and three dimensions, it was successful.
Two-dimensional conformal field theories were thoroughly analyzed
March 27, 2020 16:33 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch12 page 330
330 Mathematical Foundations of Quantum Field Theory
in numerous papers starting with the seminal paper by Belavin et al.
(1984). Another way to construct two-dimensional theories comes
from integrable models.
In all dimensions, the considerations based on supersymmetry and
superstring theory led to much deeper understanding of quantum
field theories.
We are moving forward!
March 30, 2020 11:40 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-ch13 page 331
Chapter 13
Methods of Quantum Field Theoryin Statistical Physics
13.1 Quantum statistical mechanics
The equilibrium state in (classical or quantum) statistical mechanics
can be defined as a state maximizing the entropy under some given
conditions.
In quantum mechanics, we represent a state by the density
matrix K; the entropy of this state is defined by the formula
S = −TrK logK. Denoting the Hamiltonian by H, we can say that
the equilibrium state maximizes the entropy for a given mean energy
E = Tr HK. (More precisely, we can say that this is an equilibrium
state of the canonical ensemble or a Gibbs state.) Let us check that
this state has the form
e−βH
Z, (13.1)
where β can be interpreted as the inverse temperature β = 1T and
Z = Tr e−βH is called the partition function or the statistical sum.
If the spectrum of H is discrete, then Z =∑e−βEi where Ei are the
energy levels. In the case when the ground state is non-degenerate,
we see that the equilibrium state tends to the ground state and its
entropy tends to zero when T → 0.
Note that (13.1) makes sense only if Z is finite (the operator e−βH
belongs to the trace class).
331
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332 Mathematical Foundations of Quantum Field Theory
To prove (13.1), we use the method of Lagrange multipliers, reduc-
ing our problem to finding the stationary points of the functional
L = −TrK logK − βTr(HK − E)− ζ(TrK − 1).
Calculating the variation of L, we obtain
δL = −Tr logKδK − Tr δK − β Tr HδK − ζ Tr δK,
hence at the stationary point,
− logK − 1− βH − ζ = 0.
This is equivalent to (13.1) with Z = e1+ζ .
Note that in the calculation of variation, we have used the formula
δTrφ(K) = Trφ′(K)δK, (13.2)
where φ′ stands for the derivative of the function φ. It is easy to
check this formula in the case when φ(K) = Kn; to treat the general
case, we can represent φ as a limit of polynomials.
The partition function Z does not have any physical meaning, but
there are many physical quantities that can be conveniently expressed
in terms of Z. For example, differentiating the definition of Z with
respect to β, we obtain
E = H = −∂ logZ
∂β.
The expression for the entropy in equilibrium state is
S = βE + logZ.
Introducing the notion of free energy by the formula F = E − TS,
we obtain
F = −T logZ.
If the Hamiltonian depends on a parameter λ and H(λ) = H +
λA+ · · · then using (13.2), we can write
Z(λ) = Z + (−β)λTrAe−βH + · · · .
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Methods of Quantum Field Theory in Statistical Physics 333
Here, · · · stands for higher order terms with respect to λ. Note that
A =TrAe−βH
Z
is the mean value (expectation value) of the observable A in the
equilibrium state. We will also use the notation 〈A〉 for the expecta-
tion value. The expectation values 〈A1 · · ·An〉 or, more generally,
〈A1(t1) · · ·An(tn)〉 are called correlation functions (here, Ai are
observables and Ai(t) are the corresponding Heisenberg operators).
If we want to emphasize that the correlation functions are calculated
for the equilibrium state with the temperature T = 1β , we will use
the notation 〈A1(t1) · · ·An(tn)〉β.If A and B are two observables, then
〈A(t)B〉β = 〈BA(t+ iβ)〉β. (13.3)
This equation, called the Kubo–Martin–Schwinger (KMS) condition,
can be proved by simple formal manipulations if the Hilbert space
H is finite-dimensional. In the infinite-dimensional case, one should
check that the correlation function 〈BA(t)〉β can be extended
analytically to the strip 0 ≤ =t ≤ β (here, = stands for imaginary
part).
It is easy to see that
A = −T ∂ logZ
∂λ=∂F
∂λ(13.4)
(the derivatives are calculated at the point λ = 0).
If the Hamiltonian depends linearly on a set of parameters
λ1, . . . , λk, we can calculate the correlation functions by differenti-
ating the free energy F . For example, if H = H0 +λ1A1 + · · ·+λkAk,
we have
∂2F
∂λi∂λj= 〈AiAj〉 − 〈Ai〉〈Aj〉. (13.5)
(The derivatives are calculated at the point λi = 0.) The RHS
of (13.5) is called a truncated correlation function. Higher truncated
correlation functions can be defined as higher derivatives of F. (See
Section 10.1 for the definition of truncated correlation functions in
terms of correlation functions.)
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334 Mathematical Foundations of Quantum Field Theory
If we have several integrals of motion, we can consider the
equilibrium state fixing the expectation values of all these integrals
and looking for the maximal value of entropy under these conditions.
For example, we can fix the expectation values of energy and of the
number of particles N . Then we can repeat the above consideration
by adding one more Lagrange multiplier. We obtain the following
formula for the equilibrium state:
e−β(H−µN)
Z. (13.6)
Note that we have denoted the new Lagrange multiplier as −βµto agree with standard notation (µ can be identified with chemical
potential).
13.1.1 Examples
Let us consider the multidimensional harmonic oscillator as an
example. Introducing the creation and annihilation operators
obeying CCR, we can represent the Hamiltonian in the form
H =∑
i ωia+i ai + C, where C =
∑ωi/2 is the energy of the ground
state. We shift the energy scale assuming that the ground state has
zero energy: C = 0. Then the energy levels are∑niωi where ni is a
non-negative integer. It is easy to calculate the partition function
Z = Π1
1− e−βωi,
the free energy
F =1
β
∑log(1− e−βωi),
and the mean value of energy
H =∑
ωini,
where ni = 1eβωi−1
.
The same formulas work for non-interacting identical bosons. In
the case of non-interacting identical fermions, we have very similar
formulas. The creation and annihilation operators, in this case, obey
CAR and the energy levels are given by the formula∑ωini where
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Methods of Quantum Field Theory in Statistical Physics 335
ni = 0, 1. The formulas for the partition function, free energy, and
mean energy are given by
Z = Π(1 + e−βωi),
F = − 1
β
∑log(1 + e−βωi),
H =∑
ωini,
where ni = 1eβωi+1
.
13.2 Equilibrium states of translation-invariant
Hamiltonians
In momentum representation, one can write a translation-invariant
Hamiltonian in the form
H =∑m,n
∫Hm,n(k1, . . . , km | l1, . . . , ln)
× a+(k1) · · · a+(km)a(l1) · · · a(ln)dmkdnl, (13.7)
where Hm,n(k1, . . . , km | l1, . . . , ln) contains a factor δ(k1 + · · · + km− l1 − · · · − ln). (We assume that the arguments belong to RD.)
Even in the simplest case, when H =∫ω(p)a+(p)a(p)dp, the
formula for the equilibrium state makes no sense (the integral for
the partition function diverges). Moreover, as we have seen, the
formula (13.7) in general does not define an operator on Fock
space. However, the Hamiltonian with volume cutoff HΩ specifies an
operator in Fock space under some mild conditions and we can define
an equilibrium state and correlation functions. In particular, we
can consider the correlation functions βwΩn (k1, ε1, t1, . . . , kn, εn, tn) =
〈aε1k1(t1) · · · aεnkn(tn)〉β generalizing Wightman functions. (Wightman
functions are correlation functions for T = 0.)
We can define correlation functions in infinite volume βwn =
limβ wΩn by taking the limit Ω → ∞ (removing the volume cutoff).
As in Section 8.2, the limit is understood in the sense of generalized
functions. Under some mild conditions, one can prove that the
correlation functions in infinite volume obey KMS condition (13.3)
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336 Mathematical Foundations of Quantum Field Theory
(one should verify that these functions can be analytically extended
to the strip 0 ≤ =t ≤ β).
Note that the limit at hand does not always exist, or, better, there
exist many limits, i.e. many equilibrium states in infinite volume. The
limiting procedure we have used can be slightly modified in many
ways: we can use various volume cutoffs or add to the Hamiltonian
HΩ terms that tend to zero as Ω tends to ∞. If the limit exists,
we say that it gives correlation functions in the equilibrium state
(more precisely, in one of equilibrium states.) We can repeat with
minimal changes the considerations of Sections 8.1 and 8.2. Again,
as in Section 8.2, we can construct for every β, a Hilbert space H =
H(β), a dense subset D, a vector Φ, operators H, P, and operator
generalized functions aε(k, t) in such a way that
βwn(k1, ε1, t1, . . . , kn, εn, tn) = 〈aε1(k1, t1) · · · aεn(kn, tn)Φ,Φ〉
(this is the Reconstruction theorem). The only difference is that the
spectral condition should be replaced by the KMS condition.
The operator generalized functions aε(k, t) obey the Heisenberg
equations. This allows us to say that the objects we have constructed
specify an operator realization of the equilibrium state. (The defini-
tion of an operator realization is the same as in Section 8.1, but
the condition that Φ is a ground state is replaced by the KMS
condition.)
13.3 Algebraic approach to quantum theory
Quantum mechanics can be formulated in terms of an algebra
of observables. The starting point of this formulation is a unital
associative algebra A over C (the algebra of observables). One
assumes that this algebra is equipped with antilinear involution
A→ A∗. One says that a linear functional ω on A specifies a state if
ω(A∗A) ≥ 0 (i.e. if the functional is positive). The space of states will
be denoted by C. If ω(1) = 1, we say that the state is normalized.
The probability distribution ρ(λ) of a real observable A = A∗ in
a normalized state ω should obey the relation ω(An) =∫λnρ(λ)dλ.
(In general, this formula does not specify the probability distribution
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Methods of Quantum Field Theory in Statistical Physics 337
uniquely. However, if for every continuous function f of a real
argument, we can define an element f(A) of the algebra A, then
the probability distribution is specified uniquely by the formula
ω(f(A)) =∫f(λ)ρ(λ)dλ. In particular, f(A) is well defined for every
continuous function f if A is a C∗-algebra.)
In the standard exposition of quantum mechanics, the algebra
of observables consists of operators acting on a (pre-)Hilbert space.
Every vector x having a unit norm specifies a state by the formula
ω(A) = 〈Ax, x〉. (More generally, a density matrix K defines a state
ω(A) = TrAK.) This situation is in some sense universal: for every
state ω on A, one can construct a (pre-)Hilbert space H and a
representation of A by operators on this space in such a way that
the state ω corresponds to a vector in this space.
To construct H, one defines inner product on A by the formula
〈A,B〉 = ω(A∗B). The space H can be obtained from A by means of
factorization with respect to zero vectors of this inner product. The
inner product on A descends toH providing it with the structure of a
pre-Hilbert space. The state ω is represented by a vector Φ of H that
corresponds to the unit element of A. An operator of multiplication
from the left by an element C ∈ A descends to an operator C acting
on H; this construction gives a representation of A in the algebra
of bounded operators on H. The algebra of operators of the form C
where C ∈ A will be denoted by A(ω). (In our definition, H is a
pre-Hilbert space; taking its completion, we obtain a representation
of A by operators in Hilbert space H.) The above construction is
called Gelfand–Naimark–Segal (GNS) construction.
Although every state of the algebra A can be represented by a
vector in Hilbert space, in general, it is impossible to identify Hilbert
spaces corresponding to different states.
Time evolution in the algebraic formulation is specified by a one-
parameter group α(t) of automorphisms of the algebra A preserving
the involution. This group acts in the obvious way on the space of
states. If ω is a stationary state (a state invariant with respect to time
evolution), then the group α(t) descends to a group U(t) of unitary
transformations of the corresponding space H. The generator H of
U(t) plays the role of the Hamiltonian; it can be considered as a self-
adjoint operator in H. The vector Φ representing ω obeys HΦ = 0.
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338 Mathematical Foundations of Quantum Field Theory
We say that the stationary state ω is a ground state if the spectrum
of H is non-negative.
Every element B of the algebra A specifies two operators acting
on linear functionals on A:
(Bω)(A) = ω(BA), (13.8)
(Bω)(A) = ω(AB∗). (13.9)
Note that the first of these operators is denoted by the same
symbol as the element. The operator B commutes with the operator
B′. These operators do not preserve positivity, but the operator
BB does. This means that the operator sending ω into BB(ω)
acts on the space of states. If a state ω corresponds to a vector
Φ in a representation of A (i.e. ω(A) = 〈AΦ,Φ〉), then the state
corresponding to the vector BΦ is equal to BBω.
13.3.1 Quantum field theory and statistical
physics in Rd
Symmetries of quantum theory in the algebraic formulation are
automorphisms of the algebra A commuting with the involution and
the evolution automorphisms α(t). We are especially interested in the
case when among the symmetries are operators α(x, t), where x ∈ Rdand t ∈ R, that are automorphisms of A obeying α(x, t)α(x′, t′) =
α(x + x′, t+ t′). We will use the notation A(x, t) for α(x, t)A, where
A ∈ A.
These automorphisms can be interpreted as space–time transla-
tions. One should expect that starting with a formal translation-
invariant Hamiltonian, one can construct an algebra A and space–
time translations α(x, t) in such a way that the corresponding
equilibrium states can be identified with the equilibrium states of the
preceding section (this statement can be justified in the framework
of perturbation theory).
We say that the algebra A and space–time translations specify
a quantum theory in Rd. We will define particles as elementary
excitations of the ground state; the theory of these particles and their
collisions is a quantum field theory through an algebraic approach.
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Methods of Quantum Field Theory in Statistical Physics 339
The action of a translation group on A induces an action of this
group on the space of states. In relativistic quantum field theory,
one should have an action of the Poincare group on the algebra of
observables (hence on states).
Let us now consider a state ω that is invariant with respect to
the translation group. We will define (quasi-)particles as “elementary
excitations” of ω. To consider collisions of (quasi-)particles, we should
require that ω satisfies a cluster property in some sense.
The weakest form of cluster property is the following condition:
ω(A(x, t)B) = ω(A)ω(B) + ρ(x, t), (13.10)
where A,B ∈ A and ρ is small in some sense for x→∞. For example,
we can impose the condition that∫|ρ(x, t)|dx < c(t), where c(t) has
at most polynomial growth. Note that (13.10) implies asymptotic
commutativity in some sense: ω([A(x, t), B]) is small for x→∞.To formulate a more general cluster property, we introduce the
notion of correlation functions in the state ω :
wn(x1, t1, . . . ,xn, tn) = ω(A1(x1, t1) . . . An(xn, tn)),
where Ai ∈ A. These functions generalize Wightman functions of
relativistic quantum field theory. We consider the corresponding
truncated correlation functions wTn (x1, t1, . . . ,xn, tn).
We have assumed that the state ω is translation-invariant; it
follows that both correlation functions and truncated correlation
functions depend on the differences xi − xj , ti − tj . We say that
the state ω has the cluster property if the truncated correlation
functions are small for xi − xj → ∞. A strong version of the
cluster property is the assumption that the truncated correlation
functions tend to zero faster than any power of min ‖xi − xj‖. Then
its Fourier transform with respect to variables xi has the form
νn(p2, . . . ,pn, t1, . . . , tn)δ(p1 + · · · + pn), where the function νn is
smooth. We will later need a weaker form of cluster property that
can be formulated as a requirement that the function νn is three
times continuously differentiable with respect to p2, . . . ,pn.
Instead of cluster property, one can impose a condition of
asymptotic commutativity of the algebra A(ω). In other words, one
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340 Mathematical Foundations of Quantum Field Theory
should require that the commutator [A(x, t), B], where A,B ∈ A is
small for x → ∞. For example, we can assume that for every n the
norm of this commutator is bounded from above by Cn(t)(1+‖x‖)−n,
where the function Cn(t) has at most polynomial growth. (This
property will be called strong asymptotic commutativity.) Note
that in relativistic quantum field theory, in the Haag–Araki or the
Wightman formulation, strong asymptotic commutativity can be
derived from locality.
Let us show how one can define one-particle excitations of the
state ω and the scattering of (quasi-)particles.
The action of the translation group on A generates a unitary
representation of this group on the pre-Hilbert space H constructed
from ω. Generators of this representation P and −H are identified
with the momentum operator and the Hamiltonian. The vector in
the space H that corresponds to ω will be denoted by Φ. If Φ is
a ground state, we say that it is the physical vacuum. If ω obeys
the KMS condition ω(A(t)B) = ω(BA(t+ iβ)), we say that ω is an
equilibrium state with the temperature T = 1β .
13.3.2 Particles and quasiparticles
We say that a state σ is an excitation of ω if it coincides with ω at
infinity. More precisely, we should require that σ(A(x, t))→ ω(A) as
x → ∞ for every A ∈ A. Note that the state corresponding to any
vector AΦ, where A ∈ A is an excitation of ω; this follows from the
cluster property.
One can define a one-particle state (a one-particle excitation of
the state ω or elementary excitation of ω) as a generalized H-valued
function Φ(p) obeying PΦ(p) = pΦ(p), HΦ(p) = ε(p)Φ(p). (More
precisely, for some class of test functions f(p), we should have
a linear map f → Φ(f) of this class into H obeying PΦ(f) =
Φ(pf), HΦ(f) = Φ(ε(p)f), where ε(p) is a real-valued function
called dispersion law. For definiteness, we can assume that test
functions belong to the Schwartz space S(Rd).) Let us fix an element
B ∈ A such that BΦ = Φ(φ). (We assume that φ is smooth and does
not vanish anywhere.) Recall that we consider H as a pre-Hilbert
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Methods of Quantum Field Theory in Statistical Physics 341
space obtained by the GNS construction, therefore, every element of
H can be represented in the form BΦ.
Note that for every function g(x, t) ∈ S(Rd+1), we have
B(g)Φ = Φ(gf ), (13.11)
where B(g) =∫g(x, t)B(x, t)dxdt, gf (p) = g(p,−ε(p))f(p), and g
stands for the Fourier transform of g.
