schwinger j

QuantumKinematicsand Dynamics

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QuantumKinematics

and Dynamics

JULIAN SCHWINGERUniversity of California, Los Angeles

This book was originally published as part of theFrontiers in Physics Series, edited by David Pines,

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2 3 4 5 6 7 8 9 10

ADVANCED BOOK CLASSICS

David Pines, Series Editor

Anderson, P.W., Basic Notions of Condensed Matter PhysicsBethe H, and Jackiw, R,, Intermediate Quantum Mechanics, Third EditionCowan, G. and Pines, D., Compkxity: Metaphors, Models, and Realityde Germes, P.G., Superconductivity of Metals and Alloysd'Bspagnat, B., Conceptual Foundations of Quantum Mechanics, Second EditionFeynman, R., Photon-Hadron InteractionsFeynman, R., Quantum ElectrodynamicsFeynman, R., Statistical MechanicsFeynman, R., The Theory of Fundamental ProcessesGeH-Mann, M. and Ne'eman, Y., The Eightfold WayKhalatnikov, I. M. An Introduction to the Theory of SuperfluidityMa, S-K., Modern Theory of Critical PhenomenaMigdal, A. B., Qualitative Methods in Quantum TheoryNegele, J. W, and Orland, H,, Quantum Many-Particle SystemsNozieres, P., Theory of Interacting Fermi SystemsNozieres, P. and Pines, D., The Theory of Quantum LiquidsParisi, G., Statistical Field TheoryPines, D., Elementary Excitations in SolidsPines, D., The Many-Body ProblemQuigg, C, Gauge Theories of the Strong, Weak, and Electromagnetic InteractionsSchrieffer, J.R., Theory of Superconductivity, RevisedSchwinger, J-, Particles, Sources, and Fields, Volume ISchwinger, J., Particles, Sources, and Fields, Volume IISehwinger, J., Particles, Sources, and Fields, Volume IIISchwinger, J., Quantum Kinematics and DynamicsWyld, H.W., Mathematical Methods for Physics

Editor's Foreword

Perseus Publishing's Frontiers in Physics series has, since 1961, madeit possible for leading physicists to communicate in coherent fashiontheir views of recent developments in the most exciting and activefields of physics—without having to devote the time and energyrequired to prepare a formal review or monograph. Indeed,throughout its nearly forty year existence, the series has emphasizedinformality in both style and content, as well as pedagogical clarity.Over time, it was expected that these informal accounts would bereplaced by more formal counterparts—textbooks or monographs—as the cutting-edge topics they treated gradually became integratedinto the body of physics knowledge and reader interest dwindled.However, this has not proven to be the case for a number of thevolumes in the series: Many works have remained in print on an on-dernand basis, while others have such intrinsic value that the physicscommunity has urged us to extend their life span,

The Advanced Book Classics series has been designed to meet thisdemand. It will keep in print those volumes in Frontiers in Physicsthat continue to provide a unique account of a topic of lastinginterest And through a sizable printing, these classics will be madeavailable at a comparatively modest cost to the reader.

The late Nobel Laureate Julian Schwinger was not only one of thegreat theoretical physicists of our time, but also one of the greatpedagogues of the past century. His lectures were legendary for theiralmost unique combination of clarity and elegance. I am accordinglyvery pleased that the publication in Advanced Book Classics ofQuantum Kinematics and Dynamics will continue to make his lectureson this topic readily accessible to future generations of the scientificcommunity.

David PinesCambridge, EnglandMay, 2000

Vita

Julian SchwingerUniversity Professor, University of California, and Professor of Physics at the Universityof California, Los Angeles since 1972, was born in New York City on February 12,1918.Professor Schwinger obtained his Ph.D. in physics from Columbia University in 1939. Hehas also received honorary doctorates in science from five institutions: Purdue University(1961), Harvard University (1962), Brandeis University (1973), Gustavus AdolphusCollege (1975), and the University of Paris (1990). In addition to teaching at theUniversity of California, Professor Schwinger has taught at Purdue University (1941 -43),and at Harvard University (1945-72). Dr, Schwinger was a Research Associate at theUniversity of California, Berkeley, and a Staff Member of the Massachusetts Institute ofTechnology Radiation Laboratory. In 1965 Professor Schwinger became a co-recipient(with Richard Feynman and Sin Itiro Tomonaga) of the Nobel Prize in Physics for workin quantum electrodynamics. A National Research Foundation Fellow (1939-40) and aGuggenheim Fellow (1970), Professor Schwinger was also the recipient of the C. L.Mayer Nature of Light Award (1949); the First Einstein Prize Award (1951); a J. W. GibbsHonorary Lecturer of the American Mathematical Society (1960); the National Medal ofScience Award for Physics (1964); a HumboJdt Award (1981); the Premio Citta diCastiglione de Sicilia (1986); the Monie A. Ferst Sigma Xi Award (1986); and theAmerican Academy of Achievement Award (1987).

Special Preface

The first two chapters of this book are devoted to Quantum Kinematics. In 1985 I had theopportunity to review that development in connection with the celebration of the 100thanniversary of Hermann Weyl's birthday, (See the last footnote of Chapter 2.) Inpresenting my lecture (Hermann Weyl and Quantum Kinematics, in Exact Sciences andTheir Philosophical Foundations, Verlag Peter Lang, Frankfurt am Main, 1988, pp. 107-129), I felt the need to alter only one thing: the notation. Lest one think this rather trivial,recall that the ultimate abandonment, early in the 19th century, of Newton's method offluxions in favor of the Leibnitzian calculus, stemmed from the greater flexibility of thelatter's notation.

Instead of the symbol of measurement: M(a', b'), I now write: I a'b'l , combiningreference to what is selected and what is produced, with an indication that the act ofmeasurement has a beginning and an end. Then, with the conceptual analysis of I a'b' Iinto two stages, one of annihilation and one of creation, as symbolized by

the fictitious null state, and the symbols HP and <fy can be discarded.As for Quantum Dynamics, I have long regretted that these chapters did not contain

numerous examples of the practical use of the Quantum Action Principle in solvingphysical problems. Perhaps that can be remedied in another book, on Quantum Mechan-ics. There is, however, a cornucopia of action applications in the non-operator context ofSource Theory. (See Particles, Sources, and Fields, Vols. I, 2, 3, Advanced BookClassics.)

Los Angeles, California J. S.April 1991

Foreword

Early in 1955 I began to write an article on the Quantum Theory of Fields. The introductioncontained this description of its plan. "In part A of this article a general scheme of quantumkinematics and dynamics is developed within the nonrelativistic framework appropriateto systems with a finite number of dynamical variables. Apart from specific physicalconsequences of the rclativistic invariance requirement, the extension to fields in part Bintroduces relatively little that is novel, which permits the major mathematical features ofthe theory of fields to be discussed in the context of more elementary physical systems."

A preliminary and incomplete version of part A was used as the basis of lecturesdelivered in July, 1955 at the Les Houches Summer School of Theoretical Physics. Workon part A ceased later that year and part B was never begun. Several years after, I usedsome of the material in a series of notes published in the Proceedings of the NationalAcademy of Sciences. And there the matter rested until, quite recently, Robert Kohler(State University College at Buffalo) reminded me of the continuing utility of the LesHouches notes and suggested their publication. He also volunteered to assist in thisprocess. Here is the result. The main text is the original and still incomplete 1955manuscript, modified only by the addition of subheadings. To it is appended excerpts fromthe Proc. Nat. Acad. of Sciences articles that supplement the text, together with two papersthat illustrate and further develop its methods.

JULIAN SCHWINGERBelmont, Massachusetts1969

Contents

I The Algebra of Measurement 1

I.1 Measurement Symbols 21.2 Compatible Properties. Definition of State 51.3 Measurements that Change the State 71.4 Transformation Functions 91.5 The Trace 121.6 Statistical Interpretation 141.7 The Adjoint 171.8 Complex Conjugate Algebra 191.9 Matrices 19

1.10 Variations of Transformation Functions 221.11 Expectation Value 251.12 Addendum: Non-Selective Measurements 26

II The Geometry of States 29

2.1 The Null State 292.2 Reconstruction of the Measurement Algebra 322.3 Vector Algebra 352.4 Wave Functions 372.5 Unitary Transformations 402.6 Infinitesimal Unitary Transformations 442.7 Successive Unitary Transformations 462.8 Unitary Transformation Groups. Translation and Rotations 482.9 Reflections S3

2.10 Continuous Spectra 54

x.v

2

2.11 Addendum: Operator Space 562.12 Addendum: Unitary Operator Bases 62

III The Dynamical Principle 73

3.1 The Action Operator 743.2 Lagrangian Operator 763.3 Stationary Action Principle 773.4 The Hamiltonian Operator 793.5 Equations of Motion. Generators 803.6 Commutation Relations 833.7 The Two Classes of Dynamical Variables 863.8 Complementary Variables of the First Kind 973.9 Non-Hermitian Variables of the First Kind 103

3.10 Complementary Variables of the Second Kind 106

IV The Special Canonical Group 113

I. VARIABLES OF THE FIRST KIND 1144.1 Differential Operators 1154.2 Schrodinger Equations 1194.3 The q p Transformation Function 1204.4 Differential Statements of Completeness 1224.5 Non-Hermitian Canonical Variables 1254.6 Some Transformation Functions 1264.7 Physical Interpretation 1304.8 Composition by Contour Integration 1334.9 Measurements of Optimum Compatibility 140

II. VARIABLES OF THE SECOND KIND 1434.10 Rotation Group 1434.11 External Algebra 1454.12 Eigenvectors and Eigenvalues 148

III. UNIFICATION OF THE VARIABLES 1524.13 Constructive Use of the Special Canonical Group 1524.14 Transformation Functions 1564.15 Integration 1664.16 Differential Realizations 170

V Canonical Transformations 173

5 1 Group Properties and Superfluous Variables 1755.2 Infinitesimal Canonical Transformations 178

Contentsxvi

5.3 Rotations. Angular Momentum 1825.4 Translations. Linear Momentum 1855.5 Transformation Parameters 1875.6 Hamilton-Jacobi Transformation 1905.7 Path Dependence 1915.8 Path Independence 1945.9 Linear Transformations 195

VI Groups of Transformations 201

6.1 Integrability Conditions 2026.2 Finite Matrix Representation 2046.3 Subgroups 2076.4 Differential Forms and Composition Properties 2096.5 Canonical Parameters 2116.6 An Example, Special Canonical Group 2166.7 Other Parameters. Rotation Group 2196.8 Differential Operator Realizations 2266.9 Group Volume 228

6.10 Compact Groups 2316.11 Projection Operators and Invariants 2336.12 Differential Operators and the Rotation Group 2386.13 Non-Compact Group Integration 2436.14 Variables of the Second Kind 2476.15 Reflection Operator 2496.16 Finite Operator Basis 2506.17 Addendum: Derivation of the Action Principle 2546.18 Addendum Concerning the Special Canonical Group 2596.19 Addendum: Quantum Variables and the Action Principle 275

VII Canonical Transformation Functions 285

7.1 Ordered Action Operator 2857.2 Infinitesimal Canonical Transformation Functions 2877.3 Finite Canonical Transformation Functions 2937.4 Ordered Operators. The Use of Canonical

Transformation Functions 2977.5 An Example 2997.6 Ordered Operators and Perturbation Theory 3027.7 Use of The Special Canonical Group 3067.8 Variational Derivatives 3097.9 Interaction of Two Sub-Systems 317

7.10 Addendum: Exterior Algebra and the Action Principle 321

Contents xvii

xviii Contents

VIII Green's Functions 331

8.1 Incorporation of Initial Conditions 3318.2 Conservative Systems. Transforms 3358.3 Operator Function of a Complex Variable 3378.4 Singularities 3408.5 An Example 3418.6 Partial Green's Function 343

IX Some Applications And Further Developments 347

9.1 Brownian Motion of a Quantum Oscillator 3479.2 Coulomb Green's Function 374

CHAPTER ONETHE ALGEBRA OF

1.1 Measurement Symbols 21.2 Compatible Properties. Definition of 5

State1.3 Measurements that Change the State 71.4 Transformation Functions 91.5 The Trace 121.6 Statistical Interpretation 141.7 The Adjoint 171.8 Complex Conjugate Algebra 191.9 Matrices 191.10 Variations of Transformation Functions 221.11 Expectation Value 251.12 addendum: Non-Selective Measurements 26

The classical theory of measurement is built

upon the conception of an interaction between

the system of interest and the measuring apparatus

that can be made arbitrarily small, or at least

precisely compensated, so that one can speak mean-

ingfully of an idealized measurement that disturbs

1

MEASUREMENT

2 QUANTUM KINEMATICS AND DYNAMICS

no property of the system. But it Is character-

istic of atomic phenomena that the interaction

between system and instrument is not arbitarily

small. Nor can the disturbance produced by the

interaction be compensated precisely since to some

extent it is uncontrollable and unpredictable.