It follows that we can always assume that B =∫α(x, t)
A(x, t)dxdt, where α(x, t) ∈ S(Rd+1), A ∈ A. We will say that an
operator B satisfying this assumption and transforming Φ into one-
particle state is a good operator. For a good operator, B(x, t) =∫α(x′ − x, t′ − t)A(x′, t′)dx′dt′, hence this expression is a smooth
function of x and t.
We assume that Φ(f) is normalized (i.e. 〈Φ(f),Φ(f ′)〉 = 〈f, f ′〉).Note that it is possible that there exist several types of (quasi-)
particles Φr(f) with different dispersion laws. We say that the space
spanned by Φr(f) is one-particle space and denote itH1. If the theory
is invariant with respect to spatial rotations, then the infinitesimal
rotations play the role of components of angular momentum. If d = 3,
a particle with spin s can be described as a collection of 2s + 1
functions Φr(p) obeying PΦr(p) = pΦr(p), HΦr(p) = ε(p)Φr(p).
The group of spatial rotations acts in the space spanned by these
functions; this action is a tensor product of the standard action
of rotations of the argument and irreducible (2s + 1)-dimensional
representation. (Here, s is half-integer, the representation is two-
valued if s is not an integer.) In relativistic theory, an irreducible
subrepresentation of the representation of the Poincare group in Hspecifies a particle with spin.
13.3.3 Scattering
Let us assume that we have several types of (quasi-)particles defined
as generalized functions Φk(p) obeying PΦk = pΦk(p), HΦk(p) =
εk(p)Φk(p), where the functions εk(p) are smooth and strictly
convex. Take some good operators Bk ∈ A, obeying BkΦ = Φ(φk).
Define Bk(f, t), where f is a function of p as∫f(x, t)Bk(x, t)dx and,
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342 Mathematical Foundations of Quantum Field Theory
as earlier, f(x, t) is a Fourier transform of f(p)e−iεk(p)t with respect
to p.
Let us consider the vectors
Ψ(k1, f1, . . . , kn, fn | t1, . . . , tn) = Bk1(f1, t1) · · · Bkn(fn, tn)Φ.
(13.12)
We assume that f1, . . . , fn have compact support. Let us introduce
the notation vi(p) = ∇εki(p). The set of all vectors vi(p) with
fi(p) 6= 0 will be denoted Ui. We assume that the sets Ui (the closures
of Ui) do not overlap. This assumption will be called NO condition
in what follows.
Then assuming the strong cluster property or asymptotic com-
mutativity (precise formulation will be given below), one can prove
that the vector (13.12) has a limit as ti → ∞ or ti → −∞. The set
spanned by these limits will be denoted D+ or D−.Note that the assumption that the sets Ui do not intersect (NO
condition) can be omitted if the space–time dimension is≥ 4. In these
dimensions, we can drop the NO condition defining the sets D±.The existence of the limit of the vectors (13.12) allows us to
define Møller matrices. We introduce the space Has as a Fock
representation for the operators a+k (f), ak(f) obeying canonical
commutation relations (CCR).
We define Møller matrices S− and S+ as operators defined on Hasand taking values in H by the formula
Ψ(k1, f1, . . . , kn, fn | ±∞) = S±(a+k1
(f1φk1) · · · a+kn
(fnφkn)θ).
(13.13)
This formula specifies S± on a dense subset of the Hilbert space Has.These operators are isometric, hence they can be extended to Hasby continuity.
One can say that the vector
e−iHtΨ(k1, f1, . . . , kn, fn | ±∞)
= Ψ(k1, f1e−iεk1 t, . . . , kn, fne
−iεkn t | ±∞) (13.14)
describes the evolution of a state corresponding to a collection
of n particles with wave functions f1φ1e−iεk1 t, . . . , fnφne
−iεkn t as
t→ ±∞.
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Note that we use the notation ak(f) =∫f(p)ak(p)dp, a+
k (f) =
(ak(f))∗ =∫f(p)a+
k (p)dp, hence a+k (f) =
∫f(p)a+(p)dp.
The existence of the limit in (13.12) can be derived from the strong
cluster property if we impose an additional condition B∗kΦ = 0 or the
condition B∗kΦ = Φk(νk).
Let us sketch the proof that the limit in (13.12) exists, assuming
for simplicity that the times ti are equal: ti = t. Let us denote the
LHS of (13.12) as Ψ(t). It is sufficient to prove that the norm of the
derivative of this vector with respect to t is a summable function:∫‖Ψ(t)‖dt <∞.Note that ‖Ψ(t)‖2 can be expressed in terms of correlation
functions, hence in terms of truncated correlation functions. If we
assume the strong cluster property, then ‖Ψ(t)‖ tends to zero faster
than any power of |t|, this implies the existence of the limit and the
fact that ‖Ψ(±∞)−Ψ(t)‖ also tends to zero faster than any power.
If we are interested only in the existence of the limit, it is sufficient
to assume a weaker version of cluster property (three continuous
derivatives of the function νn). This is sufficient to prove that
‖Ψ(t)‖ < C|t|−32 . Every factor in an arbitrary term of the expansion
for ‖Ψ(t)‖2 has the form
Ik,l(t) = 〈( ˙Bi1(fi1 , t))
∗ · · · ( ˙Bik(fik , t))
∗
× ˙Bj1(fj1 , t) · · ·
˙Bjl(fjl , t)〉
T ,
where the dots above the operators denote differentiation with
respect to time, which can enter in one of the first k operators and in
one of the last l operators. We can suppose without loss of generality
that k ≥ 1, l ≥ 1 (otherwise Ik,l = 0; this follows from the remark
that one-particle states are orthogonal to Φ.). If each factor in the
given term has k = l = 1, then the term is equal to zero because one
of the factors either has the form
〈( ˙Bi(fi, t))
∗ ˙Bj(fj , t)〉T = 〈( ˙
Bi(fi, t))∗ ˙Bj(fj , t)Φ,Φ〉,
or the form
〈(Bi(fi, t))∗Bj(fj , t)〉T = 〈(Bi(fi, t))∗Bj(fj , t)Φ,Φ〉,
and both expressions are zero.
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In all other cases, the factor tends to zero as t → ∞ faster than
any power of t. This follows from Lemma 10.2 of Section 10.1.
If the space–time dimension is ≥ 4, we can prove the existence
of the limit of vectors (13.12) without the NO condition; the proof
generalizes the considerations used in Section 10.1 for d = 3.
The definition of S± that we gave specifies these operators as
multi-valued maps (for example, we can use different good operators
in the construction and it is not clear whether we get the same
answer). However, we can check that this map is isometric and every
multi-valued isometric map is really single-valued (see Section 10.1).
In particular, this means that the definition does not depend on the
choice of good operators.
To prove that the map is isometric, we express the inner product
of two vectors of the form Ψ(t) in terms of truncated correlation
functions. Only two-point truncated correlation functions survive in
the limit t → ±∞. This allows us to say that the map is isometric
(see Section 10.1 for a more detailed proof in a slightly different
situation).
Let us define in- and out-operators by the formulas
ain(f)S− = S−a(f), a+in(f)S− = S−a
+(f),
aout(f)S+ = S+a(f), a+out(f)S+ = S+a
+(f).
(For simplicity of notations, we consider the case when we have only
one type of particles. If we have several types of particles, in- and
out-operators as well as the operators a+, a are labeled by a pair
(k, f), where f is a test function and k characterizes the type of
particle.) These operators are defined on the image of S− and S+
correspondingly. One can check that
a+in(f φ) = lim
t→−∞B(f, t), a+
out(f φ) = limt→∞
B(f, t). (13.15)
The limit is understood as a strong limit. It exists on the set of
vectors of the form (13.13) (with NO assumption for d < 3). The
proof follows immediately from the fact that taking the limit ti →∞
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in the vectors (13.12), we can first take the limit for i > 1 and then
the limit t1 →∞.The formula (13.15) can be written in the following way:
a+in(f φ)Ψ(f1, . . . , fn | −∞) = Ψ(f, f1, . . . , fn | −∞),
a+out(f φ)Ψ(f1, . . . , fn |∞) = Ψ(f, f1, . . . , fn |∞).
Similarly,
ain(f)Ψ(φ−1f, f1, . . . , fn | −∞) = Ψ(f1, . . . , fn | −∞), (13.16)
aout(f)Ψ(φ−1f, f1, . . . , fn |∞) = Ψ(f1, . . . , fn |∞). (13.17)
If the operators S+ and S− are unitary, we say that the theory has
a particle interpretation. In this case (and also in the more general
case when the image of S− coincides with the image of S+), we can
define the scattering matrix
S = S∗+S−.
The scattering matrix is a unitary operator in Has. Its matrix
elements in the basis |p1, . . . , pn〉 = 1n!a
+(p1) · · · a+(pn)θ (scattering
amplitudes) can be expressed in terms of in- and out-operators.
Smn(p1, . . . ,pm |q1, . . . ,qn)
= 〈a+in(q1) · · · a+
in(qn)Φ, a+out(p1) · · · a+
out(pm)Φ〉. (13.18)
Effective cross-sections can be expressed in terms of the squares
of scattering amplitudes.
Note that only when ω is a ground state can one hope that the
particle interpretations exists. In other cases, instead of a scattering
matrix and cross-sections, one should consider inclusive scattering
matrix and inclusive cross-sections (see the following).
Formula (13.18) is proved only for the case when all values of
momenta pi,qj are distinct. More precisely, this formula should
be understood in the sense of generalized functions and as test
functions we should take collections of functions fi(pi), gj(qj) with
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346 Mathematical Foundations of Quantum Field Theory
non-overlapping U(fi), U(gj). Let us write this in more detail:
Smn(f1, . . . , fm | g1, . . . , gn)
=
∫dmpdnq
∏fi(pi)
∏gj(qj)Smn(p1, . . . ,pm |q1, . . . ,qn)
= 〈a+in(g1) · · · a+
in(gn)Φ, a+out(f1) · · · a+
out(fm)Φ〉.
Using (13.15), we obtain
Smn(f1, . . . , fm | g1, . . . , gn)
= limt→∞,τ→−∞
〈B(fmφ−1, t)∗ · · · B(f1φ
−1, t)∗B(g1φ−1, τ)
· · · B(gnφ−1, τ))Φ,Φ〉
= limt→∞,τ→−∞
ω(B(fmφ−1, t)∗ · · ·B(f1φ
−1, t)∗B(g1φ−1, τ)
· · · B(gnφ−1, τ)),
where B(f, t)∗ =∫dxB∗(x, t)f(x, t).
Note that in the same way, we can obtain a more general formula
Smn(f1, . . . , fm | g1, . . . , gn)
= limt→∞,τ→−∞
〈Bm+1(f1φ−1m+1, τ)
· · · Bm+n(gnφ−1m+n, τ))Φ, B1(f1φ
−11 , t) · · ·Bm(fmφ
−1m , t)Φ〉
= limt→∞,τ→−∞
ω(Bm(fmφ−1m , t)∗ · · ·B1(f1φ
−11 , t)∗Bm+1(g1φ
−1m+1, τ)
· · ·Bm+n(gnφ−1m+n, τ)), (13.19)
where Bi are different good operators and BiΦ = Φ(φi).
13.3.4 Asymptotic behavior of 〈Q(x, t)Ψ,Ψ′〉
Our next goal is to calculate the asymptotic behavior as t → ∞ of
the expression 〈Q(x, t)Ψ,Ψ′〉 where Q ∈ A.
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We assume that the NO condition is satisfied and that vectors
Ψ,Ψ′ belong to D+. It is sufficient to consider the case
Ψ = limt→∞
Ψ(t),Ψ(t) = Ψ(f1, . . . , fn | t),
Ψ′ = limt→∞
Ψ′(t),Ψ(t) = Ψ(f ′1, . . . , f′m | t).
(To simplify the notations, we consider the case of one type of
particles.) We assume that the strong cluster property is satisfied.
Then the differences between Ψ and Ψ(t), and between Ψ′ and Ψ′(t)
are negligible for t → ∞ (we neglect terms tending to zero faster
than any power of t, more precisely terms that are less than Cnt−n
where Cn does not depend on x). We see that it is sufficient to study
the asymptotic behavior of
〈Q(x, t)Ψ(t),Ψ′(t)〉
= 〈Q(x, t)B(f1, t) · · ·B(fn, t)Φ, B(f ′1) · · ·B(f ′1, t) · · ·B(f ′m, t)Φ〉
= ω(B∗(f ′m, t) · · ·B∗(f ′1, t)Q(x, t)B(f1, t) · · ·Bn(fn, t)).
We decompose this expression in terms of truncated correlation
functions. It is clear that every non-negligible truncated function
should contain equal number of B’s and B∗’s. Using this remark
and (13.17), we obtain
〈Q(x, t)Ψ,Ψ′〉 = 〈Ψ,Ψ′〉M +
∫dpdp′〈aout(p)Ψ, aout(p
′)Ψ′〉N(p,p′)
+
∫dp〈aout(p)Ψ,Ψ′〉T1(p)
+
∫dp′〈Ψ, aout(p
′)Ψ′〉T2(p′) +R, (13.20)
where R is negligible,
M = ω(Q) = 〈QΦ,Φ〉, N(p,p′) = 〈Q(x, t)Φ(p),Φ(p′)〉,
T1(p) = 〈Q(x, t)Φ(p),Φ〉, T2(p′) = 〈Q(x, t)Φ,Φ(p′)〉.
Similar formulas can be written for t→ −∞.
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13.3.5 Scattering theory from asymptotic
commutativity
In this section, we derive the existence of limit in (13.12) from
asymptotic commutativity of the algebra A(ω). This approach is
useful in relativistic theory where asymptotic commutativity follows
from locality.
Let us assume that the operators Bk and Bl asymptotically
commute:
‖[Bk(x, t), Bl(x′, t)]‖ <C
1 + |x− x′|a,
where a > 1. We also suppose the condition of asymptotic com-
mutativity is satisfied for the spatial and time derivatives of these
operators.
Instead of these conditions, we can assume that Bk =∫gk(x, t)Ak(x, t)dxdt and
‖[Ak(x, t), Al(x′, t′)]‖ <C(t− t′)
1 + |x− x′|a,
where gk are functions from Schwartz space, C(t) is a function of at
most polynomial growth and a > 1.
The statement we need follows from the following fact:∫‖[Bki(fi, t),
˙Bkj (fj , t)]‖dt <∞. (13.21)
To check this, we note that Ψ(t) is a sum of n terms; every term
contains a product of several operators B and one operator˙B. The
estimate for the norm of Ψ(t) follows from (13.21) and (13.11) and
the remark that the norms of operators Bki(fi, t) are bounded. (We
change the order of operators in the summand under consideration in
such a way that˙B is from the right. Then we note that
˙B(f, t)Φ = 0
as follows from (13.11).)
Equation (13.21) remains to be proved. First of all, we note that
one can prove the estimate
|fk(x, t)| ≤ C|t|−D2 (13.22)
(the proof for d = 3 given in Section 10.2 works for any d).
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Let us consider the integral∫dxdx′fki(x, t)fkj (x
′, t)[Bki(x, t),˙Bkj (x
′, t)]. (13.23)
First of all, we consider the integral (13.23) over the domain Γ(t)
defined in the following way. Take non-overlapping compact sets G ⊂Rd, G′ ⊂ Rd. We say that the point (x,x′) belongs to Γ(t) if there
exist points v ∈ G,v′ ∈ G′ such that |v − xt | < C|t|−ρ, |v′ − x′
t | <C|t|−ρ where ρ = 1
2−ε. (We will apply our estimate in the case when
G is the set of points of the form ∇εki(p) where p belongs to the
support of fki and G′ is defined in similar way using the function
fkj .) For large t, the volume of this domain is less than t2d(1−ρ) and
the norm of the integrand is less than t−d × t−a. (We have used the
fact that the distance between x and x′ grows linearly with t in the
integration domain, hence the norm of the commutator in (13.23) is
less than t−a.) This allows us to say that the norm of the integral
does not exceed t−a+ε.
To estimate the integral (13.21) over the complement to Γ(t), we
consider the integral∫exp(−iε(p)t+ ipx)µ
(tρ(v(p)− x
t
))f(p)dp, (13.24)
where µ(x) is a smooth function equal to zero for |x| ≤ ν1 and equal
to one for |x| ≥ ν2. One can prove that for every r, the absolute value
of this integral does not exceed C|t|−(1−2ρ)r and for |x| > bt, it does
not exceed C|x|−r|t|−(1−2ρ)r.
The integral (10.24) is equal to f(x, t) if tρ(v(p)− xt ) ≥ ν2 for p ∈
supp f. This means that the estimate of this integral can be applied
to at least one of the factors fki(x, t), fkj (x′, t) in the integrand of
the integral (13.23) on the complement to the domain Γ(t). Using
this fact and the inequality∫dx|fk(x, t)| ≤ C|t|
d2 ,
we obtain that the norm of the integral (13.23) over the complement
to Γ(t) tends to zero faster than any power of |t| as t → ±∞.Combining this estimate with the estimate for the integral over Γ(t)
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350 Mathematical Foundations of Quantum Field Theory
and taking ε < a−1, we obtain the estimate for the norm of (13.23):
for large |t|, it is less than |t|−b where b > 1. The same arguments
work if the dot (time derivative) in (13.23) is removed.
To prove (13.21), we note that calculating˙B(f, t) we should take
into account that f depends on t. It is easy to check that
˙B(f, t) =
∫dxg(x, t)B(x, t) +
∫dxf(x, t)
˙B(x, t),
where g(p) = −iε(p)f(p). Using this expression, we obtain that the
commutator in (13.21) can be written as a sum of (13.23) and similar
expression with removed dots. Both summands do not exceed |t|−bwith b > 1 for large |t|. This implies (13.21).
Note that the proof based on asymptotic commutativity also
allows us to estimate the speed of convergence to the limit. In the
case of strong asymptotic commutativity, the difference between Ψ(t)
and Ψ(±∞) tends to zero faster than any power of t.