Accordingly, a measurement on one property can

produce unavoidable changes in the value previously

assigned to another property, and it is without

meaning to speak of a microscopic system possessing

precise values for all its attributes. This con-

tradicts the classical representation of all phys-

ical quantities by numbers. The laws of atomic

physics must be expressed, therefore, in a non-

classical mathematical language that constitutes

a symbolic expression of the properties of micro-

scopic measurement.

1.1 MEASUREMENT SYMBOLS

We shall develop the outlines of this math-

ematical structure by discussing simplified phys-

ical systems which are such that any physical quan-

tity A assumes only a finite number of distinct

THE ALGEBRA OF MEASUREMENT 3

values, a',..aM. In the most elementary type of

measurement, an ensemble of independent similar

systems is sorted by the apparatus into subensembles,

distinguished by definite values of the physical

quantity being measured. Let M(a') symbolize the

selective measurement that accepts systems possess-

ing the value a1 of property A and rejects all

others. We define the addition of such symbols to

signify less specific selective measurements that

produce a subensemble associated with any of the

values in the summation, none of these being dis-

tinguished by the measurement.

The multiplication of the measurement symbols

represents the successive performance of measure-

ments (read from right to left). It follows from

the physical meaning of these operations that add-

ition is commutative and associative, while multi-

plication is associative. With 1 and 0 symbol-

izing the measurements that, respectively, accept

and reject all systems, the properties of the

elementary selective measurements are expressed by


Indeed, the measurement symbolized by M(a')

accepts every system produced by M(a') and rejects

every system produced by M(a") , a" a* , while

a selective measurement that does not distinguish

any of the possible values of a' is the measure-

ment that accepts all systems.

According to the significance of the measure-

ments denoted as 1 and 0 , these symbols have the

algeoraic properties

and

which justifies the notation. The various properties

of 0, M(a') and 1 are consistent, provided multi-

plication is distributive. Thus,

THE ALGEBRA OF MEASUREMENT

1.2 COMPATIBLE PROPERTIES. DEFINITION OF STATE

Two physical quantities h^ and A2 are

said to be compatible when the measurement of one

does not destroy the knowledge gained by prior

measurement of the other. The selective measure-

ments MCai) and Mia,) / performed in either

order, produce an ensemble of systems for which one

i

can simultaneously assign the values a to A^i

and a2 to A2 . The symbol of this compound

measurement is

5

The introduction of the numbers 1 and 0 as multi-

pliers, with evident definitions, permits the multi-

plication laws of measurement symbols to be combined

in the single statement

where

' '

QUANTUM KINEMATICS AND DYHAMICS

then describes a complete measurement, which is

such that the systems chosen possess definite values

for the maximum number of attributes; any attempt

to determine the value of still another independent

physical quantity will produce uncontrollable

changes in one or more of the previously assigned

values. Thus the optimum state of knowledge con-

cerning a given system is realized by subjecting

it to a complete selective measurement. The

systems admitted by the complete measurement

M(a') are said to be in the state a' . The

symbolic properties of complete measurements are

also given by (1.1), (1.2) and (1.3).

6

By a complete set of compatible physical quantities,

A , ...A , we mean that every pair of these quan-J. J fc

tities is compatible and that no other quantities

exist, apart from functions of the set A , that

are compatible with every member of this set. The

measurement symbol

THE ALGEBRA OF MEASURMENT

1.3 MEASUREMENTS THAT CHANGE THE STATE

A more general type of measurement incorporates

a disturbance that produces a change of state. The

symbol M(a', a") indicates a selective measure-

ment in which systems are accepted only in the

state a" and emerge in the state a1 , The meas-

urement process M(a') is the special case for

which no change of state occurs >

The properties of successive measurements of the

type M(a', a") are symbolized by

for, if a" a"1 , the second stage of the compound

apparatus accepts none of the systems that emerge

from the first stage, while if a" - a11*, all such

systems enter the second stage and the compound

measurement serves to select systems in the state

a'" and produce them in the state a1. Note that

if the two stages are reversed, we have

7

which differs in general from (1.12). Hence the

multiplication of measurement symbols is noncommu-

tative.

The physical quantities contained in one com-

plete set A do not comprise the totality of phy-

sical attributes of the system. One can form other

complete sets, B, C, ..., which are mutually in-

compatible, and for each choice of non-interfering

physical characteristics there is a set of selective

measurements referring to systems in the appropriate

states, M(b', b"), M(c', c"), ... . The most

general selective measurement involves two incompat-

ible sets of properties. We symbolize by M(a', b')

the measurement process that rejects all impinging

systems except those in the state b', and permits

only systems in the state a1 to emerge from the

apparatus. The compound measurement

M(a', b')M(c', d') serves to select systems in the

state d1 and produce them in the state a 1, which

is a selective measurement of the type M(a", d1} .

But, in addition, the first stage supplies systems

in the state c1 while the second stage accepts


THE ALGEBRA. OF MEASUREMENT 9

only systems in the state b' . The examples of

compound measurements that we have already consid-

ered involve the passage of all systems or no sys-

tems between the two stages, as represented by the

multiplicative numbers 1 and 0. More generally,

measurements of properties B, performed on a sys-

tem in a state c' that refers to properties in-

compatible with B, will yield a statistical dis-

tribution of the possible values. Hence, only a

determinate fraction of the systems emerging from

the first stage will be accepted by the second

stage. We express this by the general multiplication

law

where <b' c')> is a number characterizing the

statistical relation between the states b1 and c'.

In particular,

1.4 TRANSFORMATION FUNCTIONS

Special examples of (1.14) are


which shows that measurement symbols of one type can

be expressed as a linear combination of the measure-

ment symbols of another type. The general relation

is

and

We infer from the fundamental measurement symbol

property (1.3) that

and similarly

Prom its role in effecting such connections, the


totality of numbers (a1 b1) is called the trans-

formation function relating the a - and the

b-descriptions, where the phrase "a -description"

signifies the description of a system in terms of

the states produced by selective measurements of

the complete set of compatible physical quantities

A.

A fundamental composition property of trans-

formation functions is obtained on comparing

On identifying the a - and c -descriptions this

becomes

with

namely


and similarly

which means that N> the total number of states ob-

tained in a complete measurement, is independent of

the particular choice of compatible physical quanta^

ties that are measured. Hence the total number of

can simultaneously assign the values a toA^

Arbitrary numerical multiples of measurement symbols

in additive combination thus form the elements of

a linear algebra of dimensionality N - the

algebra of measurement. The elements of the measure-

ment algebra are called operators.

1.5 THE TRACE

The number { a'|b') can be regarded as a

linear numerical function of the operator M(b'» a').

We call this linear correspondence between operators

and numbers the trace,

As a consequence, we observe that

which verifies the consistency of the definition

(1.27). In particular,

Hence, despite the noncommutativity of multiplication,

the trace of a product of two factors is independent

of the multiplication order. This applies to any


and observe from the general linear relation (1.20)

that

The trace of a measurement symbol product is

which can be compared with

where the numbers X(a') and X(b') can be given

arbitrary non-zero values. The elementary measure-

ment symbols M(a') and the transformation function

<a'|a"> are left unaltered. In view of this arbi-

trariness, a. transformation function <a'|b'> can-

not, of itself, possess a direct physical inter-

pretation but must enter in some combination that

remains invariant under the substitution (1.34).


two elements X, Y, of the measurement algebra,

A special example of (1.30) is

1.6 STATISTICAL INTERPRETATION

It should be observed that the general multi-

plication law and the definition of the trace are

preserved if we make the substitutions

THE ALGEBRA OP MEASUREMENT 15

The appropriate basis for the statistical

interpretation of the traasformation function can

be inferred, by a consideration of the sequence

of selective measurements M(b')M{a')M(b'), which

differs from M(b') in virtue of the disturbance

attendant upon the intermediate A-measurement.

Only a fraction of the systems selected in the

initial B-measurenten-t is transmitted through the

complete apparatus. Correspondingly, we have the

symbolic equation

and, for the A-measurement that does not distinguish

among any of the states, there appears

is invariant under the transformation (1.34). If

we perform an A-measureinent that does not distin-

guish between two (or more) states, there is a

related additivity of the numbers p(a', b') ,

where the number


whence

These properties qualify p(a', b1) for the role of

the probability that one observes the state a1 in a,

measurement performed on a system known to be in

the state b1. But a probability is a real, non-

negative number. Hence we shall impose an admissi-

ble restriction on the numbers appearing in the

measurement algebra, by requiring that <a' b"> and

<b'|a'> form a pair of complex conjugate numbers

To maintain the complex conjugate relation (1.40),

the numbers X(a') of (1.34) must obey

and therefore have the form

for then


in which the phases ¥»(a') can assume arbitrary

real values.

1.7 THE ADJOINT

Another satisfactory aspect of the probability

formula (1»36) is the symmetry property

Let us recall the arbitrary convention that accom-

panies the interpretation of the measurement sym-

bols and their products - the order of events is

read from right to left (sinistrally). But any

measurement symbol equation is equally valid if

interpreted in the opposite sense (dextrally).

and no physical result should depend upon which

convention is employed. On introducing the dextral

interpretation, <a'|b'> acquires the meaning

possessed by <b' a'> with the sinistral conven-

tion. We conclude that the probability connecting

states a1 and b' in given sequence must be con-

structed symmetrically from <a'|b'> and <b' a'> .

The introduction of the opposite convention for

measurement symbols will be termed the adjoint

operation, and is indicated by t . Thus,


and

In particular,

which characterizes M(a') as a self-adjoint or

Hermitian operator. For measurement symbol pro-

ducts we have

in which X is an arbitrary number.

The significance of addition is uninfluenced by the

adjoint procedure, which permits us to extend

these properties to all elements of the measurement

algebra:

or equivalently.


1.8 COMPLEX CONJUGATE ALGEBRA

The use of complex numbers in the measurement

algebra Implies the existence of a dual algebra in

which all numbers are replaced^ by the complex con-

jugate numbers. No physical result can depend upon

which algebra is employed. If the operators of

the dual algebra are written X* , the correspon-

dence between the two algebras is governed by the

laws

It has the algebraic properties

1.9 MATRICES

The measurement symbols of a given descrip-

tion provide a basis for the representation of an

The formation of the adjoint within the complex

conjugate algebra is called transposition,


2arbitrary operator by N numbers, and the abstract

properties of operators are realized by the combina-

torial laws of these arrays of numbers, which are

those of matrices. Thus

and in particular

shows that

The elements of the matrix that represents X can

be expressed as

The sum of the diagonal elements of the matrix is

the trace of the operator. The corresponding

defines the matrix of X in the a-description or

a-representation, and the product

If we set X or Y equal to 1 , we obtain

examples of the connection between the matrices of


basis in the dual algebra is M{a', a")* , and the

matrices that represent X* and X are the complex

conjugate and transpose, respectively, of the matrix

representing X . The operator X = XT* , an

element of the same algebra as X , is represented

by the transposed, complex conjugate, or adjoint

matrix.

The matrix of X is the mixed ab-representation

is defined by

where

The rule of multiplication for matrices in mixed

representations is

On placing X = Y = 1 we encounter the composition

property of transformation functions, since


a given operator in various representations. The

general result can be derived from the linear rela-

tions among measurement symbols. Thus,

1.10 VARIATIONS OF TRANSFORMATION FUNCTIONS

As an application of mixed representations,

we present an operator equivalent of the fundamental

properties of transformation functions:

which is achieved by a differential characteriza-

tion of the transformation functions. If &<a'\b'>

The adjoint of an operator X , displayed in the

mixed ab-basis, appears in the ba-basis with the

matrix


and <S<b*|c'> are any conceivable infinitesmal

alteration of the corresponding transformation

functions, the implied variation of <a'|c'> is

which is the definition of an infinitesimal operator

<SW . . If infinitesimal operators 6W, and6W are defined similarly, the differential proper-acty (1,66) becomes the matrix equation

and also

One can regard the array of numbers o<a'|b'> as

the matrix of an operator in the ab-representation.

We therefore write

from which we infer the operator equation


Thus the multiplicative composition law of trans-

formation functions is expressed by an additive compo-

sition law for the infinitesimal operators 6W.

On identifying the a- and b~ descriptions in

(1.70), we learn that

or

Indeed, the latter is not an independent condition

on transformation functions but is implied by the

composition property and the requirement that

transformation functions, as matrices, be nonsingu-

lar. If we identify the a- and c- descriptions we

are informed that

Now

which expresses the fixed numerical values of the.

transformation function


which must equal

and therefore

The complex conjugate property of transformation

functions is thus expressed by the statement that

the infinitesimal operators <§W are Hermitian.

1.11 EXPECTATION VALUE

The expectation value of property A for sys-

tems in the state b1 is the average of the possible

values of A, weighted by the probabilities of oc-

curence that are characteristic of the state b1

On using (1.33) to write the probability formula

as

the expectation value becomes


where the operator A is

The correspondence thus obtained between operators

and physical quantities is such that a function

f(A) of the property A is assigned the operator

£{A) , and the operators associated with a complete

set of compatible physical quantities form a com-

plete set of commuting Hermitian operators. In

particular, the function of A that exhibits the

value unity in the state a' , and zero otherwise,

is characterized by the operator M(a')

1.12 ADDENDUM: NON-SELECTIVE MEASUREMENTS1"

tReproduced from the Proceedings of the National

Academy of Sciences Vol. 45, pp. 1552-1553 (1959).

1552 PHYSICS: J. SCHWINGEK PJROC. N. A, 8.

where the real phases w are independent, randomly distributed quantities. Theuncontrollable nature of the disturbance produced by a measurement thus finds itsmathematical expression in these random phase factors. Since a nonselectivemeasurement does not discard systems we must have

The physical operation symbolized by M(a') involves the functioning of an ap-paratus capable of separating an ensemble into subensembles that are distinguishedby the various values of a', together with the act of selecting one subenserable andrejecting the others. The measurement process prior to the stage of selection,which we call a nonselective measurement, will now be considered for the purpose offinding its symbolic counterpart. It is useful to recognize a general quantitativeinterpretation attached to the measurement symbols. Let a system in the statee' be subjected to the selective M(b') measurement and then to an A -measurement.The probability that the system will exhibit the value b' and then a', for the re-spective properties, is given by

There are examples of the relation between the symbol of any selective measure-ment and a corresponding probability,

Now let the intervening measurement be nonselective, which is to say that the ap-paratus functions but no selection of systems is performed. Accordingly,

which differs from

by the absence of interference terms between different br states. This indicatesthat the symbol to be associated with the nonselective ^-measurement is

If, in contrast, the intermediate B-measurement accepts all systems without dis-crimination, which is equivalent to performing no .B-measurenient, the relevantprobability is

each of which can also be extended to all types of selective measurements, and tononseleetive measurements (the adjoint form is essential here). The expectationvalue construction shows that a quantity which equals unity if the properties A,B,. . . S successively exhibit, in the sinistral sense, the values a', 6', . . . «', and iszero otherwise, is represented by the Hermitian8 operator (M(a'). . . M(s'))t-(M(a1) ,. M(«')).

Measurement is a dynamical process, and yet the only time concept that hasbeen used is the primitive relationship of order. A detailed formulation of quan-tum dynamics must satisfy the consistency requirement that its description of theinteractions that constitute measurement reproduces the symbolic characterizationsthat have emerged at this elementary stage. Such considerations make explicitreference to the fact that all measurement of atomic phenomena ultimately involvesthe amplification of microscopic effects to the level of macroscopic observation.

Further analysis of the measurement algebra leads to a geometry associated withthe states of systems.

1 Thia development has been presented In numerous lecture series since 1951, but is heretoforeunpublished.

* Here we bypass the question of the utility of the real number field. According to a comment inTHESE PROCMEBINOS, 44, 223 (I9S8), the appearance of complex numbers, or their real equivalents,may be an aspect of the fundamental matter-antimatter duality, which can hardly be discussed atthis stage.! Compare P. A. M. Dirae, Be», Mad. Phys, 17, 19S (1945), where n<»n-Hermitian operators

and complex "probabilities" are introduced.

and

Voi. 4S, 1989 ACKNOWLEDGMENT: I. OLKIN 1SS3

which corresponds to the unitary property of the Ms operators,

It should also be noted that, within this probability context, the symbols of theelementary selective measurements are derived from the nonseleetive symbol by re-placing all but one of the phases by positive infinite imaginary numbers, which is anabsorptive description of the process of rejecting subenaembles.

The general probability statement for successive measurements is

which is applicable to any type of observation by inserting the appropriate meas-urement symbol. Other versions are

CHAPTER TWO

THE GEOMETRY OF STATES

2.1 The Null State 292.2 Reconstruction of the Measurement 32

Algebra2.3 Vector Algebra 352.4 Wave Functions 372.5 Unitary Transformations 402.6 Infinitesimal Unitary Transformations 442.7 Successive Unitary Transformations 462.8 Unitary Transformation Groups- 48

Translations and Rotations2.9 Re2.10 Continuous Spectra 542.11 Addendum: Operator Space 562.12 Addendum: Unitary Operator Bases 62

2.1 THE NULL STATE

The uncontrollable disturbance attendant upon

a measurement implies that the act of measurement

is indivisible. That is to say, any attempt to

trace the history of a system during a measurement

process usually changes the nature of the measure-

29

53


ment that is being performed. Hence, to conceive

of a given selective measurement M(a" , b 1) as a

compound measurement is without physical implica-

tion. It is only of significance that the first

stage selects systems in the state b' , and that

the last one produces them in the state a' ; the

interposed states are without meaning for the

measurement as a whole. Indeed, we can even invent

a non-physical state to serve as the intermediary.

We shall call this mental construct the null state

0 , and write

The measurement process that selects a system in

the state b1 and produces it in the null state,

can be described as the annihilation of a system

in the state b1 ; and the production of a system

in the state a1 following its selection from the

null state,

can be characterized as the creation of a system

THE GEOMETRY OF STATES 31

in the state a1 . Thus, the content of (2.1) is

the indiscernability of M(a" , b1} from the com-

pound process of the annihilation of a system in

the state b* followed by the creation of a system

in the state a' ,

and

whereas

and

The extension of the measurement algebra to

include the null state supplies the properties of

the f and * symbols. Thus


Some properties of M{0) are

The fundamental arbitrariness of measurement sym-

bols expressed by the substitution (1.34),

2.2 RECONSTRUCTION OF THE MEASUREMENT ALGEBRA

and

Furthermore, in the extended measurement algebra,

implies the accompanying substitution

in which we have effectively removed ¥>(0) by

expressing all other phases relative to it.


The characteristics of the measurement oper-

ators M(a" f b') can now be derived from those

of the *F and * symbols. Thus

In addition, the substitution (2.13) transforms the

measurement operators in accordance with (2.12).

The various equivalent statements contained

in (2.6) show that the only significant products —

those not identically zero — are of the form

¥<f , <f¥ and X¥ , 4>X , in addition to XY ,

where the latin symbols are operators, elements of

the physical measurement algebra. According to

and

while


the measurement operator construction (2.4), all

operators are linear combinations of products

ff

and the evaluation of the products Xf , $X , and

XY reduces to the ones contained in (2,7),

Hence, in any manipulation of operators leading to

a product *¥ , the latter is effectively equal

to a number,

Accordingly,

and in particular

It should also be observed that, in any application

of 1 as an operator we have, in effect,


are designed to make this result an automatic

consequence of the notation (Dirac). In the brac-

ket notation various theorems, such as the law of

matrix multiplication (1.61), or the general for-

mula for change of matrix representation (1.63),

appear as simple applications of the expression

for the unit operator

2.3 VECTOR ALGEBRA

We have associated a ¥ and a * symbol

with each of the N physical states of a descrip-

tion. Now the symbols of one description are

linearly related to those of another description,

which shows that

The bracket symbols

and

which also implies the linear relation between

measurement operators of various types. Arbitrary

numerical multiples of f or $ symbols thus

form the elements of two mutually adjoint algebras

of dimensionality N , which are vector algebras

since there is no significant multiplication of

elements within each algebra. We are thereby pre-

sented with an N-dimensional geometry -- the geo-

metry of states — from which the measurement alge-

bra can be derived, with its properties character-

ized in geometrical language. This geometry is

metrical since the number $¥ defines a scalar

product. According to (2.20), the vectors *(a')

and ¥(a1) of the a-descriptioa provides an ortho-

normal vector basis or coordinate system, and thus

the vector transformation equations (2.26) and

(2.27) describe a change in coordinate system.

The product of an operator with a vector expresses


which characterizes f(a1) and $(a') as the

right and left eigenvectors, respectively, of the

complete set of commuting operators A , with the

eigenvalues a' . Associated with each vector

algebra there is a dual algebra in which all num-

bers are replaced by their complex conjugates.

2.4 WAVE FUNCTIONS

The eigenvectors of a given description pro-

vide a basis for the representation of an arbitrary

vector by N numbers. The abstract properties of


a mapping upon another vector in the same space,

The effect on the vectors of the a-coordinate sys-

tem of the operator symbolizing property A ,

is given by


vectors are realized by these sets of numbers,

which are known as wave functions. We write

and

If $ and 41 are in adjoint relation, $ = f + ,the corresponding wave functions are connected by

The scalar product of two vectors is

and, in particular,


which characterizes the geometry of states as a

unitary geometry. The operator ¥,$,, is repre-

sented by the matrix

and

On placing X = 1 , we obtain the relation between

the wave functions of a given vector in two dif-

ferent representations,

Note that the wave function representing ¥(b')

in the-a-description is

and wave functions that represent Xf and fX

are


2,5 UNITARY TRANSFORMATIONS

The automophisms of the unitary geometry of

states are produced by the unitary transformations

From the viewpoint of the extended measurement

algebra, <j» and ifi wave functions are matrices

with but a single row, or column, respectively.

It is a convenient fiction to assert that

every Hermitian operator symbolizes a physical

quantity, and that every unit vector symbolizes a

state. Then the expectation value of property X

in the state ¥ is given by

In particular, the probability of observing the

values a1 in an A-measurement performed on sys-

tems in the state ¥ is


applied to every vector and operator, where the

unitary operator U obeys

All algebraic relations and adjoint connections

among vectors and operators are preserved by this

transformation. Two successive unitary transfor-

mations form a unitary transformation, and the

inverse of a unitary transformation is unitary -

unitary transformations form a group. The appli-

cation of a unitary transformation to the ortho-

normal basis vectors of the a-description, which

are characterized by the eigenvector equation

yields orthonormal vectors

that obey the eigenvector equation


Hence the <a'j are the states of a new descrip-

tion associated with quantities A that possess

the same eigenvalue spectrum as the properties A ,

Since all relations among operators and vectors

are preserved by the transformation, we have

where

The equivalent forms

exhibit the a-representatives of operators and

vectors as the a-representatives of associated

operators and vectors.

The basis vectors of any two descriptions,

with each set placed in a definite order, are

connected by a unitary operator. Thus


obeys

The transformation function relating the a- and

b-representations can thereby be exhibited as a

matrix referring entirely to the a- or the b--

representations ,

and all quantities of the b-representation can be

expressed as a-representatives of associated oper-

ators and vectors,

If the two sets of properties A and B possess

the same spectrum of values, the operators A and

B are also connected by a unitary transformation.