13.3.6 Green functions and scattering: LSZ
Let us start with scattering of elementary excitations of ground state
(of particles). In this case, the scattering matrix can be expressed in
terms of on-shell values of Green functions. The Green function in
translation-invariant stationary state ω is defined by the formula
Gn = ω(T (A1(x1, t1) · · ·Ar(xr, tr))),
where Ai ∈ A and T stands for time ordering. More precisely,
this is a definition of Green function in (x, t)-representation, taking
Fourier transform with respect to x, we obtain Green functions
in (p, t)-representation; taking in these functions inverse Fourier
transform with respect to t, we obtain Green functions in in (p, ε)-
representation. Due to translation-invariance of ω, we obtain that
in (x, t)-representation, the Green function depends on differences
xi − xj , in (p, t)-representation, it contains a factor δ(p1 + · · ·+ pr).
Similarly, in (p, ε), we have the same factor and the factor
δ(ε1 + · · ·+ εr). We omit both factors talking about poles of Green
functions.
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For n = 2 in (p, ε)-representation, G2 has the form
G(p1, ε1 |A,A′)δ(p1 + p2)δ(ε1 + ε2).
Poles of function G(p, ε |A,A′) correspond to particles, the depen-
dence of the position of the pole on p specifies the dispersion law
ε(p) (we consider poles with respect to the variable ε for fixed p).
This follows from Kallen–Lehmann representation (Section 8.1), but
this can be obtained also from the following considerations.
We will prove that the scattering amplitudes can be obtained
as on-shell values of Green functions (LSZ formula). To simplify
notations, we consider the case when we have only one single-
particle state Φ(p) with dispersion law ε(p). We assume that the
elements Ai ∈ A are chosen in such a way that the projection of
AiΦ on the one-particle space has the form Φ(φi) =∫φi(p)Φi(p)dp
where φi(p) is a non-vanishing function. We introduce the notation
Λi(p) = φi(p)−1.
Let us consider Green function
Gmn = ω(T (A∗1(x1, t1) · · ·A∗m(xm, tm)Am+1(xm+1, tm+1)
· · ·Am+n(xm+n, tm+n))
in (p, ε)-representation. It is convenient to change slightly the defini-
tion of (p, t)- and (p, ε)-representation changing the signs of variables
pi and εi for 1 ≤ i ≤ m (for variables corresponding to the operators
A∗i ). This convention agrees with the conventions in Section 9.2.
Multiplying the Green function in (p, ε)-representation by∏1≤i≤m
Λi(pi)(εi + ε(pi))∏
m<j≤m+n
Λj(pj)(εj − ε(pj)),
and taking the limit εi → −εi(pi) for 1 ≤ i ≤ m and the limit εj →εj(pj) for m < j ≤ m+n, we obtain on-shell Green function denoted
by σmn. We prove that it coincides with scattering amplitudes:
σm,n(p1, . . . ,pm+n) = Smn(p1, . . . ,pm |pm+1, . . . ,pm+n). (13.25)
First of all, we note that the on-shell Green function can be expressed
in terms of the asymptotic behavior of the Green function in (p, t)-
representation: if for t→ ±∞ and fixed p, this behavior is described
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352 Mathematical Foundations of Quantum Field Theory
by linear combination of exponent eiλt, then the location of poles is
determined by the exponent indicators λ and the coefficients in front
of exponents determine the residues. Using this statement, one can
show that
limt→∞,τ→−∞
ω(Bm(fmφ−1m , t)∗ · · ·B1(f1φ
−11 , t)∗Bm+1(g1φ
−1m+1, τ)
· · ·Bm+n(gnφ−1m+n, τ)), (13.26)
where Bi are good operators can be expressed in terms of on-shell
Green function as∫dm+npf1(p1) . . . fm(pm)g1(−pm+1)
. . . gn(−pm+n)σm,n(p1, . . . ,pm+n).
Using (13.19), we obtain (13.25) in the case when Ai are good
operators. We will show that the general case can be reduced to
the case when operators Ai are good.
Let us suppose that one-particle spectrum does not overlap with
multi-particle spectrum. (We represent the space H as a direct sum
of one-dimensional space H0 spanned by Φ, one-particle space H1
and the space M called multi-particle space. Our condition means
that the joint spectrum of P and H in M (multi-particle spectrum)
does not overlap with the joint spectrum of P and H in H0 + H1.
If the theory has particle interpretation, then our condition means
that ε(p1 +p2) < ε(p1)+ε(p2).1 Then we can construct an operator
B transforming Φ into one-particle state (a good operator) using the
formula B =∫α(x, t)A(x, t)dxdt where A ∈ A and the projection
of AΦ onto one-particle state does not vanish. Namely, we should
assume that the support of α(p, ω) (of the Fourier transform of α)
does not intersect the multi-particle spectrum and does not contain 0.
Moreover, the operator we constructed obeys B∗Φ = 0.
1The physical meaning of this condition: the energy conservation law forbids thedecay of a particle. This condition is not always satisfied, however, stability of aparticle is always guaranteed by some conservation laws. Our considerations canbe applied in this more general situation.
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Now, we can apply this construction to the operators Ai and
verify that on-shell Green functions corresponding to good operators
Bi coincide with Green functions corresponding to the operators
Ai. (We note that the correlation functions of operators Bi can be
expressed in terms of correlation functions of operators Ai. As we
mentioned already, on-shell Green function can be expressed in terms
of asymptotic behavior as t → ±∞ of the Green function in (p, t)-
representation. It is easy to find the relation between this behavior
for the Green functions of operators Ai and the behavior for the
Green functions of operators Bi.)
For relativistic theories with a mass gap, one can derive asymp-
totic commutativity and cluster property from Haag–Araki or Wight-
man axioms. This proves the existence of scattering matrix. One can
prove that scattering matrix is Lorentz-invariant; let us sketch the
proof of this fact.
The operator B(f, t) used in our construction can be written as an
integral of f(x, t)B(x, t), where f(x, t) is a positive frequency solution
of the Klein–Gordon equation over the hyperplane t = constant . We
will define the operator B(f, ρ) integrating the same integrand over
another hyperplane ρ. One can replace the operators B(f, t) by the
operators B(f, ρ) in (13.12) and prove that the expression we obtained
has a limit if the hyperplanes ρi tend to infinity in time direction (for
example, if they have form αt+ax = constant and the constant tends
to infinity). This means that we can use B(f, ρ) in the definition of
Møller matrices and scattering matrix. The same arguments that were
used to prove that the limit does not depend on the choice of good
operators can be applied to verify that the new construction gives
the same Møller matrices. This implies Lorentz invariance because
the Lorentz group acts naturally on operators B(f, ρ).
13.3.7 Generalized Green functions; the inclusive
scattering matrix
Let us define generalized Green functions (GGreen functions) in the
state ω by the following formula where Bi ∈ A:
Gn = ω(MN),
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354 Mathematical Foundations of Quantum Field Theory
where
N = T (B1(x1, t1) · · ·Bn(xn, tn))
stands for chronological product (times decreasing) and
M = T opp(B∗1(x′1, t′1) · · ·B∗n(x′n, t
′n))
stands for antichronological product (times increasing).
One can give another definition of GGreen functions introducing
the operator
Q = T (B1(x1, t1) · · ·Bn(xn, tn)B1(x′1, t′1) · · · Bn(x′n, t
′n)),
where the operators Bi, Bi act on the space of linear functionals on
A. (Recall (13.8) and (13.9) that operators B and B act on linear
functionals defined on A; they transform ω(A) into ω(BA) and in
ω(AB∗) correspondingly.) It is easy to check that
Gn = (Qω)(1).
Let us define inclusive S-matrix as on-shell GGreen function. We
will show that in the case when the theory has particle interpretation,
inclusive cross-section can be expressed in terms of inclusive S-
matrix.
Recall that the inclusive cross-section of the process (M,N) →(Q1, . . . , Qm) is defined as a sum (more precisely, a sum of integrals)
of effective cross-sections of the processes (M,N) → (Q1, . . . , Qm,
R1, . . . , Rn) over all possible R1, . . . , Rn. If the theory does not have
particle interpretation, this formal definition of inclusive cross-section
does not work, but still the inclusive cross-section can be defined in
terms of probability of the process (M,N → (Q1, . . . , Qn+ something
else)) and expressed in terms of inclusive S-matrix.
Let us consider the expectation value
ν(a+out,k1
(p1)aout,k1(p1) · · · a+out,km
(pm)aout,km(pm)), (13.27)
where ν is an arbitrary state. This quantity is the probability density
in momentum space for finding m outgoing particles of the types
k1, . . . , kn with momenta p1, . . . ,pm plus other unspecified outgoing
particles. It gives inclusive cross-section if ν = ν(t) describes the
evolution of a state represented as a collection of incoming particles.
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Methods of Quantum Field Theory in Statistical Physics 355
It will be convenient to consider more general expectation value
ν(a+out,k1
(f1)aout,k1(g1) · · · a+out,km
(fm)aout,km(gm)), (13.28)
where fi, gi are test functions. This expression can be understood as
a generalized function
ν(a+out,k1
(p1)aout,k1(q1) · · · a+out,km
(pm)aout,km(qm));
we get (13.27) taking pi = qi.
As earlier, we assume that we have several types of (quasi-)particles
Φk(p) obeying PΦk = pΦk(p), HΦk(p) = εk(p)Φk(p), where the
functions εk(p) are smooth and strictly convex. Good operators
Bk ∈ A obey BkΦ = Φ(φk). The operator Bk(f, t), where f is a
function of p, defined as∫f(x, t)Bk(x, t)dx (as always f(x, t) is a
Fourier transform of f(p)e−iεk(p)t with respect to p).
Now we can calculate (13.28). First of all, we take as ν the state
corresponding to the vector (13.14). We are representing this vector
in terms of good operators Bi using the formula
Ψ(k1, f1, . . . , kn, fn |∞) = limt→−∞
Ψ(k1, f1, . . . , kn, fn | t)
= limt→−∞
Bk1(f1, t) · · · Bkn(fn, t)Φ.
The corresponding state ν considered as linear functional on A can
be expressed in terms of the state ω corresponding to Φ. (We should
use the remark that the state corresponding to the vector AΦ can
be written as AAω.) Expressing the out-operators by the formula
(13.15), we obtain the expression of (13.28) in terms of GGreen
functions on-shell.
13.4 L-functionals
In the approach of Section 13.2, for every equilibrium state, we should
construct a Hilbert space depending on the temperature, CCR or
CAR are represented in this space, the equilibrium state is described
by a vector in the space. In this section, we will describe a formalism
of L-functionals (positive functionals on Weyl algebra) that can be
used to describe the states corresponding to vectors and density
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356 Mathematical Foundations of Quantum Field Theory
matrices in all representations of CCR. This formalism can be used to
calculate physical quantities for equilibrium state in the framework
of an analog of Feynman diagram technique.
Note that there exists an obvious generalization of all these results
to the case of fermions. In this case, L-functionals are positive
functionals on the algebra with generators satisfying canonical
anticommutation relations (CAR) (Clifford algebra), α and α∗ in
the definition of L-functional are anticommuting variables.
Let us consider a representation of CCR in Hilbert space H. Here,
we understand CCR as relations
[ak, a+l ] = ~δkl, [ak, al] = [a+
k , a+l ] = 0,
where k, l run over a discrete set M. To a density matrix K (or
more generally, to any trace class operator in H), we can assign a
functional LK(α∗, α) defined by the formula
LK(α∗, α) = Tr e−αa+eα∗aK. (13.29)
Here, αa+ stands for∑αka
+k and α∗a for
∑α∗kak, where k runs
over M. This formula makes sense if αα∗ =∑|αk|2 < ∞. (This
follows from the remark that e−αa++α∗a = e~αα
∗/2e−αa+eα∗a is a
unitary operator in H). We can apply (13.29) also in the case when
K is an arbitrary operator of trace class (not necessarily a density
matrix).
One can say that LK is a generating functional of correlation
functions.
One can also consider a more general case when CCR are written
in the form
[a(k), a+(k′)] = ~δ(k, k′), [a(k), a(k′)] = [a(k)+, a(k′)+] = 0,
k, k′ run over a measure space M . We are using the exponential
form of CCR; in this form, a representation of CCR is specified
as a collection of unitary operators e−αa++α∗a obeying appropriate
commutation relations. Here, α(k) is a complex function on the
measure space M , the expressions of the form α∗a, αa+ can be
written as integrals∫α∗(k)a(k)dk,
∫α(k)a+(k)dk over M. In the
space of CCR a(k), a+(k′) become Hermitian conjugate generalized
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Methods of Quantum Field Theory in Statistical Physics 357
operator functions. We always assume that α is square integrable,
then the expression (13.29) is well defined.
Knowing LK , we can calculate 〈A〉L = TrAK where A is an
element of the unital associative algebra A generated by ak, a+l
(Weyl algebra). Hence, we can consider LK as a positive linear
functional 〈A〉L on the algebra A. (We consider A as an algebra
with involution +, positivity means that 〈A+A〉L ≥ 0. Recall that
positive linear functional ω on unital algebra with involution is called
a normalized state if ω(1) = 1.)
A functional L(α∗, α) satisfying the condition L(−α∗,−α) =
L∗(α, α∗) and positivity condition is called physical L-functional. We
say that such a functional is normalized if L(0, 0) = 1. Normalized
physical L-functionals are in one-to-one correspondence with states
on A. We will consider also the space L of all functionals L(α∗, α) (of
all L-functionals). Every trace class operator K in a representation
space of CCR specifies an element of L by the formula (13.29).
We can define an antilinear involution L → L on the space L of
functionals L(α∗, α) by the formula
(L)(α∗, α) = L∗(−α,−α∗). (13.30)
Physical L-functionals are invariant with respect to this involution.
It is easy to check that
LK = LK+ .
Every normalized vector Φ ∈ H specifies a density matrix
and hence a physical L-functional. Conversely, every physical
L-functional corresponds to a vector in some representation of CCR,
given by GNS construction (see the preceding section). One can
characterize this representation by the requirement that it contains
a cyclic vector Φ obeying L(A) = 〈AΦ,Φ〉. However, an L-functional
can be obtained from many density matrices in many representations
of CCR.
An action of Weyl algebra A on L can be specified by operators
b+(k) = ~c+1 (k)− c2(k), b(k) = c1(k)
obeying CCR. Here, c+i (k) are multiplication operators by α∗k, αk
and ci(k) are derivatives with respect to α∗k, αk. This definition is
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358 Mathematical Foundations of Quantum Field Theory
prompted by relations
La(k)K = b(k)LK , La+(k)K = b+(k)LK . (13.31)
Applying the involution L→ L, we obtain another representation of
A on L. It is specified by the operators
b+(k) = −~c+2 (k) + c1(k), b(k) = −c2(k),
obeying CCR and satisfying
LKa+(k) = b(k)LK , LKa(k) = b+(k)LK . (13.32)
More generally, for any A ∈ A, we have an operator acting in
L denoted by the same symbol and obeying A(LK) = LAK (the
left multiplication by A in A specifies the action of A on linear
functionals). We can define also an operator A using the formula
A(L) = A(L). If L = LK , then AL = LKA+ . It is easy to check that
A+B = A+ B, AB = AB, AB = BA. Operators A specify another
action of Weyl algebra on L that commutes with the original action;
one can say that the direct sum of two Weyl algebras acts on L.
Note that the space of physical L-functionals is not invariant with
respect to the operators A and A, however, it is invariant with respect
to the operators AA where A ∈ A. It is easy to check that
AALK = LAKA+ .
Let us consider a Hamiltonian H in a space of representation of
CCR. We will write H in the form
H =∑m,n
∑ki,lj
Hm,n(k1, . . . , km | l1, . . . , ln)a+k1· · · a+
kmal1 · · · aln .
(13.33)
There are two operators in L corresponding to H:
H =∑m,n
∑ki,lj
Hm,n(k1, . . . , km | l1, . . . , ln)b+k1 · · · b+kmbl1 · · · bln
(13.34)
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Methods of Quantum Field Theory in Statistical Physics 359
(we denote it by the same symbol) and
H =∑m,n
∑ki,lj
Hm,n(k1, . . . , km | l1, . . . , ln)b+k1 · · · b+kmbl1 · · · bln .
(13.35)
The equation of motion for the L-functional L(α∗, α) has the form
i~dL
dt= HL = HL− HL. (13.36)
(We introduced the notation H = H − H.) It corresponds to the
equation of motion for density matrices; this follows from the formula
HLK = LHK−KH+ .
Note that often the equations of motion for L-functionals make
sense even in the situation when the equations of motion in the
Fock space are ill defined. This is related to the fact that vectors
and density matrices from all representations of CCR are described
by L-functionals. This means that by applying the formalism of
L-functionals, we can avoid the problems related to the existence of
inequivalent representations of CCR. It is well known, in particular,
that these problems arise for translation-invariant Hamiltonians; in
perturbation theory, these problems appear as divergences related to
infinite volume. Therefore, in the standard formalism, it is necessary
to consider at first a Hamiltonian in finite volume Ω (to make volume
cutoff or, in another terminology, infrared cutoff) and to take the
limit Ω → ∞ in physical quantities (see Sections 8.2 and 13.2 for
more details). In the formalism of L-functionals, we can work directly
in infinite volume.
In general, the ground state of formal Hamiltonian (13.7) does
not belong to Fock space, but the corresponding L-functional is well
defined. (Note that we always assume that ultraviolet divergences are
absent.) Similarly, the equilibrium states for different temperatures
belong to different Hilbert spaces, but all of them are represented by
well-defined L-functionals.
In what follows, we consider (13.33) as a formal expression;
we assume that it is formally Hermitian. There is no necessity to
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360 Mathematical Foundations of Quantum Field Theory
assume that (13.33) specifies a self-adjoint operator in one of the
representations of CCR.
Let us make some remarks about adiabatic evolution in the
formalism of L-functionals. Let us take a Hamiltonian that depends
on time t, but is changing adiabatically (very slowly). More formally,
we can assume that the Hamiltonian depends on a parameter g and
g = h(at) where a → 0. Let us take a family of L-functionals L(g),
where L(g) is a stationary state of the Hamiltonian H(g), i.e.
H(g)L(g) = 0.
It is obvious that L(h(at)) obeys the equation of motion up to
terms tending to zero as a → 0. (Note that a similar statement
is wrong in the standard Hilbert space formulation of quantum
mechanics because the vector corresponding to a state is defined up to
phase factor. We can only say that for a smooth family of eigenvectors
Ψ(g), there exists a phase factor C(g) such that C(h(at))Ψ(h(at))
obeys the equation of motion up to terms tending to zero as a→ 0.)