With the ordering of basis vectors established by


corresponding eigenvalues we have

2.6 INFINITESIMAL UNITARY TRANSFORMATIONS

The definition of a unitary operator, when

expressed as

shows that a unitary operator differing infinites!-

mally from unity has the general form

where G is an infinitesimal Rermitian operator.

The coordinate vector transformation described by

this operator is indicated by

Now, according to (2.49), the change of coordinate


system, in its effect upon the representatives of

operators and vectors, is equivalent to a corres-

ponding change of the operators and vectors rela-

tive to the original coordinate system. Hence

and

where

and

The rectangular bracket represents the commutator

Since all algebraic relations are preserved, the

operator and vector variations are governed by


rules of the type

the latter is the operator that exhibits the same

properties relative to the a-description that X

possesses in the a-description. Thus the basis

vectors <a'| are the eigenvectors of A - 6A

with the eigenvalues a1

2.7 SUCCESSIVE UNITARY TRANSFORMATIONS

In discussing successive unitary transforma-

tions, it must be recognized that a transformation

which is specified by an array of numerical coeffi-

cients is symbolized by a unitary operator that de-

pends upon the coordinate system to which it is

applied. Thus, let U, and U„ be the operators

describing two different transformations on the

same coordinate system. When the first transfor-

mation has been applied, the operator that symbol-

One must distinguish between X + <SX and


izes the second transformation, in its effect, upon

the coordinate system that has resulted from the

initial transformation, is

Hence the operator that produces the complete

transformation is

The same form with the operators of successive

transformations multiplied from right to left,

applies to any number of transformations. In

particular, if one follows two transformations,

applied in one order, by the inverse of the suc-

cessive transformations in the opposite order, the

unitary operator for the resulting transformation

is

When both transformations are infinitesimal,


the combined transformation described by

is infinitesimal to the first order in each of the

individual transformations,

The infinitesimal change that the latter transfor-

mation produces in an operator is

2.8 UNITARY TRANSFORMATION GROUPS. TRANSLATIONS

AND ROTATIONS

The continual repetition of an infinitesimal

unitary transformation generates a finite unitary

which, expressed in terms of commutators, yields

the operator identity (Jacobi)


transformation. On writing the infinitesimal

Hermitian operator G , the generator of the

unitary transformation, as STG/,» , we find that

the application of the infinitesimal transformation

a number of times expressed by T/<ST yields, in

the limit St-M) ,

These operators form a one-parameter continuous

group of unitary transformations,

A number of finite Hermitian operators ^n\ /•••/

G,, , generates a k-parameter continuous groupIK)

of unitary transformations if they form a linear

basis for an operator ring that is closed under

the unitary transformations of the group. This

requires that all commutators fG/.j\ » G(')^ t>e

linear combinations of the generating operators.

There is a fundamental continuous group of

unitary transformations based upon the significance


of measurements as physical operations in three-

dimensional space. A measurement apparatus defines

a spatial coordinate system with respect to which

physical properties are specified. We express the

uniformity of space by asserting that two coordi-

nate systems, differing only in location and orien-

tation, are intrinsically equivalent. In particu-

lar, physical quantities that are analogously de-

fined with respect to different coordinate systems

exhibit the same spectra of possible values, and

the associated operators must be related by a

unitary transformation. Since the totality of

translations and rotations of a coordinate system

form a six-parameter continuous group, we infer

the existence of an isomorphic group of unitary

operators.

An infinitesimal change of coordinate system

is specified by stating that a point with coordi-

nate vector x in the initial system is assigned

the coordinate vector x - fix in the new system,

where

The infinitesimal generator of the corresponding


unitary transformation is written

with the usual association of axial vectors and

antisymmetrical tensors characteristic of three

dimensions. On comparing the two ways in which

a pair of infinitesimal coordinate changes can be

performed, in the manner of (2.68), we find that

which requires that the associated infinitesimal

generators obey the commutation relation (2.71),

Hence,

The six Hermitian operators comprised in P

and J , the generators of infinitesimal transla-

tions and rotations, respectively, are identified

as the operators of total linear momentum and

total angular momentum. These physical quantities

appear measured in certain natural units - pure

numbers for angular momentum, and inverse length

for linear momentum. The connection between such

atomic units and the conventional macroscopic

standards must be found empirically. If the

latter are to be employed, a conversion factor

should be introduced, which involves the replace-

ment of P and J with fe P and ti J, respec-


where the last statement appears in three-dimen-

sional vector notation as


tively. The constant ft possesses the dimensions

of action, and its measured value is

The natural units' are preferable for general theo-

retical investigations and will be used here.

2,9 REFLECTIONS

The continuous group of transformations

among kinematically equivalent coordinate systems

can be enlarged by the operation of reflecting

the positive sense of every spatial coordinate

axis. We associate with this change of descrip-

tion the unitary reflection operator R ,

A reflection, followed by the infinitesimal dis-

placement 6e , 5u is equivalent to first per-

forming the displacement -<Se , Sw , and then

the reflection. Accordingly,


or

2.10 CONTINUOUS SPECTRA

The general mathematical structure of quan-

tum mechanics as the symbolic expression of the

laws of atomic measurement has been developed in

the context of physical systems possessing a

finite number of states. We shall comment only

briefly on the extension of these considerations

to systems with infinite numbers of states, and

properties exhibiting a continuous spectrum of

possible values. In any measurement of such a

property, systems displaying values within a cer-

tain range are selected, and the concept of state

now refers to the specification of a complete set

of compatible quantities within arbitrarily small

intervals about prescribed values. We symbolize

such states by la'>. , A<a* ' anc^ express

their completeness by

THE GEOMETRY OP STATES 55

The change of normalization,

This is the continuum analogue of the property

in which Aa1 is the product of eigenvalue inter-

vals for each continuous property, now yields, in

the limit Aa'-M) ,

if all members of set A have continuous spectra.

We deduce, for any vector ¥ represented by an

arbitrary wave function

that

which is the operational definition of the delta

function ,

and generally, in all formal relations referring

to continuous spectra, integrals replace summa-

tions. In particular, the probability that a

measurement on a system in the state f will

yield one of a set of states appears as an integral

over that set, / da" f^(a')| , so that

2.11 ADDENDUM: OPERATOR SPACE1"


Academy of Sciences, Vol. 46, pp 261-265 (I960).


can be described as the probability of encounter-

ing a system with properties A in the infinitesi-

mal range da' about a1 .

Vox,. 46, I960 PHYSICS: J. SCHWINGEK 2t»t

The geometry of states provides the elements of the measurement algebra withthe geometrical interpretation of operators on a vector space. But operators con-

262 PHYSICS: J, SCHWWGER Pnoe. N. A, S,

sidered In themselves also form a vector space, for the totality of operators is closedunder addition and under multiplication by numbers. The dimensionality of thisoperator space is Nt according to the number of linearly independent measurementsymbols of any given type. A unitary scalar product is defined iu the operatorspace by the number

The probability relating two states appears as a particular type of operator spacetransformation function.

The connection with the M(a', V) basis is described by the transformation function

Let X(a), « = \ ,. N*t he the elements of an arbitrary orthonormal basis,

One can also verify the composition property of transformation functions,

and

which is such that

can now be viewed as the transformation connecting two orthonormal bases. Thischange of basis is described by the transformation function

and the general linear relation between measurement symbols,

characterizes the M ( < t , p ) basis as orthonormal.

The trace evaluation

which has the properties

Vol.. 46, WOT

We also have

PHYSICS: J. 8CHWINQER 2(iH

and the transformation function property

and

On altering the basis the components of a given operator change in accordance with

For measurement symbol bases this becomes the law of matrix transformation.

the elements of the ab-naatrix representation of X. The scalar product in operatorspace is evaluated as

For the basis M(a', b'), the components are

defines the components

acquires the matrix form

If we multiply the latter by the &-matrix of an arbitrary operator Y, the summationwith respect to ft' and b" yields the o-matrix representation of the operator equation

the validity of which for arbitrary Y is equivalent to the completeness of the oper-ator basis X(a). Since the operator set Z(a)t also forms an orthonormal basiswe must have

and the particular choice Y = I/AT gives

The expression of an arbitrary operator relative to the orthonormal basis X(a),

264 PHYSICS: J. 8CHWINOER PBOC, N. A. S.

There are two aspects of the operator space that have no counterpart in thestate spaces—the adjoint operation and the multiplication of elements are defined

*in the same space. Thus

where

Otherwise expressed, the set of N measurement symbols M(a', b'), for fixed b', orfixed a', are left and right ideals, respectively, of the operator space.

The possibility of introducing Hermitian ortbonormal operator bases is illustratedby the set

For any such basis

and

and

and

Some consequences are

which generalize the adjoint and multiplication properties of matrices. The ele-ments of the operator space appear in the dual role of operator and operand on de-fining matrices by

The measurement symbol bases are distinguished in this context by the completereducibility of such matrices, in the sense of

VOL. 46, 1960 PHYSICS: J. SCHWINGER 265

which implies that a Hermitian operator X has real components relative to aHermitian basis, and therefore

Thus the subspace of Hermitian operators is governed by Euclidean geometry, anda change of basis is a real orthogonal transformation,

When the unit operator (multiplied by N~l/a) is chosen as a member of such basesit defines an invariant subspace, and the freedom of orthogonal transformationrefers to the JV2 — 1 basis operators of zero trace.

Important examples of orfhonormal operator bases are obtained through thestudy of unitary operators,

1 Sehwinger, J., these PEOCBBDINOS, 45,1542 (1959).


2.12 ADDENDUM: UNITARY OPERATOR BASES1"

fReproduced from the Proceedings of the National

Academy of Sciences, Vol. 46, pp. 570-579 (1960).

Hepriat«d from the Proceedings of the NATIONAL ACAWJSMV or ScieufcisisVol. 46, No. 4, pp. 5T0-S79. April, 1980,

UNITARY OPERATOR BASMK*

BY JULIAN SCHWINUJSK

HABVABB OMIVSRSOTf

Communicated February t, 1900

To qualify as the fundamental quantum variables of a physical system, a set ofoperators must suffice to construct all possible quantities of that system. Suchoperator? will therefore be identified as the generators of a complete operator basis.Unitary operator bases are the principal subject of this note.1

Two state vector space coordinate systems and a rule of correspondence define aunitary operator. Thus, given the two ordered sets of vectors (o*|, {&*[, k = I . . N,aad their adjoints, we construct

which are such that

and

for both Ua> and £/»„. If a third ordered coordinate system is given, (c*|, k — I . .N,we can similarly define the unitary operators U,f, [/»„, which obey the compositionproperty

implying the unitary property

obeys the same multiplication law as the X(a), and the F(a) t are given by the samelinear combination of the F(«) as are the X(«)f of the X(a) set. In particular, ifX{a) is a Hermitiari basis, so also is F(a),

We cannot refrain from illustrating these remarks for the simplest of the N*-dimensional operator spaces, the quaternion space associated with a physical systempossessing only two states. If a particular choice of these is arbitrarily designatedas + and —, we obtain the four measurement symbols M(±, ±), and can then in-troduce a Hermitian orthonormal operator basis

Voi. 46, 1960 PHYSICS: J. SCSWINGER 571

A unitary operator is also implied by two orthonormal operator bases in a givenspace, that have the same multiplication properties:

Let us define

and observe that

where the latter statement follows from the remark that the F(a)t form an ortho-normal basis and therefore

We also have the adjoint relation

and in consequence

according to the completeness of the X(a) basis. Thus the operator U is unitaryif we choose

and, to within the arbitrariness of a phase constant,

The converse theorems should be noted. For any unitary operator U, the ortho-normal basis

with

572 PHYSICS: J, SCHWINGSB, Pmoc. N. A, 8

such that

Accordingly, the three operators «% k — 1,2, 3, obey

and an explicit construction is given by

the coefficients of which constitute the well-known Pauli matrices. With thesedefinitions, the multiplication properties of the & operators can be expressed as

or, equivalently, by the additional trace

where <*&,, is the alternating symbol specified by em = +1. If we now introduceany other Hermitian orthonormal operator basis F(a) = 2~1/2S'a, with *« = 1,the resulting three-dimensional orthonormal basis transformation

is real and orthogonal,

The multiplication properties of the <r-basis assert that

where, characteristic of an orthogonal transformation, det r = ± 1. If the orthog-onal transformation is proper, the multiplication properties of the ff-basis coincidewith those of the o--basis, while with an improper transformation the opposite signof t is effectively employed in evaluating the <r»»j products. Hence only in the firstsituation, that of a pure rotation, does a unitary operator exist such that

The unitary operator is constructed explicitly as!

Voiu 48, 1080 PHYSICS: J SCHWINOER 573

Let us return to the definition of a unitary operator through the mapping of onecoordinate system or another, and remark that the two of vectors can be iden-tical, apart from their ordering. Thus, consider the definition of a unitary operatorF by the cvclic permutation

On multiplying M (»*) by (o'v| and using the defining property of F, we obtain

to within the ehoiee of the factor N~l, which is such that

permits the identification of the Hermitiau operator

have all the properties required of measurement symbols. Now let us observethat the factorization of the minimum equation for F that is given by

Unitary operators can be regarded as complex functions of Hermitian operators,and the entire spectral theory of Hermitian operators can be transferred to them.If the unitary operator F has JV distinct eigenvalues, its eigenvectors constitute anorthonormal coordinate system. The adjoint of a right eigenvector |»') is the lefteigenvector (v1 associated with the same eigenvalue, and the products

is the minimum equation, the polynomial equation of least degree obeyed by thisoperator, which we characterize as being of period N. The eigenvalues of F obeythe same equation and are given by the N distinct complex numbers

Thus

until we arrive at

which indicates the utility of designating the same state by any of the integers thatare congruent with respect to the modulus N, The repetition of F defines linearlyindependent unitary operators,

where

874 PHYSICS: J. 8CHWINQSR PROC, N. A, S.

from which follows

Then, with a convenient phase convention for (a*| v") we get

which is also expressed by

the elements of the transformation functions that connect the given coordinatesystem with the one supplied by the eigenvectors of the unitary operator thatcyclically permutes the vectors of the given system.

Turning to the new coordinate system {»*(, we define another unitary operatorby the cyclic permutation of this set. It is convenient to introduce U such that

which is equivalent to

This operator is also of period ff,

and has the same spectrum as V,

After using the property U"~l = U~l to write the corresponding measurementsymbol as

we follow the previous procedure to construct the eigenvectors of U,

Thus, the original coordinate system is regained and our results can now be statedas the reciprocal definition of two unitary operators and their eigenvectors,

The relation between the two coordinate systems is given by

and, supplementing, the periodic properties

Vox,. 48, 1960 PHYSICS: J, 8CHWINOBR 67S

we infer from the comparison

that

Each of the unitary operators U and F is a function of a Hermitian operator thatin itself forms a complete set of physical properties. It is natural lo transfer thisidentification directly to the unitary operators which are more accessible than theimplicit Hermitian operators. Accordingly, we now speak of the statistical rela-tion between the properties U and F, as described by the probability

As a consequence of the latter result, we also have

which are invariant under this substitution when combined with m -*• », n -*• — m.One proof of completeness for the operator basis generated by U and F depends

upon the following lemma: If an operator commutes with both U and F it is neces-sarily a multiple of the unit operator. Since U is complete in itself, such an opera-tor must be a function of U. Then, according to the hypothesis of eomniutatrvitywith V, for each k we have

The latter could be emphasized by choosing the elements of the operator basis as

and therefore together supply the foundation for a full description of a physicalsystem possessing N states. Both of these aspects are implied in speaking of U andF as a complementary pair of operators.* Incidentally, there is complete symmetrybetween U and F, as expressed by the invariance of all properties under the sub-stitution

for this asserts that the intervening non-selective w-measurement has destroyedall prior knowledge concerning w-states. Thus the properties U and F exhibit themaximum degree of incompatibility. We shall also show that U and F are thegenerators of a complete orthonormal operator basis, such as the set of Nf operators

The significance of this result can be emphasized by considering a measurement se-quence that includes a nonselective measurement, as in

576 PHYSICS: J. SCHWIN0MR PBOC. N. A. S,

and this function of U assumes the same value for every state, which identifies itwith that multiple of the unit operator. Now consider, for arbitrary F,

It is interesting to notice that a number of the powers C/*, k = 1 . , N — I , canhave the period N. This will occur whenever the integers fc and N have no commonfactor and thus the multiplicity of such operators equals $(AT), the number of in-

which is a kind of ergodic theorem, for it equates an average over all spectral trans-lations to an average over all states. The explicit reference to operators can beremoved if F(U, V) is constructed from terms which, like the individual operatorsX(mn), are ordered with U standing everywhere to the left of F, Then we canevaluate a matrix element of the operator equation, corresponding to the stateshf and I tf), which gives the numerical relation

which exhibits the unitary transformations that produce only cyclic spectral trans-lations. Now, if F is given as an arbitrary function of U and F, the completenessexpression of the operator basis reads

and observe that left and right hand multiplication with U and U~ , respectively,or with F and F~J, only produces a rearrangement of the summations. Accord-ingly, this operator commutes with U and F. On taking the trace of the resultingequation, the multiple of unity is identified with ir Y, and we have obtained

the statement of completeness for the ]V2-dimensional operator basis X(mn). Al-ternatively, we demonstrate that these JV2 operators are orthonormal by evaluating

The unit value for m = m', n — n' is evident. If m ^ TO', the difference TO' — mcan assume any value between N — I and — (N — 1), other than zero. When thetrace is computed in the v-representation, the operator [/«'-»> changes each vector(vk\ into the orthogonal vector {»*+»>-»' j and the trace vanishes. Similarly, if n 7*n' and the trace is computed in the «-representation, each vector {«* is convertedby Vn'~" into the orthogonal vector (t**'H1/~n| and the trace equals zero.

One application of the operator completeness property is worthy of attention.We first observe that

Vol., 48, I960 PHYSICS: J, SCHWINGSM 877

tegers less than and relatively prime to N. Furthermore, to every such power of Uthere can be associated a power F', also of period N, that obeys with 17* the sameoperator equation satisfied by U and V,

Furthermore,

so that I/i, Fi and Ut, Vt constitute two independent pairs of complementary oper-ators associated with the respective periods JVi and Nt. We also observe that theN = JViJVz independent powers of U can be obtained as

since all of these are distinct powers owing to the relatively prime nature of N\ andNt. With a similar treatment for F, we recognize that the members of the ortho-normal operator basis are given in some order by

It is seen that Ui, Yt are of period N\t while Ut, V% have the period Nt, and that thetwo pairs of operators are mutually commutative, as illustrated by

with

where the integers N\ and Nt are relatively prime, and define

This requires the relation

and the unique solution provided by the Fermat-Euler theorem is

The pair of operators U*, V1 also generate the operator basis X(mn), in some per-muted order,

We shall now proceed to replace the single pair of complementary operators U, Vby several such pairs, the individual members of which have smaller periods thanthe arbitrary integer N. This leads to a classification of quantum degrees of free-dom in relation to the various irreducible, prime periods. Let

578 PHYSICS; J, SCSWINGSR PEOC, N A. S.

together with the commutativity of any two operators carrying different sub-scripts. The orthonormality of the NfN<? — N* operators X'(?»i«»»j»z} can nowbe directly verified. Also, by an appropriate extension of the proceeding discus-sion, we obtain the transformation function

with

The continuation of the factorization terminates in

Equivatently, we replace

where

Another approach to this commutative factorization of the operator basis pro-ceeds through the construction of the eigenvalue index k = 0.. N — 1 with the aidof a pair of integers,

which gives

by

On identifying these vectors as the eigenvectors of sets of two commutative unitaryoperators, we can deine U\,t, Vi,» by the reciprocal relations

which reproduce the properties

We shall not carry out the necessary operations for v -*• «, which evidently yieldthe well-known pair of complementary properties with continuous spectra. Oneremark must be made, however. In this approach one does not encounter thesomewhat awkward situation in which the introduction of continuous spectra re-quires the construction of a new formalism, be it expressed in the language ofDirac's delta function, or of distributions. Rather, we are presented with the directproblem of finding the nature of the subspaees of physically meaningful states andoperators for which the limit v -*• <*> can be performed uniformly,

* Publication assisted by the Office of Scientific Research, United States Air Force.1 For the notation and concepts used here see these PROCEEDINGS, 45, 1542 (1959), and 46,287

(1«60).2 The absence in the available literature of an explicit statement of this simple, general result

is rather surprising. The inverse calculation giving the three-dimensional rotation matrix interms of the elements of the unitary matrix is very we!! known (rotation parametrizations ofBuler, Cay ley-Klein), and the construction of the unitary matrix with the aid of Euleriftn anglesis also quite familiar.

* Operators having the algebraic properties of U and F have long been known from the work ofWeyl, H., Theory of Groups and Quantum Mechanics (New York: B. P. Button Co., 1932), chap.4, sect. 14, but what has been lacking is an appreciation of these operators as generators of a com-plete operator basis for any If, said of their optimum incompatibility, as summarized in the at-tribute of complementarity. Nor bag it been clearly recognized that an a priori classification ofall possible typee of physical degrees of freedom emerges from these considerations.

and where

Vol. 46, 1960 PHYSICS: J, SCHWINGER 579

where / is the total number of prime factors in N, including repetitions. We callthis characteristic property of N the number of degrees of freedom for a systempossessing N states. The resulting eommutatively factored basis

is thus constructed from the operator bases individually associated with the / de-grees of freedom, and the pair of irreducible complementary quantities of each de-gree of freedom is classified by the value of the prime integer v — 2, 3, 5, . . . «.In particular, for v — 2 the complementary operators U and V are anticommutativeand of unit square. Hence, they can be identified with <n and <r», for example, andthe operator basis is completed by the product —iUV — <r\.

The characteristics of a degree of freedom exhibiting an infinite number of statescan be investigated by making explicit the Hermitian operators upon which U andV depend,

CHAPTER THREETHE DYNAMICAL PRINCIPLE

3.1 The Action Operator 743.2 Lagrangian Operator 763.3 Stationary Action Principle 773.4 The Hamiltonian Operator 793.5 Equations of Motion. Generators 803.6 Commutation Relations 833.7 The Two Classes of Dynamical 86

Variables3.8 Complementary Variables of the 97

First Kind3.9 Non-Hermitian Variables of the 103

First Kind3.10 Complementary Variables of the 106

Second Kind

A measurement is a physical operation in

space and in time. The properties of a system

are described in relation to measurements at a

given time, and no value of the time is intrin-

sically distinguished from another by the results

of measurements on an isolated physical system.

73


Hence the operators symbolizing analogous proper-

ties at different times must be related by a uni-

tary transformation. The propagation in time of

the disturbance produced by a measurement implies

that physical quantities referring to different

times are incompatible, in general. Accordingly,

complete sets of compatible properties will per-

tain to a common time, and the characterization of

a state requires the specification of the values

of these quantities (together with a spatial

coordinate system) and of the time. The trans-

formation function relating two arbitrary des-1 H

criptions thus appears as <a_t.|a2t2> , special

cases of which are <a't[b't> , connecting two

different sets of quantities at the same time, and

<a't1|allt_> , which relates analogous properties

at different times. The connection between states

at two different times involves the entire dynami-

cal history of the system in the interval. Hence

the properties of specific systems must be con-

tained completely in a dynamical principle that

characterizes the general transformation function.

3.1 THE ACTION OPERATOR

for consecutive transformations. We now state our

fundamental dynamical postulate: There exists a

special class of infinitesimal alterations for

which the associated operators SW12 are obtained

by appropriate variation of a single operator,

the action operator W,,., »

and has the additive combinatorial property

where 5W is an infinitesimal Hermitian opera-J.1&

tor with the additivity property

THE DYNAMICAL PRINCIPLE 75

Any infinitesimal alteration of the trans-1 i "formation function <a,t,|a~ t_> can be expressed

as [Eq. (1.68)1

It is consistent with the properties of the infini-

tesimal operators to assert that the action opera-

tor is Hermitian,


3.2 LAGRANGIAN OPERATOR

If one views the transformation from the a2t2

description to the a, t^ description-s>s occurring

continuously in time through an infinite succession

of infinitesimally differing descriptions, the

additivity property of action operators asserts that

where

since <a't|a"t> has fixed numerical values. On

writing

the action operator acquires the general form

in which L(t) , the Lagrangian operator, is a

Hermitian function of some fundamental dynamical


variables x (t) in the infinitesimal neighbor-€*

hood of time t . There is no loss of generality

in taking the operators xa(t) to be Hermitian,

and, for our present purposes, we suppose their

number to be finite. The conceivable objects of

variation in the action operator are the terminal

times t, and t- , the dynamical variables, and

the structure of the Lagrangian operator.