13.4.1 Translation-invariant Hamiltonians in the
formalism of L-functionals; one-particle states
In what follows, we consider translation-invariant Hamiltonians.
We say that an L-functional σ is an excitation of a translation-
invariant L-functional ω if it coincides with ω at infinity. More
precisely, we should require that σ(Txα∗, Txα)) → ω(α∗, α) as
x→∞. Here, Tx stands for spatial translation.
We assume that the L-functional ω has cluster property. The
weakest form of cluster property is the requirement that 〈ATxB〉ω −〈A〉ω〈B〉ω tends to zero as x→∞ for A,B ∈ A.
If Φ is a vector corresponding to ω in GNS representation space
H, then the state corresponding to any vector AΦ where A ∈ A is
an excitation of ω; this follows from cluster property.
We can define elementary excitations (called particles if ω is a
ground state and quasiparticles if ω is a general translation-invariant
stationary state) in the same way as in Section 13.3.2.
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Methods of Quantum Field Theory in Statistical Physics 361
13.4.2 Quadratic Hamiltonians
Let us consider as an example the simplest translation-invariant
Hamiltonian
H0 =
∫ω(k)a+(k)a(k)dk, (13.37)
where k runs over Rd. It can be approximated by a Hamiltonian of
the form∑ωka
+k ak having a finite number of degrees of freedom.
The equilibrium state of the latter Hamiltonian can be represented
by density matrix Ω(T ) in the Fock space; it is easy to check that
akΩ(T ) = e−~ωkT Ω(T )ak, a
+k Ω(T ) = e
~ωkT Ω(T )a+
k . (13.38)
Applying (13.31) and (13.32), we obtain equations for the cor-
responding L-functional; taking the limit, we obtain for the
L-functional corresponding to the equilibrium state in infinite volume
c1(k)LT = e−~ω(k)T (−~c+
2 (k) + c1(k))LT , (13.39)
hence
c1(k)LT = −n(k)c+2 (k)LT , c2(k)LT = −n(k)c+
1 (k)LT , (13.40)
where
n(k) =~
e~ω(k)T − 1
. (13.41)
We obtain
LT = e−∫α∗(k)n(k)α(k)dk. (13.42)
If we are interested in equilibrium state for given density, we should
replace ω(k) with ω(k) − µ in (13.41) (here, µ stands for chemical
potential).
The Hamiltonian H governing the evolution of L-functionals can
be written in the form
H =
∫ω(k)b+(k)b(k)dk −
∫ω(k)b+(k)b(k)dk
=
∫~(ω(k)c+
1 (k)c+1 (k)dk − ω(k)c+
2 (k)c2(k))dk.
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362 Mathematical Foundations of Quantum Field Theory
It follows that the vector generalized functions Φ1(k) = c+1 (k)Φ,
Φ2(k) = c+2 (k)Φ in the space of GNS representation correspond-
ing to (13.42) are one-(quasi)particle excitations. (Note that in this
statement, n(k) in the formula (13.42) is an arbitrary function;
then the corresponding state is not an equilibrium, but it is still
a stationary translation-invariant state.)
Very similar considerations allow us to prove that the L-functional
corresponding to an equilibrium state of general quadratic Hamil-
tonian H = a+Ma+ + a+Na + aRa is Gaussian (has the form
eα∗Aα∗+α∗Bα+αCα).
13.4.3 Perturbation theory
Let us assume now that the Hamiltonian H is represented as a sum of
quadratic Hamiltonian H0 and interaction Hamiltonian Hint = gV.
Then in the formalism of L-functionals, one can introduce in the
standard way the evolution operator U(t, t0), the evolution operator
in the interaction picture S(t, t0) and the operator Sa, the analog
of adiabatic S-matrix. If Ha governs the evolution of L-functional
for the Hamiltonian H0 + h(at)Hint, then we denote by Ua(t, t0) the
operator transforming the L-functional at the moment t0 into the
L-functional at the moment t (the evolution operator). The operator
Sa(t, t0) is defined by the formula
Sa(t, t0) = ei~ H0tUa(t, t0)e−
i~ tH0t0 .
(We assume that h(t) is a smooth function equal to 1 in the
neighborhood of 0 and to 0 in the neighborhood of infinity. It is
increasing for negative t and decreasing for positive t.) The operator
Sa is defined as Sa(∞,−∞).
The perturbation theory for operators Sa(t, t0) can be constructed
in the standard way: we apply the formula
Sa(t, t0) = T exp
(− i~h(at)Hint(t)
).
(We use the notation A(t) = ei~ H0tAe−
i~ H0t.)
If we are interested only in the action of these operators on func-
tionals represented by polynomials of α, α∗ with smooth coefficients
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Methods of Quantum Field Theory in Statistical Physics 363
tending to zero at infinity (smooth functionals in the terminology of
Tyupkin (1973)), the diagram techniques can be described as follows.
The vertices come from − i~h(at)Hint. To find the propagator, we
calculate the T -product of two operators of the form b(+)i (t) and
express it in normal form with respect to the operators c+i , ci (i.e.
the operators ci are from the right). The propagator (that can be
considered as 4 × 4 matrix) is equal to the numerical part of this
expression. In other words, the propagator is given by the formula
〈T (b(k1, t1, σ1)b(k2, t2, σ2)〉L=1.
Here, b(k, 0, σ) is one of the operators b+, b, b+, b.
Let us define the adiabatic generalized Green functions (GGreen
functions) by the formula
Gan(k1, t1, σ1, . . . , kn, tn, σn)
= 〈T (b(k1, t1, σ1) · · · b(kn, tn, σn)Sa(∞,−∞))〉L=1 (13.43)
As usual, the perturbative expansion for these functions can be
constructed by the same rules as for adiabatic S-matrix, but the
diagrams have n external vertices.
Note that Sa(∞,−∞)1→ 1 and Sa(0,−∞)1→ L as a→ 0. Here,
L denotes the L-functional corresponding to the ground state of the
Hamiltonian H. (This follows immediately from similar statement
for the adiabatic evolution operators Ua(∞,−∞) and Ua(0,−∞)
and from the fact that the L-functional L = 1 corresponds to the
ground state of H0.) We obtain that the adiabatic GGreen function
Gan(k1, t1, σ1, . . . , kn, tn, σn) tends to the GGreen function
Gn(k1, t1, σ1, . . . , kn, tn, σn) = 〈T (b(k1, t1, σ1) · · ·b(kn, tn, σn))〉L(13.44)
as a→ 0.
(Here, we use the notation
b(k, t, σ) = S−1(t, 0)b(k, t, σ)S(t, 0) = U−1(t, 0)b(k, 0, σ)U(t, 0)
for the analog of Heisenberg operators.) This means that we can
construct the perturbation expansion for the GGreen function
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364 Mathematical Foundations of Quantum Field Theory
Gn(k1, t1, σ1, . . . , kn, tn, σn) taking the limit a → 0 in the diagrams
for Gan(k1, t1, σ1, . . . , kn, tn, σn). The only modification of diagrams is
in internal vertices: now, these vertices are governed by − i~Hint.
Similar procedure can be applied for the calculation of the action
of operators Sa(t, t0) on the space of functionals represented as a
product of a smooth functional and Gaussian functional Λ = eλ,
where λ is a quadratic expression in terms of α∗, α. We assume that
Λ is translation invariant and stationary with respect to the evolution
corresponding to the Hamiltonian H0, i.e. H0Λ = 0. (Here, H0 is a
translation-invariant quadratic Hamiltonian not necessarily of the
form (13.37).) It is easy to check imposing some non-degeneracy
conditions that there exist such operators c+i (k), ci(k), i = 1, 2
obeying CCR that Λ can be characterized as a functional satisfying
the conditions ci(k)Λ = 0, i = 1, 2,Λ(0, 0) = 0. Then the perturbative
expression for the action of Sa(t, t0) on the space under consideration
can be obtained by means of the diagram technique with propagators
described in the same way as for Λ = 1, the only difference is that
instead of normal form with respect to the operators c+i (k), ci(k),
we should consider normal form with respect to the operators
c+i (k), ci(k), i = 1, 2. Equivalently, we can define the propagator by
the formula
〈T (b(k1, t1, σ1)b(k2, t2, σ2)〉Λ.
Again, we can define adiabatic GGreen functions
Gan(k1, t1, σ1, . . . , kn, tn, σn)Λ
= 〈T (b(k1, t1, σ1) · · · b(kn, tn, σn)Sa(∞,−∞))〉Λ (13.45)
corresponding to Λ and introduce the diagram technique for their
calculation. The GGreen functions corresponding to Λ can be defined
either as limits of adiabatic GGreen functions as a → 0 or by the
formula
Gn(k1, t1, σ1, . . . , kn, tn, σn)Λ = 〈T (b(k1, t1, σ1) · · ·b(kn, tn, σn))〉Λ,(13.46)
where Λ = lima→0 Ua(0,−∞)Λ denotes the stationary state of
the Hamiltonian H that we obtain from the stationary state Λ of
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Methods of Quantum Field Theory in Statistical Physics 365
the Hamiltonian H0 adiabatically switching the interaction on. The
diagrams representing the functions (13.46) have n external vertices,
the internal vertices come from − i~Hint, the propagator is equal to
〈T (b(k1, t1, σ1)b(k2, t2, σ2)〉Λ.
In particular, we can take H0 of the form (13.37) and
Λ = e−∫α∗(k)n(k)α(k)dk. (13.47)
(All translation-invariant Gaussian functionals that are stationary
with respect to the Hamiltonian (13.37) have this form.) Then we
obtain the following formulas for the propagator:
〈T (b+(t)b(τ)〉Λ = θ(t− τ)ei~ω(k)(t−τ)n(k)
+ θ(τ − t)ei~ω(k)(t−τ)(n(k) + ~)r,
〈T (b+(t)b+(τ)〉Λ = ei~ω(k)(t−τ)(n(k) + ~),
〈T (b+(t)b(τ)〉Λ = 〈T (b+(t)b(τ)〉Λ,
〈T (b(t)b(τ)〉Λ = ei~ω(k)(t−τ)n(k).
All other entries vanish.
It follows from the above formulas that the diagrams we con-
structed coincide with the diagrams of Keldysh and TFD formalisms
(see Chu and Umezawa (1994) for review of both formalisms).
We have noted already that the L-functional corresponding to an
equilibrium state of quadratic Hamiltonian is Gaussian. Assuming
that the equilibrium state of the Hamiltonian H0 + gV can be
obtained from the equilibrium state of H0 by means of adiabatic
evolution, we can say that the diagram technique we have described
allows us to calculate the GGreen functions in the equilibrium state.
13.4.4 GGreen functions
We have constructed the diagram technique for GGreen functions.
As in the standard technique, we can express all diagrams in terms of
connected diagrams; moreover, connected diagrams can be expressed
in terms of 1 PI diagrams. (One says that a diagram is one particle
irreducible (1 PI) if it remains connected when we remove one of the
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366 Mathematical Foundations of Quantum Field Theory
edges. Calculating a 1 PI diagram, we do not take into account the
contributions of external edges.)
The contribution of a disconnected diagram is equal to the
product of the contributions of its components (up to some factor
taking into account the symmetry group of the diagram). The two-
point GGreen function
G2(k1, t1, σ1, k2, t2, σ2)Λ = 〈T (b(k1, t1, σ1)b(k2, t2, σ2))〉Λ (13.48)
obeys the Dyson equation
G2(k1, t1, σ1, k2, t2, σ2)Λ
= G2(k1, t1, σ1, k2, t2, σ2)Λ
+
∫dk′2dt
′2dσ
′2dk′′2dt′′2dσ
′′2G2(k1, t1, σ1, k
′2, t′2, σ′2)Λ
×M(k′2, t′2, σ′2, k′′2 , t′′2, σ′′2)Λ
×G2(k′′2 , t′′2, σ′′2 , k2, t2, σ2)Λ (13.49)
connecting it with the propagator and self-energy operator (mass
operator) M (the integration over discrete parameters is understood
as summation). The generalized mass operator M is defined as a sum
of 1 PI diagrams. The Green functions and mass operator can be
regarded as kernels of integral operators, hence the Dyson equation
can be represented in operator form
GΛ2 = GΛ
2 +GΛ2M
ΛGΛ2
or
(GΛ2 )−1 = (GΛ
2 )−1 +MΛ. (13.50)
It is useful to write this equation in (k, ε)-representation. (Here,
ε stands for the energy variable.) Due to translational invariance
in this representation, the operators entering Dyson equation are
operators of multiplications by a matrix function of (k, ε).We identify
the operators with these matrix functions. The (quasi-)particles are
related to the poles of the matrix function GΛ2 (k, ε). Recall that
we assume that the Hamiltonian H is represented as a sum of
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Methods of Quantum Field Theory in Statistical Physics 367
quadratic Hamiltonian H0 =∫ω(k)a+(k)a(k)dk and interaction
Hamiltonian Hint = gV. The poles of the GGreen function for
g = 0 (of the propagator) are located at the points ±ω(k) (we set
~ = 1); the dependence of these poles on g can be found in the
framework of the perturbation theory; the location of these poles will
be denoted ±ω(k | g). (Note that we cannot apply the perturbation
theory directly, but we can use it to find zeros of the RHS of (13.50).
The function ω(k | g) can be regarded as energy of (quasi-)particle.
Only in the ground state, one can hope that this function is real
(thermal quasiparticles are in general unstable).
13.4.5 Adiabatic S-matrix
The Dyson equation can also be written for adiabatic GGreen
functions; they can be used to describe the asymptotic behavior
of these functions for a → 0. As we have noted, the adiabatic
GGreen functions tend to GGreen functions as a → 0, but they do
not converge uniformly. However, the adiabatic self-energy operator
converges uniformly; this allows us to analyze the asymptotic behav-
ior of GGreen functions. (The same is true for conventional Green
functions). The reason for the uniform convergence is the fact that
matrix function M in (k, t)-representation tends to zero as t → ∞.Conventional adiabatic Green functions were analyzed by Likhachev
et al. (1970), the same method was applied by Tyupkin (1973) to
obtain the approximation for adiabatic GGreen functions. These
results were used to obtain the renormalized scattering matrix from
adiabatic scattering matrix. Note that all these considerations are
based on the assumption that the functions ω(k | g) are real, therefore
rigorously they can be applied only to the scattering of particles
(of elementary excitations of the ground state). Nevertheless, they
make sense as approximate formulas if the quasiparticles are almost
stable (we should assume that the collision time is much less than
the lifetime of quasiparticles and choose a in such a way that 1a
the lifetime of quasiparticles, but 1a the collision time).
The following statements were derived by Tyupkin (1973) from
the results of Schwarz (1967) and Likhachev et al. (1970) in the
framework of perturbation theory.
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368 Mathematical Foundations of Quantum Field Theory
The scattering matrix in the formalism of L-functionals can be
defined as an operator on the space of smooth L-functionals by the
formula
S = lima→0
VaSaVa, (13.51)
where Sa = Sa(∞,−∞) stands for the adiabatic S-matrix,
Va = exp i
∫ra(k)(c+
1 (k)c1(k)− c+2 (k)c2(k))dk
and the function ra is chosen from the requirement that S acts
trivially on one-particle states. (One can give an explicit expression
for ra in terms of one-particle energies ω(k | g); namely ra(k) =1a
∫ 0−∞(ω(k |h(τ))− ω(k))dτ.)
One can prove the existence of the limit in (13.51) in the
framework of perturbation theory. (One should impose the condition
ω(k1 + k2) < ω(k1) + ω(k2). This condition means that one-particle
spectrum does not overlap with multi-particle spectrum.)
The conventional renormalized S-matrix was related by Likhachev
et al. (1970) to the adiabatic S-matrices. (To obtain the S-matrix,
one should multiply the adiabatic S-matrix in finite volume by factors
similar to Va, take the limit when the volume tends to infinity, and
then take the limit a → 0.) Using the methods of Likhachev et al.
(1970), one can relate S to the conventional renormalized S-matrix S;
we obtain SLK = LSKS−1 . Using this formula, one can express
inclusive cross-section in terms of S (see Section 13.4.6).
13.4.6 Scattering of (quasi-)particles; inclusive
cross-section
This subsection is independent of the rest of this section (we use only
the definition of L-functional).
Let us start with the situation of quantum field theory when the
standard scattering matrix S is well defined as an operator acting
in the Fock space of asymptotic states. (Strictly speaking, we should
denote the operators acting in this space as ain(k), a+in(k), but we use
shorter notations a(k), a+(k).) We assume that the scattering matrix
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Methods of Quantum Field Theory in Statistical Physics 369
as well as Møller matrices S−, S+ are unitary. Considering the Fock
space as a representation of CCR, we assign an L-functional LK to
a density matrix K in the Fock space using the formula (13.47).
We define the scattering matrix in the space of L-functionals by the
formula
SLK = LSKS∗ . (13.52)
If the density matrix K corresponds to a vector Ψ, we can represent
the RHS of (13.52) as
Tr e−αa+eα∗aSKS∗
= Tr eα∗aSKS∗e−αa
+= 〈eα∗aSΨ, e−α
∗aSΨ〉
=∑n
∫dp1 · · · dpn〈eα
∗aSΨ|p1, . . . , pn〉〈e−α∗aSΨ|p1, . . . , pn〉,
where |p1, . . . , pn〉 =√
1n!a
+(p1) · · · a+(pn)θ constitute an orthonor-
mal basis in Fock space. (Here, θ = | 0〉 stands for Fock vac-
uum.) The expression 〈a(k1) · · · a(km)SΨ|p1, . . . , pn〉 =
√(m+n)!n!
〈SΨ|k1, . . . , km, p1, . . . , pn〉 can be interpreted as the scattering
amplitude of the process Ψ→ (k1, . . . , km, p1, . . . , pn) (after dividing
by numerical factor). We see that the LHS is expressed in terms of
scattering amplitudes. In particular, writing it in the form∑m,m′
∫dk1 · · · dkmdk′1 · · · dk′m′
(−1)m
m!m′!α(k1)
· · ·α(km)α∗(k′1)α′m′(k′m′)σm,m′(k1, . . . , km|k′1, . . . , k′m′ |Ψ),
we obtain
σm,m′(k1, . . . , km|k′1, . . . , k′m′ |Ψ)
=∑n
∫dp1 · · · dpn
√(m+ n)!