3.3 STATIONARY ACTION PRINCIPLE

For a given dynamical system, changes in a

transformation function can be produced only by

explicit alteration of the states to which it

refers. Such variations of states arise from

changes of the physical properties or of the time

involved in the definition of state, and infini-

tesimal eigenvector transformations are generated

by Hermitian operators that depend only upon dyna-

mical variables at the stated time,


Hence,

which is the operator principle of stationary ac-

tion for it asserts that the variation of the

action integral involves only dynamical variables

at the terminal tiroes. This principle implies

equations of motion for the dynamical variables

and yields specific forms for generators of infi-

nitesimal transformations. Note that the Lagran-

gian operator cannot be determined completely by

the dynamical nature of the system, since we must

be able to produce a variety of infinitesimal

transformations for a given system. Indeed, if

two Lagrangians differ by a time derivative,

the action operators are related by

and, with a given dynamical system,


and

which also satisfies the stationary action require-

ment but implies new generators of infinitesimal

transformations at times t, and t~ , as given

by

3.4 THE HAMILTONIAN OPERATOR

The development of the fundamental dynamical

variables through an infinitesimal time interval

is described by an infinitesimal unitary transfor-

mation, which implies first order differential

equations of motion for these variables. A gener-

al form of Lagrangian operator that yields first

order equations of motion is


in which A , is a numerical matrix. We shall

speak of the two parts of the Lagrangian operator

as the kinematical and the dynamical parts, re-

spectively. Note that the kinematical part has

been symmetrized with respect to transference of

the time derivative. The Hermitian requirement on

L applies to the two parts separately. Hence

H , the Hamiltonian operator, must be Hermitian,

and the finite matrix A must be skew - Hermitian.

For an isolated dynamical system, there is no ex-

plicit reference to time in H

3.5 EQUATIONS OF MOTION. GENERATORS

The action operator implied by the particular

structure of L in (3.17) is


In this form the limits of integration are objects

of variation. However, one can think of intro-

ducing an auxiliary variable T , and producing

the variations 6t, , 6t~ by altering the func-

tional dependence of t = t(t) upon T , with

fixed limits T, , T- . This procedure places

the time variable on somewhat the same footing as

the dynamical variables. Now

since

and thus the stationary action principle asserts

that

or


and yields the infinitesimal generators

at the terminal times.

The structure of the Hamiltonian operator is

as yet unspecified. If its variation is to possess

the form (3.23), with the 5x appearing only on

the left and on the right, these variations must

possess elementary operator properties character-

izing the special class of operator variations to

which the dynamical principle refers. Thus, we

should be able to displace each Sx entirely toSI

the left or to the right, in the structure of

5H , which defines the left and the right deriva-

tives Of H ,

In view of the complete symmetry between left and

right, we infer that the terms in (3.23) with 6x

on the left, and on the right, are identical. We

thus obtain


and

and, generally, left and right hand forms are

connected by the adjoint operation. We shall

assume in our discussion that the matrix A is

nonsingular. This implies that every variation

<5x appears independently in G and that eachct X

variable x obeys an explicit equation of motion,ci

3.6 COMMUTATION RELATIONS

The infinitesimal generator Gt = - H fit

where the latter must be equivalent forms of the

equations of motion. Similarly, the infinitesimal

generator G possesses the equivalent forms3v


evidently describes the unitary transformation

from a description at time t to the analogous

one at time t -f 6t , which identifies H as the

energy operator of the system. -If F is any func-

tion of the dynamical variables x(t) and of the

time, the operator F that plays the role of F

in the description referring to time t + 6t is

or

which is the general equation of motion. On plac-

ing F = H , we obtain

in agreement with (3.26), derived from the station-

ary action principle. Such consistency must also

since the numerical parameter t is not affected

by operator transformations. Hence


appear in the equations of motion for the dynamical

variables, which, requires that

and

On writing these relations in the form

we can conclude from the equality of the right

sides, and from the two equivalent expressions for

Gx , that

We shall satisfy this consistency requirement by

demanding that each variation 6x commute witha

the Hamiltonian operator,


for this enables the commutation relations (3.35)

to be stated as

and identifies the transformation generated by G2v

as the change of each dynamical variable x byOL

*s6xa . Accordingly, for an arbitrary function of

the dynamical variables at the stated time, we

have

3.7 THE TWO CLASSES OF DYNAMIC VARIABLES

The two equivalent versions of Gx , bilinear

in x and 5x , indicate that displacing a

variation <$x across any of the dynamical vari-Si

ables induces a linear transformation on these

variables,

where k_ designates a matrix. The adjoint state-a

ment is


from which we conclude that

The commutativity of each <$x with the Bamil-a

tonian operator H(x) now appears as the set of

invariance properties

which shows that the set of linear transformations

k form a group of invariance transformations fora

H . From the fundamental significance of this

group we conclude that it must apply to the com-

plete structure of the Lagrangian operator. The

kinematical term is invariant under the linear

where the latter statement expresses the Hermitian

nature of H . On forming the commutator with a

second variation 6x, we learn thatb


transformation k If3s.

Also, the two equivalent expressions for GX

imply that

which is consistent with (3.45). For this intrinsic

equivalence of the variables x and k x to bea

complete, the latter must be Hermitian operators.

Hence the matrices k are real, and obeya

The construction of the invariance group

described by the matrices k has been based upona

the particular operator variation Sx . ButC*

there must exist some freedom of linear transfor-

mation whereby new Hermitian variables are intro-

duced,


For the new choice of variables, the invariance

properties of the Hamiltonian read

so that

On the other hand, these matrices should appear

directly from the commutation properties of the

variations <Sx ,ct

By referring this statement back to the character-

istics of the <Sx , we obtain

with accompanying redefinitions of the matrix A

and of the Hamiltonian operator.

Now this result cannot be valid for arbitrary £ ,

within a certain group of transformations, unless

the matrices k, are identical for all values ofb

b that can be connected by the linear transforma-

tion. Accordingly, the dynamical variables x

must decompose into classes such that linear trans-

formations within each class only are permissible,

with the different classes distinguished by speci-

fic matrices k , In view of the freedom of in-

dependent linear transformations within each class,

the matrices k^ must maintain the decompositiona

into classes and thus contain only submatrices

characteristic of each class-of variable. The

partitioning of the dynamical variables produced

by any of the k matrices,a

combined with the variation properties



Indicate further that the submatrices of k3.

appear merely as numbers kab, labelling the

various classes. According to (3.47) / these

numbers obey

Hence the two possibilities of k = ± 1 defineclct

two distinct classes of dynamical variables»

We shall see shortly that k, _ - k,,. , and,

since the identity transformation must appear in

the group of k-transformations, we have

If we distinguish the two classes of dynamical

variables as variables of the first kind, z, ,.K

and variables of the second kind, £K , the

operator properties of the variations <Sz. ,JC

6 ?; , summarized in


are given explicitly by

requires that H be an even function of the vari-

ables of the second kind, but no restriction ap-

pears for the dependence upon the variables of the

first kind.

The property of the matrix A contained in

(3.45), together with the opposite signs of k«2

and k--, , shows that all elements of A connec-

ting the two classes of variables must be zero.

Hence A reduces completely into two submatrices

associated with the two kinds of variable, which

we shall designate as a and io , respectively.

where the curly bracket signifies the anticoKHnu-

tator

The Hamiltonian invariance transformation implied

by commutativity with the <5e ,


It follows from (3,46) that a is antisymmetrical»

and hence real,

Two complete reduction of A into a and ia

implies a corresponding additive decomposition of

the generator G »«K.

with

and

This structure of Gx is an aspect of the additive

form assumed by the kinematical term in the

Lagrangian,

while a is symmetrical and real,


which we express by calling the two sets of vari-

ables kinematieally independent. The equations of

motion for these kinematically independent sets of

variables are

We shall adopt a uniform notation to indicate the

characteristic symmetrization in the Lagrangian of

bilinear structures referring to variables of the

first kind, and the antisymnnetrization for bilinear

functions of variables of the second kind, namely

On displacing <$x to the left or to the

right in the general commutation relation (3,39),

and


we obtain

The equations resulting from the special choice

F (x) = xh can be presented as

and, on interchanging a and b in one version,

we conclude from the other that

We have already made use of the only significant

statement contained here, k-,2 m kpi * Tne com~

mutation properties of the two classes of dynami-

cal variables now appear explicitly as


It will be seen that the structure of the operators

reproduces that of the matrices, with the antisyra-

metrical, skew-Hermitian commutator appearing with

the antisymrrtetrical, imaginary matrix (l/i)a ,

while the symmetrical, Hermitian anticommutator is

related to the syymetrical, real matrix a . In

addition, the matrix a must be positive-definite

if the variables £ are to be linearly indepen-K.

dent. The explicit forms of the general commuta-

tion relation are

without distinction between left and right deriva-

tives, while, by distinguishing operators that are

even and odd functions of the variables of the

second kind, we find that

and


3.8 COMPLEMENTARY VARIABLES OF THE FIRST KIND

The matrices a and a are nonsingular,

according to the assumption made about A . For

the antisymmetrical matrix a , associated with

variables of the first kind, the remark that

shows that the number of variables of the first

kind cannot be odd. We shall designate this num-

ber as 2n, , Now the matrix defined by

is a real, symmetrical, positive-definite function

of a , and on writing

we observe that X is a real, antisymmetrieal

matrix that obeys


Furthermore, there exists a real, symmetrical

matrix p such that

The matrix p commutes with a. and anticommutes

with X . By an appropriate choice of Hermitian

variables z , all these 2n,-dimensional matrices

can be displayed in a partitioned form, with n,-

dimensional submatrices, as

pondingly partitioned sets of n., variables

z,: and z,, , k - l,...,n, . According toK K X

(3.75), these variables obey the commutation rela-

with a an n,-dimensional real, symmetrical, posi-

tive-definite matrix. We shall call the corres-

which establishes the possibility of attaining the

matrix forms


tions

With the partitioned form of a , the kine-

matical term and the infinitesimal generator, re-

ferring to the variables of the first kind, ac-

quire the forms

and

G

where the latter generates changes in z and

z(2) of h&z^ and %6z(2) , respectively.

Now let us exploit the freedom to add a time deri-

vative term to the Lagrangian with a corresponding

alteration of the infinitesimal generator, in the

manner of (3.13) and (3.16). With the choice


we obtain the new kinematical term

while the opposite sign for W gives the kinema-

tical term

and the generator

By comparison with (3,88) we recognize that G ,^%*

generates the transformation in which the operators

z are unaltered and the z are changed by

6z . The converse interpretation applies to

G ... . Thus we have divided the variables ofz k '

the first kind into two sets that are complemen-

tary, each set comprising the generators of infini-

tesimal variations of the other set. The interpre-

and the new generator


tation of the generators G ,,, and G ._. isZ £i

expressed by

from which we regain the commutation relations

(3.86), and derive the equations of motion for

the complementary variables,

in agreement with the implications of the action

principle.

A real, symmetrical positive-definite matrix

can always be reduced to the unit matrix by a real

transformation, which, applied to a. , places the

description by complementary variables in a canon-

ical form. The required transformation is pro-

duced by introducing the canonical variables

for this converts the kinematical term (3.90) into


Thus the canonical variables constitute n, com-

plementary pairs of kinematically independent dy-

namical variables of the first kind. Various

aspects of the canonical variables are: the

equations of motion,

the infinitesimal generators,

and the commutation properties of the canonical

variables,

the general commutation relations»


3.9 NON-HEBMITIAN VARIABLES OF THE FIRST KIND

It is important to recognize that the dyna-

mical theory, which has been developed in terms of

Hermitian dynamical variables, permits the intro-

duction of non-Hermitian complementary variables.

Let us define

and observe that the kinematical term (3.87) can

also be written as

which still has the same structure, with iy

and y replacing z and z , respectively.

The latter form also persists under arbitrary com-

plex linear transformations of the non-Hermitian

variables y , with appropriate redefinitions of

a. as a positive-definite Hermitian matrix. The

formal application of the previous considerations

to the non-Hermitian variables now leads, for ex-

ample, to the commutation relations


written for real a , which are precisely the re-

sults that would be obtained by combining the com-

mutation relations for the Hermitian variables, in

accordance with the definitions of the non-Hermi-

tian variables, together with

Regarded as the definition of differentiation with

respect to the non-Hermitian variables, these

equations imply that

which justifies the formal treatment in which y


and y are subjected to independent variations.

Thus the formal theory employing non-Hermitian

variables produces correct equations of motion

and commutation relations. In particular, the

canonical version is applicable, although the

canonical variables q, and p, are not self-iC JC

adjoint but rather

and

in which the pairs of equations stand in adjoint

relation.

Accordingly, the canonical equations of motion

and commutation relations can be written


3.10 COMPLEMENTARY VARIABLES OF THE SECOND KIND

The necessity of an even number of variables

and the possibility of dividing them into two com-

plementary sets also appears for variables of the

second kind. Let us observe that the real, symme-

trical, positive-definite matrix a can be reduced

to the unit matrix by an appropriate real trans-

formation of the variables, which is effectively

produced on introducing

The new canonical Bermitian variables obey

which means that two different £ -operators

anticoimnute, and that the square of each £ is

a common numerical multiple of the unit operator.

The accompanying canonical form of the equations

of motion is

while the general commutation relations become

We now want to emphasize, for a system described

by variables of the second kind, the requirement

that the complete measurement algebra be derived

from the fundamental dynamical variables, which is

the assumption that accompanies the introduction

of such variables. The linearly independent oper-

ators that can be constructed from the £;-variables

are enumerated as follows; the unit operator;

the v operators £ ; the 'sv(v-l) operatorsIs

C-^C, i K < A ; the i v ( v - l ) ( v - 2 ) operators

C 5,5 , t c < X < y ; and so forth, culminating in theK A \i

single operator £iCo »*••» € • The total num-

ber of independent operators so obtained, the

dimensionality of the £-algebra, is

But this number must also be the dimensionality of

the measurement algebra, which equals the square

of the integer N representing the total number



of states. The desired equivalence is possible

only if v is an even integer,

On dividing the Hermitian canonical variables

into two sets of equal number, £ and 5 ,

the kinematical term for the variables of the

second kind becomes

A description employing complementary variables

appears with the introduction of non-Herinitian

canonical variables of the second kind,

which converts the kinematical term into

This structure is applicable to either of the two

kinds of complementary dynamical variable, as are

THE DYNAMICAL PRINCIPiLE 109

the forms

obtained by adding suitable time derivatives.

Accordingly, the equation of motion

and the generators of infinitesimal changes in q

or p ,

can refer to either of the two types of variable.

The distinction between the two classes is impli-

cit in the relation of left and right derivatives,

and, generally, in the operator properties of the

variations <$q and 6p . Thus, for the variables

of the second kind,


The general commutation relations that express the

significance of these generators are

together with

and these statements are identical with the results

obtained directly from (3.113), the commutation

properties of the Hermitian dynamical variables.

As applications of the general commutation

relations, we regain the canonical equations of

motion, and derive the commutation properties of

the canonical non-Hermitian variables of the

second kind,


In virtue of the adjoint connection between the

canonical variables,

and

the equations of motion and commutation relations

can also be presented as

CHAPTER FOURTHE SPECIAL CANONICAL GROUP

I. VARIABLES OF THE FIRST KIND H4

4.1 Differential Operators H5

4.2 Schrodinger Equations H^4.3 The q p Transformation Function 1204.4 Differential Statements of 122

Completeness4.5 Non-Herroitian Canonical Variables 1254.6 Some Transformation Functions 1264.7 Physical Interpretation 1304.8 Composition by Contour Integration 1334.9 Measurements of Optimum Compatibility 1^0

II. VARIABLES OF THE SECOND KIND I43

4.10 Rotation Group 1434.11 External Algebra i45

4.12 Eigenvectors and Eigenvalues 148III. UNIFICATION OF THE VARIABLES i52

4.13 Constructive Use of the Special I52

Canonical Group4.14 Transformation Functions 1564.15 Integration 1664.16 Differential Realizations 170

The commutation properties of the infinites-

imal operator variations that are employed in the

fundamental dynamical principle are such as to

113


maintain the commutation relations obeyed by the

dynamical variables. Accordingly, these special

variations possess a transformation group aspect,

which we now proceed to examine.

I. VARIABLES^ OF THE FIRST KIND

For variables of the first kind, the variations

5z commute with all operators and thus appear as

arbitrary infinitesimal real numbers. If one con-

siders the generators of two independent infinites-

imal variations

their commutator can be evaluated by enploying the

generator significance of either operator,

The equivalent canonical forms are

THE SPECIAL CANONICAL GROUP 115

We thus recognize that the totality of infinitesi-

mal generators G , or G , G , togetherz cj p

with infinitesimal multiples of the unit operator

are closed under the formation of commutators and

therefore constitute the infinitesimal generators

of a group, which we call the special canonical

group.

4,1 DIFFERENTIAL OPERATOR'S

The transformations of this group can be

usefully studied by their effect on the eigenvec-

tors of the complete set of commuting Hermitian

operators provided by the n, canonical variables

q , or p , at a given time t . We first

consider the interpretation of the transformation

generated by G on the eigenvectors of the q-

description, and similarly for the p -description,

as indicated by (2.58),

Now the vector <q't| + 6<q't| is an eigenvector

of the operators q - <Sq with the eigenvalues

q1 . But, since the <$q are numerical multiples

of the unit operator, the varied, vector is des-

cribed equivalently as an eigenvector of the oper-

ators q with the eigenvalues q1 -f 5q . This

shows that the spectra of the Bermitian operators

q, form a continuum, extending from -« to « ,

and that the variation <$• <q't can be ascribedq

to a change of the eigenvalues q' by Sq . A

similar argument applies to the complementary vari-

ables p . Accordingly,

and



The adjoint relations are

Twhere the symbol 3 is used to indicate that the

conventional sense of differentiation is reversed.

For functions F (q » p) thai, can be constructed

algebraically from the variables p and arbitrary

functions of the q , we establish by induction

that

and

which supplies a differential operator realization


for the abstract operators. Similarly, for func-

tions derived by algebraic construction from the

q and. arbitrary functions of the p ,

The significance of the transformation gener-

ated by G on the eigenvectors of the q - des-

cription, and conversely, is indicated by

namely, these transformations multiply the vectors

by numerical phase factors. The transformations

of which infinitesimal real multiples of the unit

operator are the generators also multiply vectors

by phase factors, but without distinction between

the two descriptions. Hence, in response to the

operations of the special canonical group, any

and


eigenvector in the q or p descriptions with

specified eigenvalues can be transformed into one

with any other set of eigenvalues, and multiplied

with an arbitrary phase factor.

4.2 SCHR5DINGER EQUATIONS

The infinitesimal operator G. = -H <5t gen-

erates the group of time translations, whereby a

description at any one time is transformed into

the analogous description at any other time.

Thus, for the infinitesimal transformation from

the q - description at time t to the q -des-

cription at time t + <St we have

or, as an application of the differential operator

realization (4.8), for a system described by vari-

ables of the first kind,

if H is an algebraic function of the p varia-


bles. The adjoint statement can be written

The arbitrary state symbolized by f is represen-

ted by the complex conjugate pair of wave functions

4.3 THE q p TRANSFORMATION FUNCTION

The transformation function connecting the

q and p representations at a common time can be

constructed by integration of the differential

equation

and the variation of these functions in time is

thus described by the (Schrodinger) differential

equations


in which we have made direct use of the operator

properties contained in (4.10) and (4.11). Hence

The magnitude of the constant C is fixed by the

composition property

while the phase of c is an intrinsically arbi-

trary constant which can be altered freely by

common phase factor transformations of the q

states relative to the p states. Hence, with

a conventional choice of phase, we have

and then

According to the structure of this transformation

function, eigenvectors in the q and. p repre-

sentations are related by the reciprocal Fourier

transformations


4.4 DIFFERENTIAL STATEMENTS OF COMPLETENESS

Alternative forms, involving differentiation

rather than integration, can be given to these

latter expressions of completeness, either by di-

but also in mixed qp or pq representations,

With the aid of these connections, the unit opera-

tor can be exhibited, not only in terms of the

complete set of q , or p states,


rect transformation, or from the following consi-

derations. An arbitrary vector ¥ is constructed

from the states q't> and the wave function

4>(q't) as (suppressing the reference to time)

in which we have introduced the operator ^(q) ,

and recognized the structure' of the state |p'>

with eigenvalues p'=0 . Employing a similar

procedure, we find

and thus the scalar product of two vectors can be

evaluated as

The use of the differential operator realizations

(4.8) or (4.9) now gives


On abstracting from the arbitrary vectors, these

results become

and the adjoint operation produces the analogous

properties

or


The practical utility of these forms depends, of

course, upon the differentiation operations appear-

ing in an algebraic manner. The basic examples of

this type occur in the differential composition

properties of the transformation functions <q'|p'>

and <p'Iq'> ,

and

4.5 NON-HERMITIAN CANONICAL VARIABLES

The significance of the special canonical

group will now be studied for the non-Hermitian

canonical variables y , iy , which are defined

by


(In this discussion, we restrict the symbols q ,

p to represent Hermitian canonical variables).

The formal theory provides the infinitesimal gen-

erators

and these operators obey the commutation relations

To follow the pattern set by the Hermitian canoni-

cal variables, we must first construct the eigen-

vectors of the two complete sets of commuting

operators y and y

4.6 SOME TRANSFORMATION FUNCTIONS

The vectors |y't> are described in relation


to the states <q't| by the transformation func-

tion <q't y't> , which obeys the differential

equation

Now the following forms of the generators,

and

permit all operators to be replaced by their eigen-

values ,

which gives the desired transformation function.

Since the adjoint of the right eigenvector equation


we conclude that the complex conjugate of (4.41)

is

and this is a finite number. Hence the eigenvee-j. i

tors |y't> and <y t| exist for all complex

values of y' . It will be noted that the lengths

of these vectors depend upon the eigenvalues, the

is the left eigenvector equation for y ,

We can now compute the transformation function

which is the analogue of <p't q't> . In particu-

lar, the scalar product of a vector |y*t> with

its adjoint is given by


eigenvector belonging to zero eigenvalues being

uniquely distinguished as the one of minimum

length. On normalizing the latter to unity, the

constant C is determined in magnitude, and, with

a conventional choice of phase, we have

for the substitution y •*• y , y -*• -y , while

formally preserving the commutation relation, con-

verts the non-negative operator y y into the non-

positive operator -yyt , Thus, unlike |y't>

A similar discussion for the transformation

i t *function <q't|y t> gives

i t'and no vector jy t> , or any linear combination

of these vectors, possesses a finite length. This

asymmetry between y and y , in striking con-trast with the situation for Hermitian canonical

variables is evident from the commutation relation


and <y t| , the vectors |y t> and <y't| do

not exist. Yet we shall find it possible to define

vectors which, in a limited sense, are right

eigenvectors of y and left eigenvectors of y

4.7 PHYSICAL INTERPRETATION

The physical interpretation of the states

|y't> requires some comment. Since these vectors

are not normalized to unity, in general, we must

express the expectation value for such states as

Now

and, on writing

we infer that


and

Thust the states |y't> are such that neither the

q nor the p variables have definite values but,

rather, the distribution about the expectation

values corresponds to an (optimum) compromise be-

tween the complementary aspects of the two incom-

patible sets of variables. We shall designate the

normalized vectors of this description as

in which notation the transformation function

(4.47) reads

The interpretation of the transformations

produced by G , , and G , on the states

<y t , and jy't> , respectively, is quite the

same as for the Hermitian variables, which the

construction of (4.41) implicitly assumes. Thus

+Y


and generally,

Accordingly, the wave functions 4» (y t) and

(j)(y't) for a system described by non-Hermitian

canonical variables of the first kind obey the

Schrodinger equations

and

4. I

The transformations generated, by G on <y tj

and by Go. on |y't> are given byy


These changes are infinitesimal multiples of the

original vectors, but unlike the situation with

Hermitian variables, the multiplicative factors

are not necessarily imaginary numbers. The same

comment applies to the infinitesimal multiples of

the unit operator produced by commuting two gener-

ators, as in (4,36). Hence the special canonical

group for non-Hermitian variables constitutes the

totality of transformations that change an eigen-

vector with specified complex eigenvalues into one

with any other set of eigenvalues, and which is

multiplied by arbitrary complex numerical factors.