√(m′ + n)!
n!
×〈SΨ|k1, . . . , km, p1, . . . , pn〉〈SΨ|k′1, . . . , k′m′ , p1, . . . , pn〉.
If m = m′ and ki = k′i, this expression is proportional to the inclu-
sive cross-section Ψ → k1, . . . , km. If the initial state Ψ has definite
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370 Mathematical Foundations of Quantum Field Theory
momentum q, then 〈SΨ | k1, . . . , km, p1, . . . , pn〉 is a product of delta-
function δ(∑ki +
∑pj − q) coming from momentum conservation
and a function that will be denoted ρm(k1, p1, . . . , pn |Ψ). Now,
σm,m′(k1, . . . , km|k′1, . . . , k′m′ |Ψ)
= σm,m′(k1, . . . , km|k′1, . . . , k′m′ |Ψ)
× δ(k′1 + · · ·+ k′m − (k1 + · · ·+ km)),
where
σm,m′(k1, . . . , km|k′1, . . . , k′m′ |Ψ)
=∑n
∫dp1 · · · dpn
√(m+ n)!
√(m′ + n)!
n!
× ρm(k1, . . . , km, p1, . . . , pn |Ψ)ρm′(k′1, . . . , k
′n, p1, . . . , pn |Ψ)
× δ(k1 + · · ·+ km + p1 + · · ·+ pn − q).
The inclusive cross-section is proportional to σm,m(k1, . . . , km|k1,
. . . , km|Ψ) in this case.
We will call S inclusive S-matrix. Let us show that this matrix
can be calculated in terms of GGreen functions (more precisely, in
terms of on-shell values of these functions). Let us take a density
matrix K in the representation space of CCR. We assume that
the momentum and energy operators (infinitesimal spatial and time
translations) act on this space and K is translation invariant. This
allows us to define Heisenberg operators a+(k, t), a(k, t) where k
is the momentum variable and t is the time variable. Then the
corresponding GGreen function can be defined as TrBAK. Here, A
denotes chronological product (T -product) of Heisenberg operators
(the times are decreasing) and B stands for antichronological product
T opp of Heisenberg operators (the times are increasing).
If the density matrix K corresponds a vector Φ, the GGreen
function can be represented in the form
〈AΦ, B∗Φ〉,
where B∗ stands for chronological product of Hermitian conjugate
operators.
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Methods of Quantum Field Theory in Statistical Physics 371
Let us now consider the case when Φ is the ground state. Then one
can obtain an expression of S in terms of GGreen functions that is
analogous to the LSZ formula. It will be derived from some identities
that were used in the proof of LSZ, namely, we can use the identity
[· · · [S, ain(k1, σ1)] · · · ain(kn, σn)]
= (−1)nS−S∗+
∫dt1 · · · dtnLn · · ·L1T (a(k1, t1, σ1) · · · a(kn, tn, σn)),
(13.53)
where S−, S+ are Møller matrices, the scattering matrix S is
represented in terms of in-operators ain(k, 1) = a+in(k), ain(k,−1) =
ain(k), operators a(k, t, σ) are Heisenberg operators a+(k, t), a(k, t),
the operators∫dtiLi in (k, ε)-representation can be interpreted
as “on-shell operators.” (To apply the operator∫dtiLi in (k, ε)-
representation, we should multiply by iΛ(ki, σi)(εi−ω(ki)) and take
the limit εi → ω(ki). Here, ω(k) stands for the location of the pole
of the two-point Green function and Λ can be expressed in terms
of the residue in this pole. Note that in the transition to (k, ε)-
representation, we are using direct Fourier transform for σ = −1
and inverse Fourier transform for σ = 1.)
The identity (13.53) can be obtained, for example, from (32.17)
of Section 9.2 (by means of conjugation with S−).
It follows from (13.53) that Green function defined as vac-
uum expectation value of chronological product T (a(k1, t1, σ1) · · ·a(kn, tn, σn)) is related to scattering amplitude: one should take the
Fourier transform with respect to time variables and “go on-shell”
in the sense explained above. This gives the LSZ formula (see the
preceding section for more general approach). We remark that these
considerations also go through in the case when instead of vacuum
expectation value 〈0|A|0〉 where |0〉 obeys ain(k)|0〉 = 0 (represents
physical vacuum), we can take matrix elements 〈0|A|p1, . . . , pn〉where |p1, . . . , pn〉 = 1√
n!a+
in(p1) · · · a+in(pn)|0〉. This remark allows us
to express “on-shell” GGreen functions as sesquilinear combinations
of scattering amplitudes; comparing this expression with the formula
for SLK , we obtain the expression of S in terms of GGreen functions
“on-shell” (an analog of LSZ formula). Indeed, we can consider
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372 Mathematical Foundations of Quantum Field Theory
GGreen function as vacuum expectation value of chronological
product A multiplied by antichronological product B. Using the fact
that |p1, . . . , pn〉 constitute a generalized orthonormal basis, we can
say that
〈0|BA|0〉 =∑n
∫dp1 · · · dpn〈0|B|p1, . . . , pn〉〈p1, . . . , pn|A|0〉.
(13.54)
This representation allows us to express on-shell GGreen functions
in terms of scattering amplitudes. (The antichronological product is
related to the chronological one by Hermitian conjugation.)
Let us write explicit expressions obtained this way. We represent
S in normal form
S =∑r,s
1
r!s!
∫dp1 · · · dprdq1 · · · qsσr,s(p1, . . . , pr | q1, . . . , qs)a
+in(p1)
· · · a+in(pr)ain(q1) · · · ain(qs).
We assume that all pi’s are distinct and all qj ’s are distinct, then the
coefficient functions in normal form coincide with scattering ampli-
tudes 〈Sa+in(q1) · · · a+
in(qs)θ | a+in(p1) · · · a+
in(pr)θ〉. Let us take σi = 1
for i ≤ m, σi = −1 for i > m in (13.53). Introducing the notation
qi = ki−m, we obtain that
〈p1, . . . , pn|LHS|0〉 =1√n!σm+n,l(p1, . . . , pn, k1, . . . , km | q1, . . . , ql).
Here, LHS stands for the LHS of (13.53). Now, we can apply (13.53)
and (13.54) to identify on-shell GGreen functions with matrix entries
of the inclusive scattering matrix S.
Our considerations used LSZ relations that are based on the
conjecture that the theory has particle interpretation. Moreover, the
very definition of S that we have applied requires the existence of
conventional S-matrix. However, using (13.51) as the definition of
the inclusive scattering matrix S, one can prove the relation between
S and on-shell GGreen functions analyzing diagram techniques for
these objects. This proof can also be applied in the case when the
theory does not have particle interpretation. Moreover, the same
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Methods of Quantum Field Theory in Statistical Physics 373
ideas can be applied to quasiparticles considered as elementary exci-
tations of translation-invariant stationary state. These excitations
are related to the poles of the two-point GGreen function; we can
define the inclusive scattering matrix as on-shell GGreen function or
generalizing (13.51). Of course, this definition makes sense only if the
quasiparticles are (almost) stable.
In the preceding section, we have analyzed the inclusive scattering
matrix in the framework of algebraic approach to quantum theory
(see also Schwarz (2019c)).
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Appendix
A.1 Hilbert spaces
Let us consider a setH equipped with two operations: addition of two
elements and multiplication of elements by a complex number. We
say that H is a linear space if the following conditions are satisfied:
(1) x+y = y+x; (2) (x+y)+z = x+(y+z); (3) λ(x+y) = λx+λy;
(4) λ(µx) = (λµ)x, (λ+µ)x = λx+µx; (5) 0 ·x = 0, 1 ·x = x (where
x, y, z ∈ H; λ, µ are complex numbers). The elements of the linear
space are called vectors. We say the space is spanned by a subset S if
every element of the space can be represented as a linear combination
of the elements of the subset S. The dimension of a linear space is
the minimal number of vectors spanning the space. If this number
is infinite, then we say that the space is infinite dimensional. The
symbol 0 denotes the vector satisfying the conditions x + 0 = x for
all x in H.
A function on H taking values in the set of complex numbers is
called a linear functional if f(λx+ µy) = λf(x) + µf(y).
We say that a linear space is pre-Hilbert if it is equipped with
the inner product 〈x, y〉, satisfying the following axioms: (1) 〈x, y〉 =
〈y, x〉; (2) 〈λx, y〉 = λ〈x, y〉; (3) 〈x+y, z〉 = 〈x, z〉+〈y, z〉; (4) 〈x, x〉 ≥0 and 〈x, x〉 = 0 only if x = 0 (here, x, y, z ∈ H and λ is a complex
number).
We say that a bijective map is isomorphic if it preserves all
operations defined in a pre-Hilbert space.
The number ‖x‖ =√〈x, x〉 is called the norm of vector x. The
vector x is normalized if ‖x‖ = 1. Two vectors are orthogonal if
〈x, y〉 = 0.375
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376 Mathematical Foundations of Quantum Field Theory
One says that the sequence xn ∈ H convergences to the element
x ∈ H (denoted by x = limxn), if limn→∞ ‖x−xn‖ = 0. We say that
the sequence xn ∈ H weakly converges to the vector x ∈ H (denoted
x = wlimxn) if for all elements y ∈ H the sequence 〈xn, y〉 tends
to 〈x, y〉.A sequence is called a Cauchy sequence if limm,n→∞ ‖xm−xn‖= 0.
The set M ⊂ H is closed in H if every point x ∈ H that can be
represented as a limit of a sequence xn ∈ M is itself an element of
M . A set M is dense in H if every vector x ∈ H can be represented
as a limit of elements of M .
A pre-Hilbert space H is called Hilbert space if every Cauchy
sequence converges. Every pre-Hilbert space can be embedded in a
Hilbert space. In other words, for every pre-Hilbert space H, one
can construct a Hilbert space H, called the completion of H, and an
isomorphic map α of the space H onto a dense subset of the space
H (isomorphic embedding of H into H).
The completion H is unique in the following sense: if H1 and H2
are two completions and α1 and α2 are isomorphic embeddings of Hinto H1 and H2, then there exists an isomorphic map α of H1 onto
H2 satisfying αα1 = α2.
A pre-Hilbert space is called separable if it contains a countable
dense subset. We will consider only separable spaces.
A.2 Systems of vectors in a pre-Hilbert vector space
A subset M ⊂ H is called a linear subspace if every linear
combination λx+ µy with x, y ∈M is also in M .
A system of vectors ξa ∈ H is called orthonormal if 〈ξα, ξβ〉 = δαβ(we consider systems that are either finite or countable). A orthonor-
mal system is called an orthonormal basis if linear combinations of
elements of this system are dense in H.The quantities 〈x, ξa〉 are called the Fourier coefficients of x in the
orthonormal basis ξa. For every vector x ∈ H, the series∑
a〈x, ξa〉ξa(called the Fourier series of x in the orthonormal basis ξa) converges
to the vector x. The scalar product 〈x, y〉 can be expressed in terms
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Appendix 377
of the Fourier coefficients as follows:
〈x, y〉 =∑α
〈x, ξα〉 〈y, ξα〉.
In particular, 〈x, x〉 =∑
α | 〈x, ξα〉 |2.
If H is a Hilbert space, then the sequence cα is a sequence of
Fourier coefficients of vector x if and only if∑
α |cα|2 <∞.
A.3 Examples of function spaces
1. The space l2n, also denoted as Cn, is the n-dimensional space of
rows of n complex numbers equipped with the inner product
〈x, y〉 =
n∑i=1
xiyi
(here, x = (x1, . . . , xn) ∈ Cn, y = (y1, . . . , yn) ∈ Cn).
2. The space l2 consists of sequences of complex numbers x =
(x1, . . . , xi, . . . ), satisfying the condition∑∞
i=1 |xi|2 < ∞. Addi-
tion and multiplication by a complex number are defined
coordinate-wise and the scalar product is defined as 〈x, y〉 =∑∞i=1 xiyi.
The spaces l2n and l2 are Hilbert spaces.
If H is a Hilbert space and ξα is an orthonormal basis for H,
then we can construct an isomorphic map to the space l2n (if His finite dimensional) and l2 (if H is infinite dimensional). The
isomorphism transforms a vector x to its sequence of Fourier
coefficients in the basis ξα, obtained as 〈x, ξα〉.3. The space C2(Er) consists of functions on an r-dimensional space
Er that are continuous and square integrable (i.e. the functions
satisfy the condition∫|f(ξ1, . . . , ξr)|2dξ1, . . . , dξr <∞). Multipli-
cation and addition of functions are defined in the standard way.
The scalar product of two functions f and g in C2(Er) is given by
〈f, g〉 =
∫f(ξ1, . . . , ξr)g(ξ1, . . . , ξr)dξ1, . . . , ξr
=
∫f(ξ)g(ξ)dξ.
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378 Mathematical Foundations of Quantum Field Theory
The space C2(Er) is a pre-Hilbert space. Its completion is L2(Er).
4. The space C2(Er × S), where S denotes a finite set, consists
of functions f(ξ, s) that are continuous with respect to the
variable ξ and are square integrable (i.e. satisfying the condition∑s
∫|f(ξ, s)|2dξ < ∞). Here, ξ = (ξ1, . . . , ξr) ∈ Er; s ∈ S, dξ =
dξ1 . . . dξr. The scalar product is defined by
〈f, g〉 =∑s
∫f(ξ, s)g(ξ, s)dξ.
The function f(ξ, s) can be considered as a column of k functions
fs(ξ) = f(ξ, s) depending on the variable ξ ∈ Er (k is the number
of elements in S).
The completion of C2(Er × S) is denoted by the symbol
L2(Er × S).
Let us fix a family B of subsets of M that contains with every
two subsets A,B the subsets A ∪ B,A ∩ B, and A \ B (we call this
family a ring of subsets). Let us also assume that the set M is a
countable union of subsets belonging to the family B. A countably
additive measure µ on a family B is specified by an assignment of
non-negative numbers µ(A) to every set A ∈ B.
We require that for any countable sequence of sets A,A1, . . . ,
An, . . . from B we have µ(A) =∑∞
i=1 µ(Ai) if
(1) A = ∪∞i=1Ai,
(2) the sets A1, . . . , An, . . . are pairwise disjoint. (In other words, we
assume that the measure is countably additive.)
The setM , equipped with the family B and the countably additive
measure µ, specifies a measure space.
A set R ⊂ M (not necessarily belonging to B) is called a set of
measure zero if for every ε > 0, there exists a set A ∈ B, such that
R ⊂ A and µ(A) < ε. One says that a sequence of functions fn(x)
converges to a function f(x) almost everywhere on a set M if the
set of points x ∈M , such that the sequence fn(x) does not converge
to f(x), is a set of measure zero. In general, if a relation is satisfied
everywhere except a measure zero set, then we say that it is satisfied
almost everywhere.
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Appendix 379
For functions defined on a measure space, one can define a notion
of Lebesgue integral. A function f defined on a set A ∈ B is
called a simple function, if the set of values of the function is finite
y1, . . . , yn and the sets Ai consisting of points where the function
f takes the value yi belong to the family B. The Lebesgue integral
of a simple function f is defined by the formula∫Af(x)dµ =
r∑i=1
yiµ(Ai).
A function f(x) that can be represented as a limit of a sequence of
simple functions converging almost everywhere is called measurable.
In what follows, we consider only measurable functions. Let us say
that two measurable functions are equivalent if their difference is
equal to zero almost everywhere. We do not distinguish equivalent
measurable functions.
If a measurable function f is bounded, then it can be represented
as a limit of a sequence of simple functions fn(x) that are bounded
from above by a constant, |fn(x)| ≤ C. The Lebesgue integral of the
function f on a set A ∈ B can be defined by∫Af(x)dµ = lim
n→∞
∫Afn(x)dµ.
One can prove that this limit always exists and does not depend on
the choice of the sequence fn(x).
If the function f is unbounded and non-negative, then one can
define its Lebesgue integral by the relation∫Af(x)dµ = lim
n→∞
∫Afn(x)dµ,
where fn(x) = f(x) for f(x) ≤ n and fn(x) = 0 for f(x) ≥ n.
If the set A does not belong to the family B, but can be
represented as A =⋃∞i=1Ai, where Ai ∈ B, then the Lebesgue
integral of a non-negative function f over A is defined by∫Af(x)dµ = lim
n→∞
∫A1∪···∪An
f(x)dµ.
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380 Mathematical Foundations of Quantum Field Theory
(For any non-negative measurable function, the Lebesgue integral
exists, but is not necessarily finite.)
A measurable function is called Lebesgue summable on the set
A ⊂ M , if the Lebesgue integral of the function |f(x)| is finite. For
a summable real function, the Lebesgue integral is defined by the
relation ∫Af(x)dµ =
∫A|f(x)|dµ−
∫A
(|f(x)| − f(x))dµ.
If the function f(x) is complex, then the Lebesgue integral is defined
by∫A Re(f(x))dµ + i
∫A Im(f(x))dµ, where Re(f(x)), Im(f(x)) are
real and imaginary parts of f(x).
We note some important properties of the Lebesgue integral:
1. Let fn(x) be a sequence of functions converging almost everywhere
to the function f(x) and suppose there exists a function g(x)
dominating the functions fn(x) (i.e. |fn(x)| ≤ g(x)). Then
limn→∞∫A fn(x)dµ =
∫A f(x)dµ.
This theorem allows us to interchange the limit and Lebesgue
integration.
2. Let M1 and M2 be measure spaces. Then in the space M1 ×M2
consisting of the pairs (x1, x2), where x1 ∈M1 and x2 ∈M2, one
can construct a countably additive measure µ defined by µ(A1 ×A2) = µ1(A1)µ2(A2) (here, µi denotes the measure on Mi; the
measure µ is called the product measure of µ1 and µ2).
If the function f(x1, x2) is a function on the product M1 ×M2
that is summable with respect to the measure µ, then for almost
all x2 the function f(x1, x2) is a summable function of x1 ∈ M1.
The double integral of the function f (i.e. the integral over the
measure µ) can be expressed as the repeated integral∫A1×A2
f(x1, x2)dµ =
∫A2
dµ2
∫A1
f(x, x2)dµ1
(Fubini’s theorem).
Using the Lebesgue integral, one can introduce a notion of measure
for some subsets that do not belong to the family B. Namely, for
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every subset A ⊂ M , one can construct a function χA(x) that is
equal to 1 for x ∈ A and 0 for x 6∈ A and set µL(A) =∫M χA(x)dµ.
This measure µL is called the Lebesgue measure. It is defined for
the set A if the function χA is measurable (then the set A is called
measurable).
For every measure set M , we can construct a space L2(M) of
square-integrable functions1 (i.e. measurable functions such that the
Lebesgue integral of the function |f2(x)| over the set M is finite).
The scalar product on the space L2(M) can be defined by means of
the Lebesgue integral2
〈f, g〉 =
∫f(x)g(x)dµ.
One can prove that the space L2(M) is a Hilbert space.
For the Euclidean space Er, we take the standard volume as a
measure. Then, for functions that are integrable in the standard
sense, the conventional integral (Riemann integral) coincides with
the Lebesgue integral. The space L2(Er) is the completion of the
space C2(Er).
If S is a finite set, then a measure of its subset A is by definition
the number of points in A. The space L2(S) is isomorphic to the
space l2k where k is the number of points in S. In the set Er × S,
we define the measure as the product of measure in Er and S. The
space L2(Er×S) constructed with this measure is the completion of
the space C2(Er × S).
A.4 Operations with Hilbert spaces
The direct sum H1 + H2 of Hilbert spaces H1 and H2 is a Hilbert
space, whose elements consist of pairs (h1, h2) where h1 ∈ H1, h2 ∈H2, with addition, multiplication by a complex number, and the
1Two equivalent measurable functions specify the same element of the space ofL2(M).2Integrating over the whole measure space M , we use the notation
∫φ(x)dµ
instead of the notation∫Mφ(x)dµ.
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382 Mathematical Foundations of Quantum Field Theory
scalar product defined by
(h1, h2) + (h′1, h′2) = (h1 + h′1, h2 + h′2),
λ(h1, h2) = (λh1, λh2),⟨(h1, h2), (h′1, h
′2)⟩
=⟨h1, h
′1
⟩+⟨h2, h
′2
⟩.
Let us assume that subspaces H1 and H2 of the Hilbert space Hsatisfy the following conditions: (1) every two vectors h1 and h2 are
orthogonal and (2) every h ∈ H can be represented as a sum h =
h1 + h2 of two vectors h1 ∈ H1 and h2 ∈ H2. Then one says that
the Hilbert space H is represented as a direct sum of subspaces H1
and H2. In this case, one can construct a natural isomorphism of
the direct sum H1 + H2 and the space H, transforming the vector
(h1, h2) ∈ H1 +H2 into the vector h1 + h2 ∈ H.
The direct sum∑
iHi of a countable sequence of Hilbert spaces
H1, . . . ,Hi, . . . is defined as the space of sequences (h1, . . . , hi, . . . ),
satisfying the condition∑
i ‖hi‖2 < ∞ (here, hi ∈ Hi). Linear
combination and scalar product are defined by
λ(h1, . . . , hi, . . . ) + λ′(h′1, . . . , h′i, . . . )
= (λh1 + λ′h′1, . . . , λhi + λ′h′i, . . . ),⟨(h1, . . . , hi, . . . ), (h
′1, . . . , h
′i, . . . )
⟩=∑i
⟨hi, h
′i
⟩.
The Hilbert space H is said to be a tensor product of Hilbert spaces
H1 and H2, if there exists a bilinear map α(h1, h2) mapping H1×H2
to H and having the following properties:
(1) 〈α(h1, h2), α(h′1, h′2)〉 = 〈h1, h
′1〉 · 〈h2, h
′2〉;
(2) the set of linear combinations of vectors of the form α(h1, h2) is
dense in H. We say that a map α(h1, h2) from H1 ×H2 into His bilinear if it is linear with respect to h1 for a fixed h2 and is
linear with respect to h2 for fixed h1.
The tensor product is defined by the above properties up to a natural
isomorphism. (If H and H′ are two tensor products of H1 and H2
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and α1 and α2 are corresponding bilinear maps, then there exists an
isomorphism λ of Hilbert spaces H and H′, such that λα = α′.)
The tensor product of spaces H1 and H2 will be denoted by the
symbol H1 ⊗H2 and the vector α(h1, h2) by the symbol h1 ⊗ h2.
It is easy to check that the tensor product of the spaces l2m and
l2n is isomorphic to the space l2mn. The space l2mn can be realized as
a space of m × n matrices. Then the bilinear map α assigns to the
vectors (x1, . . . , xm) ∈ l2m and (y1, . . . , yn) ∈ l2n the matrix A with
the elements Aij = xiyj .
The tensor product of n copies of space H is called the nth tensor
power of the space and is denoted by ⊗Hn. The formal definition is
via taking repeated tensor products
⊗Hn = (. . . ((H⊗H)⊗H) · · · ⊗ H).
Every permutation π of the indices (1, . . . , n) naturally corresponds
to an isomorphic map ρπ of the space ⊗Hn on itself.
The subspace Hns of the space ⊗Hn consisting of vectors that
satisfy the condition ρπx = x for all permutations π is called an
nth symmetric power of H. The nth antisymmetric power of Hna of
the space H is defined as the subspace consisting of vectors obeying
ρπx = (−1)γ(π)x for all π (where γ(π) stands for the parity of the
permutation π).
Note that
L2(M1)⊗ L2(M2) = L2(M1 ×M2)
(the measure in the space M1 × M2 is defined as the product of
measures in M1 and M2). The bilinear map α in the definition of the
tensor product sends the functions f1 ∈ L2(M1), f2 ∈ L2(M2) to the
function f(x1, x2) = f1(x1)f2(x2) ∈ L2(M1 ×M2). The symmetric
(antisymmetric) power of the space L2(M) can be realized as the
space of symmetric (antisymmetric) square integrable functions of
the variables x1, . . . , xn ∈M .
A.5 Operators on Hilbert spaces
We consider linear operators acting on a Hilbert space H1 and taking
values in a Hilbert space H2 as maps from dense subspace D ⊂
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384 Mathematical Foundations of Quantum Field Theory
H1 into H2, satisfying the condition of linearity (i.e. the operator
A sends every vector x ∈ D to the vector Ax ∈ H2 and satisfies
A(λ1x1 +λ2x2) = λ1Ax1 +λ2Ax2 where x1, x2 ∈ D). The domain of
definition for the operator A is denoted by DA. The set of vectors in
H2 that can be written as Ax for x ∈ H1 is called the range of A or
the image of A.
We say that the operator A is bounded if there exists a number
K, such that ‖Ax‖ ≤ K‖x‖ for all x ∈ DA. The operator norm of
A is defined as supx‖Ax‖‖x‖ . If the operator A is bounded, then it can
be continuously extended to the whole space H1 (i.e. there exists a
unique bounded operator on H1 that coincides with A on DA).
The multiplication operator by the number λ will be denoted by
the same symbol. In particular, the identity operator, considered as
a multiplication operator by 1, will be denoted by the symbol 1.
A linear combination C = λ1A1 +λ2A2 of operators A1 and A2 is
defined by Cx = λ1A1x+λ2A2x. Operator multiplication F = A1A2
of operators A1 and A2 is defined as composition Fx = A1(A2x).
The linear combination of operators A1 and A2 acting from H1
into H2 is defined if the intersection of their domains of definition
DA1 and DA2 is everywhere dense in H1. Similarly, the product of
operators A1 and A2 is defined if A2 maps H0 into H1, A1 maps H1
into H2 and the set of x ∈ DA2 for which A2x ∈ DA1 is dense in H0.
If the range of an operator A, transforming H1 into H2, is dense
in H2 and the operator is injective (only 0 ∈ H1 goes to 0 ∈ H2),
then we can define the inverse operator A−1, for which A−1y = x, if
y = Ax.
If the operator A transforms H1 into H2 and the operator B
transforms H2 into H1, then the two are called conjugate if 〈Ax, y〉 =
〈x,By〉 for all x ∈ DA, y ∈ DB.
If two operators B1 and B2 are both conjugate with the operator
A and both are defined on the vector y ∈ H2, then B1y = B2y (this
follows from the density of DA in H1). We will denote by A∗ the
conjugate operator to A that is defined for all y ∈ H2 for which there
exists z ∈ H1 that satisfies 〈Ax, y〉 = 〈x, z〉 (in other words, A∗ has
the maximum domain of definition of all operators conjugate to A).
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Note that we demand that an operator is defined on a dense set,
therefore, the operator A∗ does not always exist.
If the operator A is bounded, then the operator A∗ always exists
and is bounded, with ‖A∗‖ = ‖A‖.The operator U transforming H1 into H2 is called an isometry
if it preserves inner products: 〈Ux,Uy〉 = 〈x, y〉. For an operator
to be an isometry, it is necessary and sufficient for it to satisfy
U∗U = 1. An isometry is called a unitary operator if it is surjective
(the range coincides with H2). The condition for unitarity can be
written in the form UU∗ = U∗U = 1. A unitary operator defines
an isomorphism between H1 and H2, while an isometry defines an
isomorphism between H1 and a subspace of H2.
If H1 = H2 = H, then operators on H and their associated
conjugate operators are defined on the same space H. In this case,
we define the following notions. An operator A is called Hermitian if
it is conjugate to itself (i.e. 〈Ax, y〉 = 〈x,Ay〉 for all x, y ∈ DA). An
operator A is called self-adjoint if A∗ = A.
If A is Hermitian, B is self-adjoint, and DA ⊂ DB and Ax = Bx
for all x ∈ DA, then the operator B is called a self-adjoint extension
of A. If a Hermitian operator has only one self-adjoint extension,
then it is called essentially self-adjoint.
Self-adjoint operators are important in quantum mechanics
because they correspond to physical quantities. Usually, in physics
books, only Hermitian operators are discussed; the notion of the self-
adjoint operator is identified with the notion of the Hermitian opera-
tor. This does not cause many issues, since most Hermitian operators
encountered in physics turn out to be essentially self-adjoint. Note,
however, that if a Hermitian operator is not essentially self-adjoint,
then a choice of self-adjoint extension in physics problems leads to
different results (quite often this choice corresponds to a choice of
boundary conditions.)
An operator A is called bounded from below if there exists a
constant K such that 〈Ax, x〉 ≥ K 〈x, x〉 for any vector x ∈ DA. If
K = 0, then we call A positive semidefinite, and if for x 6= 0 the
expression 〈Ax, x〉 is positive, then we call A positive definite.
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Hermitian operators that are bounded from below always have
self-adjoint extensions. One of these extensions is called the
Friedrichs extension Aµ; it can be defined for the operator A by
the following extremal property: if B is a self-adjoint extension of
A and λ is a number such that B + λ is positive definite, then Aµsatisfies
⟨(Aµ + λ)−1x, x
⟩≤⟨(B + λ)−1x, x
⟩. Friedrichs extension
Aµ is bounded from below if A is bounded from below (in fact, if
〈Ax, x〉 ≥ K 〈x, x〉 then 〈Aµx, x〉 ≥ K 〈x, x〉).The concrete operators we consider in this book are often given
as formal expressions that combine simpler operators (such as
multiplication and differentiation in the space L2(En) or annihilation
and creation operators on Fock space). For full rigor, one should
specify the domain of all operators considered. Instead, however, for
simplicity, we agree that operators in the space L2(En) are defined
on the space S(En) of rapidly decaying smooth functions (more
detailed definition of S(En) is given in A.6) and operators in the
Fock space F (L2(En)) are defined on the set of Fock states specified
by a sequence of functions fk(x1, . . . , xk) ∈ S(Enk) (we assume that
only a finite number of these functions are not equal to 0).
Saying that a formal operator expression specifies a self-adjoint
operator, we have in mind that this expression specifies a Hermitian
operator that is either essentially self-adjoint or is bounded from
below. In the latter case, the self-adjoint operator defined by the
expression at hand is the Friedrichs extension of the corresponding
Hermitian operator.
When the formal operator expression does not define a Hermitian
operator or specifies a Hermitian operator that does not have a self-
adjoint extension, then we say that this formal expression does not
specify a self-adjoint operator.
We say that the operator P is a projection operator if P = P ∗
and P 2 = P . Projections on a Hilbert space are in one-to-one
correspondence with the subspaces of the Hilbert space (every
projection corresponds to its range).
Let us consider the example of the operator a(x) of multiplication
by a measurable function a(x) in the space L2(M) where M is a
measure space (the operator a(x) transforms f(x) into the function
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Appendix 387
a(x)f(x) and is defined on the functions f(x) ∈ L2(M) such that
the function a(x)f(x) is square integrable). It is easy to check that
(a(x))∗ =ˆ
a(x). It follows that the operator a(x) is (1) self-adjoint
if the function a(x) is real, (2) unitary if |a(x)| = 1, and (3) is a
projection if the function a(x) takes only the values 0 and 1.
Note that the operator a(x) can be considered on a smaller
domain; the adjoint operator does not depend on the choice of the
domain. It follows from this remark that for real a(x) the operator
a(x) is essentially self-adjoint on every domain.
The operator of multiplication is universal in some sense. Namely,
one can prove the following important theorem.
For every self-adjoint or unitary operator A in the space H, one
can find a measure space M and an isomorphism α of the space
H onto the space L2(M) that transforms the operator A into a
multiplication operator by a function a(x) (i.e. αAα−1 = α(x)). The
function a(x) is real if A is self-adjoint and |a(x)| = 1 if A is a unitary
operator.
The theorem allows us to define a function of a self-adjoint or a
unitary operator. Namely, if φ(λ) is a function of a real variable, then
we can define the operator φ(A), where A is a self-adjoint operator,
as the operator of multiplication by φ(a(x)) using the isomorphism α
described above (more precisely, if αAα−1 = a(x), then αφ(A)α−1 =
φ(a(x))). (We assume that the function φ(a(x)) is measurable; this
is always the case if the function φ(λ) is Borel measurable, i.e. it
is measurable with respect to the family B generated by open sets,
their countable intersections, and unions.)
Note the following obvious relations:
1. If h(λ) = γf(λ) + µg(λ); r(λ) = f(λ)g(λ); s(λ) = f(g(λ)), then
h(A) = γf(A) + µg(A); r(A) = f(A)g(A); s(A) = f(g(A)).
2. If B = U−1AU , where U is a unitary operator on the space H,
then φ(B) = U−1φ(A)U .
3. If φ(λ) is a bounded function and |φ(λ)| ≤ M , then φ(A) is a
bounded operator with ‖φ(A)‖ ≤M .
4. [φ(A)]∗ = φ(A); if the function φ(λ) is real, then the operator φ(A)
is self-adjoint, and if |φ(λ)| = 1, then φ(A) is a unitary operator.
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In particular, if A is self-adjoint, then the operator exp(itA) is
unitary.
If φ is a measurable function on the circle |z| = 1, and the operator
A is unitary, then the operator φ(A) can be defined precisely in the
same way and has similar properties.
If A is a self-adjoint operator, it is convenient to consider
projections Eµ = eµ(A), where eµ(λ) is a function equal to 1 for
λ ≤ µ and 0 for λ > µ. Projections Eµ are called spectral projections
of the operator A. It follows from the relation 1 =∫deµ that
A =
∫ ∞−∞
µdEµ
(this formula is called the spectral decomposition of the operator
A; for finite a and b, the integral∫ ba µdEµ is defined as the limit of
integral sums ∑µk(Eµk+1
− Eµk),
where µ0 = a < µ1 < · · · < µn = b; ∆µk = µk+1 − µk → 0.
The integral∫∞−∞ µdEµ is defined as the strong limit
∫ ba µdEµ where
a→∞, b→ −∞.
We say that a real number λ does not belong to the spectrum of
a self-adjoint operator A, if for some ε > 0 we have Eλ+ε = Eλ−ε.
It is easily checked that the spectrum of the operator a(x) of
multiplication by the function a(x) in the space L2(M) coincides
with the range of a(x). Recall that we do not distinguish equivalent
measurable functions, therefore, the range of measurable functions
should be defined in such a way that the set does not change when we
replace a function by an equivalent function. We say that the number
λ belongs to the range of the function a(x) if for every function b(x)
differing from a(x) only on a set of measure 0 and every ε > 0, there
exists an x ∈M such that λ− ε < b(x) < λ+ ε.
Using this statement and the previously defined isomorphism
transforming a self-adjoint operator into an operator of multiplica-
tion by a function, it is easy to prove the following statements:
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Appendix 389
1. If the number β doesn’t belong to the spectrum of a self-adjoint
operator H, then ∫exp(−iβa) exp(iHa)da = 0.
More precisely, for any x, y ∈ H∫exp(−iβa) 〈exp(iHa)x, y〉 da = 0.
2. Let us suppose that for every smooth finite function χ(ω) with
support on (a, b) we have∫χ(t) exp(iHt)dt = 0,
where
χ(t) =
∫exp(−iωt)χ(ω)dω.
Then the spectrum of the operator H lies outside the interval (a, b).
Two operators A and B, defined on the whole space H, are called
commuting if AB = BA. The same definition can be used in the
case where operators A and B are defined on the same set D and
transform it into itself. However, for self-adjoint operators, one should
use another definition of commuting operators. One says that two
self-adjoint operators A and B in the Hilbert space H commute if
φ(A)ψ(B) = ψ(B)φ(A) for all bounded functions φ(λ), ψ(λ) (if this is
the case, we say that AB = BA). Note that the operators φ1(A) and
φ2(A) commute (here, φ1(λ) and φ2(λ) are arbitrary real functions
and A is a self-adjoint operator).
One can prove the following theorem.
For any family of self-adjoint commuting operatorsA1, . . . , An, . . .
in a Hilbert space H, one can find a measure space M and an isomor-
phism α between the spaceH and the space L2(M) transforming each
of these operators into the operator of multiplication by a function
(i.e. αAiα−1 = ai(x)).
As in the case of one operator, this theorem allows us to
define functions of families of operators. Namely, if A1, . . . , An are
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390 Mathematical Foundations of Quantum Field Theory
commuting self-adjoint operators and f(λ1, . . . , λn) is a measurable
function of n real variables, then f(A1, . . . , An) is defined as the
operator corresponding to the operator of multiplication by the
function f(a1(x), . . . , an(x)) when we apply the isomorphism α.
If ∆ is an open set in n-dimensional space, we define e∆(x1,
. . . , xn) to be the function that is equal to 1 if (x1, . . . , xn) ∈ ∆
and 0 otherwise.
The projection operators E∆ = e∆(A1, . . . , An) are called the
spectral projections of the family of commuting self-adjoint operators
A1, . . . , An.
We say that the point (λ1, . . . , λn) of an n-dimensional space
belongs to the spectrum (more precisely, the joint spectrum) of the
family of commuting self-adjoint operators A1, . . . , An, if for every
neighborhood U of the point (λ1, . . . , λn), the spectral projection
EU = eU (A1, . . . , An) does not equal 0.
If the operators A1, . . . , An can be realized in the space L2(M)
as operators of multiplication by the functions a1(x), . . . , an(x), then
the joint spectrum of these operators consists of all points of the form
(a1(x), . . . , an(x)) (more precisely, the point (λ1, . . . , λn) belongs to
the spectrum if and only if for all functions bi(x) equivalent to the
functions ai(x) and any neighborhood U of the point (λ1, . . . , λn)
there exists a point x ∈M such that (b1(x), . . . , bn(x)) ∈ U).
Let us discuss the question of the convergence of a sequence of
operators. We define three types of convergence, assuming for brevity
that all operators are bounded.
A sequence of operators An converges to the operator A uniformly
(or in norm), if ‖A−An‖ → 0.
A sequence of operators An converges to the operator A strongly
if limAnx = Ax for any vector x.
A sequence of operators An converges to the operator A weakly
if lim 〈Anx, y〉 = 〈Ax, y〉 for any vectors x and y.
Convergence in norm will be denoted by the symbol A = nlimAnor An =⇒ A, strong convergence will be denoted by the symbol
A = slimAn or An → A, and weak convergence will be denoted by
A = wlimAn or An A.
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Appendix 391
It is clear that convergence in norm implies strong convergence
and strong convergence implies weak convergence.
Note the following statements.
(a) if for a dense set of vectors x the limit slimAnx exists and the
sequence ‖An‖ is bounded, then the sequence of operators An is
strongly convergent;
(b) if the limit lim 〈Anx, y〉 exists for all x from a dense subset X
and for all y from a dense subset Y and the sequence ‖An‖ is
bounded, then the sequence An is weakly convergent;
(c) if An → A,Bn → B and the sequence ‖An‖ is bounded, then
AnBn → AB (the analog of this statement for weak convergence
does not hold); if An → A and the sequence ‖A−1n ‖ is bounded,
then A−1n → A−1;
(d) if An is a sequence of unitary operators and An =⇒ A, then
the operator A is unitary; if An → A, then the operator A is
isometric; if An A, then we can only say that ‖A‖ ≤ 1;
(e) if An A, then A∗n A∗ (for strong convergence, the analogous
statement does not hold; however, if An is a sequence of unitary
operators that strongly converges to the unitary operator A, then
A∗n = A−1n → A∗ = A−1).
Very often, it is useful to consider a family of operators transforming
a fixed linear subspace D that is dense in the Hilbert subspace
into itself (in other words, the operators are defined on D and
their range is contained within D); the family of operators having
this property will be denoted ND. The family ND is closed with
respect to linear combination and multiplication of operators. If for
an operator A ∈ ND, there exists an operator B ∈ ND satisfying the
condition 〈Ax, y〉 = 〈x,By〉 for all x ∈ D, y ∈ D, then we say that
the operator B is D-conjugate to the operator A and we denote it
by A+.
The strong limit of a sequence A = slimAn of operators An ∈ NDis defined as the operator satisfying the condition Ax = limAnx for
all x ∈ D; the weak limit A = wlimAn is defined by the condition
〈Ax, y〉 = lim 〈Anx, y〉 for any x, y ∈ D.
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A.6 Locally convex linear spaces
A seminorm on a linear space R is a non-negative function p(x) that
satisfies the conditions p(λx) = |λ|p(x), p(x+y) ≤ p(x)+p(y) for all
x, y ∈ R and for any real number λ. A seminorm that vanishes only
for x = 0 is called a norm.
A locally convex topology on a linear space R is specified by a
system A of seminorms pα(x) that satisfies the following conditions:
(1) for every two seminorms pα, pβ ∈ A, one can find a seminorm pγsuch that pγ ≥ pα, pγ ≥ pβ; (2) for every point x ∈ R, one can find a
seminorm pα ∈ A that does not vanish at the point x. A linear space
equipped with a locally convex topology is called a locally convex
topological linear space or simply a locally convex space, and the
system A is called the defining system of seminorms. If the defining
system consists of a single seminorm (which is automatically a norm),
then the space is called a normed space.
The function f(x) on a locally convex space R is continuous in
the topology of this space if for every ε > 0 and every point x0 ∈ R,
there exists a seminorm pα in the defining system of seminorms for
R and a δ > 0 such that for any point x satisfying the condition
pα(x − x0) < δ, we have |f(x) − f(x0)| < ε. As an example of a
locally convex space, we can consider the space S(Er) of smooth
functions of r variables, having all derivatives tending to zero faster
than any polynomial.
The topology on the space S(Er) is specified by the system of
seminorms
‖φ‖α,β = supx∈Er
|x(α)D(β)φ(x)|
(here, α = (α1, . . . , αr), β = (β1, . . . , βr) are arbitrary sets of
non-negative integers, x(α) = xα11 . . . xαrr , D(β) = ∂β1+···+βr
∂xβ11 ...∂xβrr
).
We say that two systems of seminorms on a linear space are
equivalent (i.e. they specify the same topology) if every function that
is continuous in one system is necessarily continuous in the other. In
other words, one can consider different defining systems of seminorms
on the same locally convex space; among these systems, there exists
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Appendix 393
a maximal system consisting of all continuous seminorms. Instead of
using the precise term “a seminorm that is continuous with respect
to the locally convex topology in R”, we will use the shorter term “a
seminorm in R”.
A sequence ξn of elements of the locally convex space R converges
to the element ξ if for every seminorm p in R the sequence p(ξn− ξ)tends to zero. A sequence ξn is called Cauchy if for every seminorm
p in R we have limm→∞,n→∞ p(ξm − ξn) = 0. In a complete locally
convex space, every Cauchy sequence is convergent.
A.7 Generalized functions (distributions)
Physicists define generalized functions f(x) as symbols that only
make sense under the sign of an integral∫f(x)φ(x)dx, where φ(x) is
a “good” function. The mathematical formalization of this definition
is given as follows.
Let M be a measure space and R any linear subspace of the
space of measurable functions on M (the elements of R are called
test functions). A generalized function on M (more precisely, a
generalized function with numerical values) is defined as a linear
functional f(φ) on the space R. A generalized function on the space
M is denoted by f(x) and we define∫f(x)φ(x)dx = f(φ). Note that∫
f(x)φ(x)dx should be understood as another way of writing the
number f(φ).
It is often convenient to consider the space of test functions R
endowed with a topology, and generalized functions as continuous
linear functionals on R. We will consider the topology on the space
of test functions only in one situation — when defining generalized
functions of moderate growth.
Clearly, the definition of generalized functions depends on the
choice of the space of test functions R. The set of generalized
functions, corresponding to a given space of test functions R, we
denote by R′. The set R′ is naturally equipped with the structure of
a linear space. If R ⊂ L2(M), then any function g ∈ L2(M) defines
a linear functional on R by the formula g(φ) =∫M g(x)φ(x)dµ.
Moreover, this integral also makes sense for other functions g.
In other words, under certain conditions, a conventional function
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394 Mathematical Foundations of Quantum Field Theory
generates a generalized function, which will be denoted by the same
symbol.3
Generalized functions of n variables x1, . . . , xn ∈ M will be
defined as functionals f(φ1, . . . , φn), depending on n elements
φ1, . . . , φn ∈ R and linear in every argument (in other words, gener-
alized functions of n variables are defined as linear functionals on the
tensor product of n copies of the space R). We will denote a general-
ized function of n variables by the symbol f(x1, . . . , xn) and will write
that f(φ1, . . . , φn) =∫f(x1, . . . , xn)φ(x1) . . . φ(xn)dx1dx2 . . . dxn.
The set of generalized functions of n variables will be denoted by R′n.
Let us consider some examples of generalized functions.
1. Let C(Er) be the space of continuous functions on the Euclidean
space Er. By the symbols δ(x,a) and δ(x − a), we will denote
the generalized function (the functional on the space C(Er)) that
maps a function φ(x) ∈ C(Er) to φ(a), the function’s value at
the point a (the point a ∈ Er plays the role of a parameter).
In other words, the function δ(x − a) (δ-function) satisfies the
relation∫δ(x−a)φ(x)dx = φ(a). It is clear that for any subspace
R ⊂ C(Er), a δ-function is a generalized function from the set R′.
2. Let Cn(Er) be the space of n-times continuously differentiable
functions on the space Er. The generalized function δ(α)(x−a) ∈(Cn(Er))′ is defined by the relation∫
δ(α)(x− a)φ(x)dx = (−1)|α|(D(α)(φ))(a)
(here, α = (α1, . . . , αk) is a collection of non-negative integers
satisfying |α| = α1 + · · ·+ αk ≤ n, D(α)(φ) = ∂|α|
∂xα11 ...∂xαnn
φ).
3. Let us assume that a space R is contained in the space L2(M),
then we can define a generalized function of two variables
3Let us emphasize that while∫f(x)φ(x)dx is understood as another way of
writing the number f(φ) and is defined if φ is a test function and f is a generalizedfunction, in this section we use the notation
∫f(x)φ(x)dµ for the Lebesgue
integral of the function f(x)φ(x). Clearly, the Lebesgue integral is defined whenthe function f(x)φ(x) is summable. For integrals in the Euclidean space Er inboth cases, we use the same notation
∫f(x)φ(x)dx.
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Appendix 395
δ(x, y) ∈ R′2 by means of the relation∫δ(x, y)φ(x)ψ(y)dxdy =
∫φ(x)ψ(x)dµ.
4. Every operator A in the space L2(M) specifies a generalized
function of two variables A(x, y) ∈ (DA)′2 by the formula∫A(x, y)φ(x)ψ(y)dxdy =
⟨Aψ, φ
⟩,
and the function A(x, y) is called the kernel of the operator A.
The operator A can be represented as an integral operator with
the kernel A(x, y). In other words, we have
(Aψ)(x) =
∫A(x, y)ψ(y)dy. (A.1)
The equation (A.1) should be understood as an abbreviation of
the following relation:∫(Aψ)(x)φ(x)dµ =
∫A(x, y)ψ(y)φ(x)dxdy.
5. Considering numerical generalized functions on the space Er, we
restrict ourselves to generalized functions of moderate growth,
defined as continuous functionals on the space S(Er) of smooth
functions having derivatives that tend to zero at infinity faster
than any power of the norm of x. The topology in this space is
described in A.6.
The linear space of generalized functions of moderate growth is
denoted by S ′(Er).
Every numerical function a(x) that grows at infinity no faster than
a polynomial (i.e. for large enough x, we have |a(x)| ≤ C‖x‖n) and
is summable in any bounded domain specifies a generalized function
a(φ), belonging to the space S ′, by the formula
a(φ) =
∫a(x)φ(x)dx
(the integral is understood as a Lebesgue integral).
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396 Mathematical Foundations of Quantum Field Theory
Let us define some operations on generalized functions.
A generalized function f ∈ S ′ is called the limit of a sequence
fn ∈ S ′ of generalized functions, if for every function φ ∈ S,
the limit of the sequence fn(φ) is equal to f(φ) (in other words,
lim∫fn(x)φ(x)dx =
∫f(x)φ(x)dx).
The derivative fi = ∂f∂xi
of a generalized function f ∈ S ′ is defined
by the formula
fi(φ) = f
(− ∂φ∂xi
)or, in a different notation, by the formula∫
∂f(x)
∂xiφ(x)dx = −
∫f(x)
∂φ
∂xidx.
The Fourier transform of a generalized function f is defined by the
relation
f(φ) = f(φ), where φ(x) = (2π)−r2
∫exp(i 〈k, x〉)φ(k)dk.
Note that the derivative of a function f ∈ S ′ and the Fourier
transform of this function always exist and belong to the space
S ′ (this is obvious because together with the function φ ∈ S, the
functions ∂φ∂xi
and φ also belong to S).
The following important statement is called the kernel theorem.
Let us consider the functional f(φ, ψ) of φ ∈ S(Er), ψ ∈ S(En),
belonging to the space S ′(Er) for fixed ψ and S ′(En) for fixed φ.
Then, there exists one and only one functional f ∈ S ′(Er+n), obeying
f(φ⊗ ψ) = f(φ, ψ)
(the symbol φ⊗ ψ denotes the function
φ(x1, . . . , xr)ψ(y1, . . . , yn) ∈ S(Er+n)).
Vector generalized functions and operator generalized functions are
defined analogously to numerical generalized functions.
Namely, to define a vector generalized function, one should
construct for every function φ, from the space of test functions R,
a vector f(φ), from a linear space F , that linearly depends on φ.
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Appendix 397
(In other words, a vector generalized function is an operator defined
on the space R and taking values in the linear space F .)
To define an operator generalized function, we should construct
for every function φ, from the space of test functions R, an operator
A(φ) acting linearly on elements in the space H1 and transforming it
into the space H2 (we assume that all operators A(φ) have the same
domain D). One can say that an operator generalized function is an
operator defined on the space of test functions R, taking values on
the linear space of operators with domain D ⊂ H1 and taking values
in H2.
For vector and operator generalized functions, we will use the
same notation as for numerical generalized functions
f(φ) =
∫f(x)φ(x)dx.
To describe some examples of vector generalized functions, we use
the following general constructions.
Let us suppose that a linear space L is embedded in a linear
space K and on the measure space M we have specified a function
f(x) with values in K. Let us also assume that for every test function
φ ∈ R there exists an integral∫f(x)φ(x)dµ and this integral4 belongs
to the space L. Then, it is clear that the correspondence f(φ) =∫f(x)φ(x)dµ can be considered as a generalized function with values
in L, generated by the conventional function f(x) with values in K.
We will use the same notation f(x) for this generalized function.
Let us consider for example the case where x ∈ Er, R = L2(Er) =
L,K = S ′(Er). Consider the functions δa(x) = δ(x−a) and φa(x) =
(2π)−r/2 exp(i 〈a, x〉) from K = S ′(Er) that depend on the parameter
a ∈ Er (i.e. we consider the functions on Er with values in S ′(Er)).If f(a) ∈ R = L2(Er), then
∫f(a)δa(x)da = f(x) ∈ L = L2(Er)
and∫f(a)φa(x)da = f(x) ∈ L = L2(Er). In other words, we have
constructed vector generalized functions δa(x), φa(x) with values in
L2(Er).
4The integral in this formula can be understood in any sense; it is importantonly that it is linear with respect to φ.
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398 Mathematical Foundations of Quantum Field Theory
Let us suppose that D is a linear subspace in H1 and the space
H2 is embedded in the space H2. Further, let us assume that to every
point x of a measure space M , we have constructed an operator A(x)
with domain D and taking values in H2. Finally, if we assume that
for every function φ, the operator A(φ) =∫A(x)φ(x)dµ transforms
D into H2, then the conventional operator function A(x) generates a
generalized operator function A(φ) with values in the set of operators
transforming D into H2.
In particular, if M = Er, R = L2(Er), D = H1 = Fn, H2 =
Fn+1, where Fn is the space of symmetric (antisymmetric) square-
integrable functions f(x1, . . . ,xn) depending on variables x1, . . . ,
xn ∈ Er, and H2 = S ′(Er(n+1)) is the space of generalized functions
of moderate growth, depending on the variables x1, . . . ,xn+1 ∈Er, then the operator function a+
n (x) on En, where a+n (x) is an
operator transforming the function f(x1, . . . ,xn) ∈ Fn into the
function√n+ 1P [f(x1, . . . ,xn)δ(xn+1 − x)] ∈ H2, specifies an
operator generalized function, acting from Fn to Fn+1. This is clear
because the operator∫φ(x)a+(x)dx transforms Fn into Fn+1 (in the
above formula, P denotes the operator of symmetrization (antisym-
metrization)). This operator generalized function is considered in
Section 3.2.
If the space of test functions R is dense in L2(M), then the vector
generalized function f(x) with values in H is called normalized (or,
more precisely, δ-normalized) if the following condition is satisfied:
〈f(x), f(y)〉 = δ(x, y)
(to make this equation precise, we should integrate it, multiplying
by test functions φ(x), ψ(y) and writing it in the form⟨∫f(x)φ(x)dx,
∫f(y)ψ(y)dy
⟩=
∫δ(x, y)φ(x)ψ(y)dxdy
=
∫φ(x)ψ(x)dµ
or, in the shorter form 〈f(φ), f(ψ)〉 = 〈φ, ψ〉).This means that for a generalized function f that is δ-normalized,
we can construct an isometric map of the space R into H, sending
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Appendix 399
the function φ ∈ R into the vector f(φ) ∈ H. This map can be
extended continuously to an isometric map of the space L2(M) into
H, with the extended map denoted by the same symbol f . If linear
combinations of vectors f(φ) are dense in H, then the isometric map
of L2(M) into H is an isomorphism (i.e. it is a unitary map onto H).
A δ-normalized generalized vector function f having this property
is called a generalized basis. In this case, the statements given above
imply that every vector a ∈ H can be expanded with respect to the
generalized basis (i.e. in other words a has a unique representation
in the form
a = f(φ) =
∫φ(x)f(x)dx,
where φ ∈ L2(M)). If a = f(φ), then the function φ(x) can be written
in the form
φ(x) = 〈a, f(x)〉 .
The generalized vector functions δa and φa defined above can be
considered as generalized bases.
If A is an operator on the space H, then the generalized function
〈x|A|y〉 = 〈Af(y), f(x)〉
is called the matrix of the operator A in the generalized basis f(x).
Note the following useful statement: let us consider a sequence
An of operators, assuming that the norms ‖An‖ form a bounded
sequence and the matrices 〈x|An|y〉 converge to the matrix 〈x|A|y〉,
then the operatorsAn weakly converge to the operatorA
[saying that
the matrices 〈x|An|y〉 converge to 〈x|A|y〉, we have in mind that for
all functions φ, ψ ∈ R, where R is a dense subspace of L2(M), the
following relation holds:
lim
∫〈x|An|y〉φ(x)ψ(y)dxdy =
∫〈x|A|y〉φ(x)ψ(y)dxdy
].
Let us fix the space of test functions R and a dense linear subspace D
in the Hilbert spaceH. We will consider only the generalized operator
functions that transform D into itself (i.e. in other words, we assume
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400 Mathematical Foundations of Quantum Field Theory
that the domain of the operator A(φ), where φ is a test function,
coincides with D and all vectors A(φ)a, where a ∈ D, are contained
in D).
The set of all operator functions of this kind will be denoted
by MD.
We call generalized operator functions A ∈ MD, B ∈ MD
conjugate if for any a ∈ D, b ∈ D
〈A(x)a, b〉 = 〈a,B(x)b〉
(more precisely, if 〈A(φ)a, b〉 =⟨a,B(φ)b
⟩). In this case, we will use
the symbol B = A+.
The product of generalized operator functions A ∈MD, B ∈MD
is defined as the generalized operator function
C(x, y) = A(x)B(y).
To be more precise, the generalized operator function C maps a test
function α(x, y) of the form
α(x, y) =∑v
φv(x)ψv(y), (A.2)
where φv, ψv ∈ R, into the operator
C(α) =
∫(α(x, y)A(x)B(y)dxdy =
∑v
A(φv)B(ψv).
In some cases, the generalized operator function C(x, y) can be
defined on a broader class of test functions than functions in the
form (A.2). As an example, consider the following statement (the
operator analog for the kernel theorem).
Let us suppose that the space of test functions is the space
S(Er). Let us fix in the set MD a subset RD, consisting of
operator generalized functions A ∈ MD, satisfying the following
conditions: (a) the functional 〈A(φ)a, b〉 for any a, b ∈ D is continuous
in the topology of the space S (in other words, the numerical
generalized functions 〈A(x)a, b〉 belong to the space S ′); (b) there
exists an operator generalized function A+ ∈ MD conjugate to A.
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Appendix 401
If A1, . . . , Ap ∈ RD, then to every function ψ ∈ S(Epr) we can assign
an operator
C(ψ) =
∫ψ(x1, . . . , xp)A1(x1) . . . Ap(xn)dpx, (A.3)
defined on the set D. Namely, we should take
C(ψ) =s∑
k=1
A1(φ(k)1 ) . . . Ap(φ
(k)p )
in the case when
ψ =
s∑k=1
φ(k)1 (x1) . . . φ(k)
p (xp), (A.4)
and in general, we should represent ψ as limit of a sequence ψn of
functions of the form (A.4), converging to ψ in the topology of S and
take
C(ψ) = slimC(ψn). (A.5)
The existence of this limit follows from the following estimates:
‖C(ψm)a− C(ψn)a‖2 = 〈C(ψm)a− C(ψn)a,C(ψm)A− C(ψn)a〉
=⟨(C+(ψm)C(ψm) + C+(ψn)C(ψn)
−C+(ψm)C(ψn)− C+(ψn)C(ψm))a, a⟩
=
∫ρm,n(x1, . . . , xp|y1, . . . , yp)
⟨A+p (xp)
. . . A+1 (x1)A1(y1) . . . Ap(yp)a, a
⟩dpxdpy,
(A.6)
where
ρm,n(x1, . . . , xp|y1, . . . , yp) = ψm(x1, . . . , xp)ψm(y1, . . . , yp)
+ψn(x1, . . . , xp)ψn(y1, . . . , yp)
−ψn(x1, . . . , xp)ψ(y1, . . . , yp)
−ψm(x1, . . . , xp)ψn(y1, . . . , yp).
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402 Mathematical Foundations of Quantum Field Theory
It is easy to see that for m→∞, n→∞, the function ρm,n tends to
zero in the topology of the space S(E2pr).
Using the kernel theorem, one can conclude from this fact that the
expression A.6 tends to zero for m → ∞, n → ∞. This means that
the sequence C(ψn)a is a Cauchy sequence, hence it is convergent
for every a ∈ D. This proves that the sequence of operators C(ψn)
is strongly convergent; hence, we can define the operator C(ψ) by
means of the formula (A.5).
Let us consider now the set D′ consisting of linear combinations
of vectors of the form C(ψ)a, where a ∈ D, and C(ψ) is an operator
of the form (A.3), constructed by means of the operator analog of the
kernel theorem. Let us show that the expression of the form (A.3)
specifies an operator with domain D′ by the formula
C(ψ)(C ′(ψ′)a) = (C(ψ)C ′(ψ′))a,
where
C ′(ψ′) =
∫ψ′(x1, . . . , xq)A
′1(x1) . . . A′q(xq)d
qx,
C(ψ)C ′(ψ′) =
∫ψ(x1, . . . , xp)ψ
′(x′1, . . . , x′q)A1(x1)
. . . Ap(xp)A′1(x′1) . . . A′q(x
′q)d
pxdqx′.
To check that this definition is correct, we should verify that the
vector C(ψ)d does not depend on the representation of the vector d
in the form d = C ′(ψ′)a. We should also check that the constructed
operator is linear. The proof is based on the following remark.
Assume that the vector ξn has the form
ξn =
s(n)∑i=1
C(i)n (φ(i)
n )ai,
ξn → 0, and the sequence
ηn =
s(n)∑i=1
(C(ψn)C(i)n (φ(i)
n ))ai,
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Appendix 403
is convergent. Then ηn → 0.
[Here, ai ∈ D,φ(i)
n ∈ S, ψn ∈ S, ψn → ψ
in the topology of S, A(i)k ∈ RD,
C(i)n (φ(i)
n ) =
∫φ(i)n (y1, . . . , yp(i))A
(i)1 (y1) . . . A
(i)p(i)(yp(i))d
p(i)y.
]This is true because for every vector ζ ∈ D
| 〈ηn, ζ〉 | = |⟨ξn, C
+(ψn)ζ⟩| ≤ ‖ξn‖‖C+(ψn)ζ‖ → 0.
A.8 Eigenvectors and generalized eigenvectors
We call a vector φ ∈ H an eigenvector of the operator A, acting on the
spaceH, if Aφ = λφ and φ 6= 0. The number λ is called an eigenvalue,
corresponding to the eigenvector φ. The set Hλ of eigenvectors for a
given eigenvalue λ is in fact a linear subspace; the dimension of this
space is called the multiplicity of the eigenvalue λ. An eigenvalue
with multiplicity one is called non-degenerate or simple.
One says that the operator A has discrete spectrum if linear
combinations of its eigenvectors are dense in the space H.
The eigenvalues of a self-adjoint operator are real; the eigenvectors
corresponding to distinct eigenvalues of a self-adjoint operator are
orthogonal.
If a self-adjoint operator has discrete spectrum, then the spectrum
of this operator is the closure of its set of eigenvalues. The spectral
projection Eµ is equal to a sum of projections on the linear subspace
Hλ, where λ ≤ µ.
Suppose that M is a measure space, R is a space of test functions
defined on M , and f ∈ R′ is a generalized vector function of the
variable x ∈ M with values in H. Then the function f is called a
generalized eigenfunction of an operator A, acting on the space H, if
Af(x) = a(x)f(x).
More precisely, the above equation means that
A
∫f(x)φ(x)dx =
∫f(x)a(x)φ(x)dx,
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404 Mathematical Foundations of Quantum Field Theory
i.e. Af(φ) = f(aφ). The last equation must be satisfied in the cases
when it makes sense (i.e. when φ ∈ R and aφ ∈ R).
The generalized vector function φa = (2π)−r/2 exp(iax), defined
in (A.6), turns out to be a generalized eigenfunction for the operators1i∂∂x1
, . . . , 1i∂∂xr
(and more broadly for differential operators with
constant coefficients):
1
i
∂
∂xkφa = akφa.
Note the following important statement.
For any commuting self-adjoint operators A1, . . . , An, acting on
the Hilbert space H, we can find a generalized basis of functions that
are eigenfunctions for each of the operators (more precisely, we can
find a generalized basis f that is δ-normalized (〈f(x), f(y)〉 = δ(x, y))
and satisfies the equation Aif(x) = ai(x)f(x) for every i = 1, . . . , n).
The proof of this statement is easily obtained if we recall that
there exists an isomorphism α of the space H and the space
L2(M), by which the operators Ai are transformed into operators of
multiplication by the function ai(x). Indeed, the generalized vector
function f can be represented by the formula
f(φ) =
∫f(x)φ(x)dx = α−1φ,
where φ ∈ L2(M).
A.9 Group representations
Suppose that for every group element g of a group G, there is a
corresponding unitary operator Tg on the Hilbert space H, such
that the identity operator corresponds to the unit of the group
and the multiplication of operators corresponds to the multiplication
operation in the group:
Te = 1, Tgh = TgTh. (A.7)
We call this correspondence a unitary representation of the group G.
A subspace H1 ⊂ H is called invariant if every operator Tgtransforms H1 into itself. A unitary representation H is called
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Appendix 405
irreducible if the space H has no non-trivial (differing from the zero
subspace and the full space H) invariant subspace.
Two unitary representations Tg and T ′g on the spaces H and H′are equivalent if there exists an isomorphism α between the spaces
H and H′ satisfying
T ′gα = αTg.
For any self-adjoint operator A, we can construct a unitary
representation of the group of real numbers E1 by assigning to every
number t ∈ E1 the operator Tt = exp(iAt) (we use addition as the
group operation on E1 so that equation (A.7) takes the form
T0 = 1, Tt+τ = TtTτ ).
Conversely, given a unitary representation Tt of the group E1 such
that for any two vectors a, b ∈ H, the function 〈Tta, b〉 is continuous
or at least measurable, then we can find a self-adjoint operator A such
that Tt = exp(iAt); this operator can be obtained by the formula
A = 1i limt→0
Tt−1t (Stone’s theorem).
A family of operators satisfying the conditions of Stone’s theorem
is called a one-parameter group of unitary operators and the operator
A is called the generator (or an infinitesimal generator) of the group.
A Poincare group P is the group of affine transformations of
Minkowski space
x′ = Λx+ a, (A.8)
preserving the space–time interval and the sign of x0.
(Minkowski space is to be understood as a four-dimensional space
with a scalar product (x, y) = x0y0 − 〈x,y〉 = x0y0 − x1y1 − x2y2 −x3y3. The space–time interval is defined as
√(x− y, x− y). We write
vectors from Minkowski space as x = (x0,x), where x0 ∈ E1,
x ∈ E3.)
The transformation (A.8), viewed as an element of the group P,
will be denoted by (Λ, a).
Let Tg be a unitary representation of the group P such that for any
vectors a, b ∈ H the function 〈Tga, b〉 is continuous in g (we will only
consider unitary representations satisfying this continuity condition).
March 30, 2020 10:29 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-x99a-app page 406
406 Mathematical Foundations of Quantum Field Theory
The group P contains a subgroup of translations (1, a), consisting
of transformations x′ = x + a. This subgroup is commutative. By
Stone’s theorem, the unitary operator T(1,a), corresponding to the
translations (1, a), can be represented in the form
T(1,a) = exp[−i(a0H − a1P1 − a2P2 − a3P3)]
= exp[−i(a0H − aP)],
where H,P1, P2, P3 are commuting self-adjoint operators.
The operator H is called the energy operator and the vector
operator P = (P1, P2, P3) is the momentum operator.
It is easy to check that the joint spectrum of the operators H,P
is invariant with respect to homogenous Lorentz transformations.
One of the simplest representations of the group P can be written
in the following way.
Let us consider the set Um in four-dimensional space specified
by the relation p20 − p2 = m2, p0 ≥ 0. To every function f ∈
L2(E3), we will assign a function f on the space Um defined by the
formula f(p0,p) = f(p)√p0. This assignment can be viewed as an
isomorphism between the space L2(E3) and some space of functions
on Um. This space of functions on Um is denoted by L2(Um); the
inner product on L2(Um) is defined by the formula
⟨f , g⟩
=
∫f(p0,p)g(p0,p)
dp
p0.
On the space L2(Um), we define a unitary representation of the group
P by the formula
(U(Λ, a)f)(p) = exp(i(a, p))f(Λ−1p) (A.9)
(to check that the formula (A.9) defines a unitary representation,
observe that the expression p−10 dp defines a Lorentz-invariant mea-
sure on Um). The resulting representation is continuous; the spectrum
of the operators H,P on the space L2(Um) coincides with the set Um.
By the isomorphism between L2(E3) and L2(Um), demon-
strated above, there exists equivalent unitary representation V (Λ, a)
March 30, 2020 10:29 Mathematical Foundations of Quantum Field Theory 9in x 6in 11122-x99a-app page 407
Appendix 407
on L2(E3):
(V (Λ, a)f) = U(Λ, a)f .
Let us describe (up to equivalence) all the irreducible representa-
tions of the group P in which the energy operator H is non-negative
and not equal to zero.
For every non-negative number m, there exists an infinite series
of unitary irreducible representations, where the joint spectrum of
the operators H,P in these representations is the set Um. The
representation of this series will be denoted by the symbols (m,σ),
where σ is a non-negative integer for positive m and any integer for
m = 0.
A representation of the type (m,σ) can be realized via unitary
operators on the space L2(E3 × S), where S is a set consisting of
2σ + 1 elements for positive m and of one element for m = 0. Here,
the energy and momentum operators on L2(E3×S) are given by the
formulas
Hf(p, s) =√
p2 +m2f(p, s), (A.10)
Pf(p, s) = pf(p, s). (A.11)
For example, the representation described above is a representation
of type (m, 1).
Let us note that in addition to the single-valued unitary rep-
resentations of P described here, there exist two-valued unitary
representations. Irreducible two-valued representations, where the
energy operator is non-negative and is not identically zero, can be
described in the same way as single-valued (with the difference that
σ must be a half-integer).
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Index
adiabatic, viii, xxxi, 43, 45, 52, 101,150, 166, 170, 253, 254, 309,362–365, 367, 368
angular momentum, xxxiv, 9–12, 341
asymptotically commutative, 194,233, 242, 292–294, 313
CAR, xxxvi, xxxviii, xxxix, 30, 75,76, 79, 82–86, 88, 93, 97, 98,109–113, 120, 123, 133, 156, 163,179, 334, 355, 356
CCR, xxviii, xxix, xxxiv, xxxvi,xxxviii, xxxix, 30, 69, 75–77, 87,90, 93, 95, 97, 98, 109–113, 120,126, 127, 133, 141, 156, 163, 179,181, 193, 277, 281–285, 290, 327
correlation function, xxvi, xxvii, xl,333, 335, 336, 343
cross section, xix, xlii, 58–61, 162,236–238, 345, 354, 368–370
decoherence, xxxi, xxxii, xliii
dressed particle, xix, 150
dressing operator, 153, 178, 229, 232,233, 294, 311
entropy, 331, 332, 334
equilibrium, xxix, xxxii, xl, xli, 181,331–336, 338, 340, 355
formal Hamiltonian, xxxviii, xxxix,126, 127, 275, 277, 327, 359
generalized function, xxii, xxxviii, 30,393–400
generalized Green function, xix, 353,363
GNS (Gelfand–Naimark–Segal)construction, xxv, xxvi, 337, 341,357, 360, 362
Green function, xxvi, xxvii, 111–114,116, 118–122, 128, 161, 327,350–353, 366, 370
Heisenberg equation, xxxiii, 6, 7, 18,84, 121, 138, 139, 143, 278–280,285, 290, 305
inclusive, xix, xlii, 345, 353, 368–370,372, 373
KMS (Kubo–Martin–Schwinger)condition, xli, 333, 335, 336, 340
LSZ(Lehmann–Symanzik–Zimmerman)formula, 163, 350, 351, 371, 372
measurable function, ix, 259,379–381, 386, 388, 390, 393
measure, xxiii, xxxii, 4, 5, 378, 380,381, 383, 388, 406
measure space, ix, xlv, xlvi, 4,378–380, 386, 387, 389, 393, 397,398, 403
413
April 1, 2020 14:21 Mathematical Foundations of Quantum Field Theory 9in x 6in 11222-x99c-index page 414
414 Mathematical Foundations of Quantum Field Theory
momentum, xxi, xxxiv, xli, xliii, xlvi,3, 4, 9–11, 14–17, 27, 29, 34–36,58–60, 62–64, 123, 124, 126, 138,142, 172, 184, 233, 275, 301, 335,407
Møller matrix, 73, 149, 196, 203, 240,249, 253, 274, 279, 307–309, 311
operator realization, viii, xxxix,126–129, 138, 139, 143–146, 153,154, 166, 184–188, 275, 277–279,286–290, 292, 293, 325, 336
oscillator, xxxv, 17, 18, 20, 21, 77,334
partition function, 331, 332, 334, 335perturbation theory, viii, xvii, xviii,
xxxvii, xxxix, 39, 44, 45, 65, 86,114, 129, 131, 170, 277, 301, 303,305–309, 311, 327, 328, 338, 359
Schrodinger equation, 1, 2, 7, 46, 63,280
soliton, xliii–xlv, 188, 189spin, 11–16, 23, 57, 123, 316, 326, 341stationary, xxvi, xxix–xxxi, 6, 7, 15,
16, 18–22, 27, 39, 43, 44, 46, 63,119, 168, 178, 209, 306, 332, 360,373
symmetry breaking, 186, 187
translation-invariant Hamiltonian,viii, xxxix, 64, 65, 68, 123–126, 128,147, 275, 311,
truncated, xl, 197–199, 333
Wightman function, xxvii, xxxix, xl,107–113, 322