4.8 COMPOSITION BY CONTOUR INTEGRATION

t'The transformation function <y t|y"t>

differs in form from <p'tjq't> only in the ab-

sence of the powers of 2 IT, and therefore obeys

the differential composition property


We infer that

or, that scalar products of arbitrary vectors can

be computed as

Now one can give the evaluation at zero of these

complex eigenvalues a contour integral form (in

this discussion we place n=l , for simplicity;

the extension to arbitrary n is immediate).

Thus,


where the integration paths encircle the origin in

the positive sense and enclose only the singularity

at the origin. Such paths can be drawn since the

existence of the limits in (4.64) requires that4, I

wave functions of the type $(y t) and $(y't)

be regular functions of the corresponding complex

variable in a neighborhood of the origin. Alterna-

tive forms obtained by partial integration are

where

These functions are regular in a- neighborhood

of the point at infinity. The integrations in


(4.66) are to be extended through a common domain

of regularity of the two factors, with the enclosed

+'region containing all singularities of <j> (y t) ,j.»

or tfi(y't) » but no singularity of ty (y t) , or

cHy't) .4. I

If the wave functions ty(y't) and $(y t)

axe related to the vector bases <y't| and4-1

jy t> / given symbolically by

and

the scalar product evaluations can be presented as

the completeness properties

It is implied by these statements that


and

where, according to the definition (4.68), and the

transformation function (4.47),

The indicated regularity domain corresponds to

that of wave functions of the types ^i(y') and

<Hy") . Indeed,

if the contour encloses y" but contains no sin-

gularity of 4>(y't) t while

if the integration path encloses a region continu-

ing all the singularities of 4>(y"t) but not the

point y1 . In the latter situation, the inte-


gral is evaluated with the aid of a circle about

the point at infinity where the function i{i(y"t)

vanishes. A similar discussion applies to the

properties of the transformation function

The symbolic construction of <y't| is made

explicit on writing

and

With these forms we can examine to what extent

i t 1

<y't| and |y t> are left eigenvector of y(t) ,

where the integration path is extended to infinity

along any path that implies convergence of this,

and subsequent integrals. Thus


and right eigenvector of y (t) , respectively.

Now

provided the contour encircles all the singulari-

ties of the integrand. The adjoint statement is

Hence, as our discussion of the asymmetry between

y and y would lead us to anticipate, <y't|

and y t> are not true eigenvectors , although

1The use of complex eigenvalues has been developedin a more formal way by P.A.M. Dirac, Cornm. Dub,Inst. for Adv. Studies, Ser. A, No. 1 (1943),without initial recognition of the asymmetry be-

or, as we can recognize from the symbolic form

(4.68)

+

+ 1

4.9 MEASUREMENTS OF OPTIMUM COMPATIBILITY

According to the eigenvector constructions

tween left and right eigenvectors of the non-Her-itdtian variables. With this procedure, the alter-native evaluations of <y*|y[y"> lead to resultsdiffering by unity and, generally, one is forcedto assume that two <|»(y') functions differingby an arbitrary non-negative power series in y1 ™describe the same state.


the failure of this property appears in a very

simple form. The consistency of the theory, in

which the action of y on its left eigenvectors

contains the additional contour integral term of

(4.81) , can be verified from the alternative eval-

uations of <y llyly" > where, acting on the right

eigenvector,

whereas the properties of the left eigenvectors

give


(4.78) and (4,79), the unit vector can be presented

as

which is the analogue of (4.24). Here the complex

t"variables y' and y are to be integrated along

orthogonal paths. If we write

the paths can be so deformed that q" and p'

are integrated independently from -« to » through

the domain of real values. This produces the form

The transformation function composition property

implied by this expression of completeness,

or, in the notation of (4.55),

can be verified by direct integration. Although

the vectors |q'p't> are complete, they are

certainly not linearly independent or the transfor-

mation function (4.56) would be a delta function.

Instead, it appears that displacing the eigenvalues

q'p' from q"p" f by amounts of the order of unity,

does not produce an essentially different state.

Changes in eigenvalues that are considerably in ex-

cess of unity do result in new states, however,

since the value of the transformation function

becomes very small. In a rough sense, there is one

state associated with each eigenvalue range

{ Aq' Ap1 )/2ir = "I . It is not necessary to con-

struct the linearly independent vectors that describe

distinct states if we are interested only in the

comparison with measurements on a classical level

(as the final stage of every measurement must be)

for then we are concerned with the probability

that the system be encountered in one of a large

number of states, corresponding to an eigenvalue

range ( Aq' Ap1 )/2ir » 1 . Within this context,



it can be asserted that

is the probability that q and p measurements,

performed with optimum compatibility on the state

f at the time t , will yield values q' and p1

lying in the intervals dq1 and dp1 , respectively.

II VA BLE j Hj ECONjp KIMD

4.10 ROTATION GROUP

Now we turn to the variables of the second

kind. The requirement that the variations Si,

antieommute with each variable of this type is

expressed most simply with the aid of the canonical

Hermitian variables £ . The enumeration of the^K

2 =4 distinct operators of the algebra shows

that there is only one operator with the property

of anticommuting with every £ • This is the

product

144 QUANTUM KINEMATICS JVND DYNAMICS

which is written as a Hermitian operator with unit

square. Thus the operator properties of the C

variables contained in (3.110) also apply to

I, - . We see that the variations 55 must be&.U * J» K

infinitesimal real multiples of the single operator

C2n+1 , say

acquires the form

On forming the commutator of two such generators

we obtain

where the

so that the infinitesimal generator


comprise a set of %v(v-l) = n(2n-l) Hermitian

operators obeying

Thus the generators Gg and their commutators can

be constructed from the operator basis provided by -the

n(2n+l) Hermitian operators £ , , where K andK A

X range from 1 to 2n-fl . This basis is com-

plete according to the commutation property

and the totality of these transformations form a

group, which possesses the structure of the (proper)

rotation group in 2n+l dimensions [compare (2.80)] .

But this is not what we shall call the special

canonical group for variables of the second kind.

4.11 EXTERNAL ALGEBRA

We have been discussing a group of inner

automorphisms - transformations, constructed from

the elements of the algebra, that maintain all

algebraic relations and Hermitian properties. It


is characteristic of the structure of the algebra

that the only operators anticommutative with every

| , K = l,...,2n , are numerical multiples of(C

£? , , and two variations formed in this way are

commutative. Hence the generator of one variation

does not commute with a second such variation and

two variations are not independent. To obtain

independent variations, the operators 6 C and

6 ' ' £ roust anticommute, and this is impossible for

inner automorphisms. But the algebraically desirable

introduction of independent variations can be

achieved - at the expense of Hermitian properties

by considering outer automorphisms, constructed

with the aid of a suitably defined external algebra.

Let e be a set of 2n completely anti-Is

commutative operators,

that commute with the elements of the physical

algebra. That anticommutativity property includes

and these operators cannot be Hermitian, nor is the

adjoint of any e - operator included in the set.


We see that the 2n operator products e 5

are completely anticommutative among themselves,

and also antieommute with every £ , <— l,...,2n

Hence variations <SE defined as numerical multi-1C

pies of e^an-n wil1 obey

and

It is the latter property that unifies the generators

for the variables of the first and second kind, and

permits the commutator of two generators,3C

and G , to be evaluated aenerallv as

The canonical form is (4.3) which, as written, applies

to either type of variable. For the variables of

the second kind, 6 'x. A 5^2'x is linearly related

to the products e e.. , and these combinationsK A

commute with all operators of the physical algebra

1


and of the external algebra. Hence, with either

kind of variable the generators of independent

variations together with their commutators form the

infinitesimal elements of a group, which is the

special canonical group,

4.12 EIGENVECTORS AND EIGENVALUES

The existence of this group for the variables

of the second kind enables one to define eigenvectors

of the complete set of anticommuting operators

q or p=iq , Let us observe first that only thealgebraic properties of the operators and variations

are involved in the infinitesimal transformation

equations

and

I 4. I

Furthermore, if q and q are quantitiesK, K

formed in the same way as the independent variations,


anticommuting among themselves and with the operators

q and q , we can assert, that

Hence if vectors exist, obeying

and

so also do the vectors

exist, and these are similarly related to the

eigenvalues q + 6q and q 4- <Sq , respectively,We shall see that that the eigenvectors associated

with zero eigenvalues certainly exist, which implies

that eigenvectors of the type |q t> and <q tj


can be constructed from the null eigenvalue states

by the operations of the special canonical group.

It should be remarked that these right and left

eigenvectors are not in adjoint-relationship, sinceI 4. I

there is no such connection between q and q

A vector obeying the equations

is an eigenvector of the Hermitian operators

q q , with zero eigenvalues.

The converse is also true, since

implies (4.110). The operators q q are commuta-fC rC

tive and indeed constitute a complete set of commut-

ing Hermitan operators for the variables of the

second kind. (These statements apply equally to the

non-Hermitian variables of the first kind.) And,

from the algebraic properties of the canonical

variables, we see that


the spectrum of each operator q q contains onlyK K

the values 0 and 1 , Hence there is a state for

which all operators q q possess the value zeroIN K,

and the state is also described by a right eigenvector

of the non-Hermitian operators q , or the adjoint

left eigenvector of the q , belonging to the set

of zero eigenvalues. We should observe here that,

unlike the situation with non-Hermitian variables

of the first kind, there is complete symmetry between

the operators q and q . In particular, the

operators q q possess zero eigenvalues, whichK, 1C

are equivalent to the unit eigenvalues of q q ,

and the zero eigenvalue, right eigenvector of the

q and left eigenvector of the q also exist.

The possibility of defining the eigenvectors |q t>

and <q't] is then indicated by the differential

equations

Q+

=

+

+

+

+


which possess interpretations analogous to those

of (4.1095.

Ill UNIFICATION OF THE VARIABLES

4.13 CONSTRUCTIVE USE OF THE SPECIAL CANONICALGROUP

In the progression from Hermitian canonical

variables of the first kind to the non-Hermitian

canonical variables of the second kind, the impor-

tance of the special canonical group has increased

to the point where one uses it explicitly to define

the eigenvectors of the canonical variables, rather

than merely investigating the effect of the trans-

formation group on independently constructed vectors

The former approach is universally applicable for,

with all type of variables, the zero eigenvalue,

right eigenvector of the canonical variables q

can be constructed and the general eigenvector de-

fined by a finite operation of the group,

The analogous general construction of left eigenvectors

is

The Hermitian variables of the first kind and the

non-Hermitian variables of the second kind permit,

in addition, the construction of the vectors

and the exceptional situation of the Hermitian

canonical variables of the first kind stems front the

unitary nature of the operator group for those vari-

ables, which deprives the zero eigenvalues of any

distinguished position. The significance of the

operations of the special canonical group on the

eigenvectors of the canonical variables is indicated

generally by


they are the totality of transformations that alter

eigenvalues and multiply eigenvectors by commutative

factors.

The eigenvalues in the equations

and

are formed by multiplying an element of the external

algebra with the member of the physical algebra that

anticommutes with every C , or eguivalently, with1C

the totality of operators q and q . The1C (C

translation of (4.91) into the language of the

canonical variables presents the anticommutative

operators as


and

+


which we shall designate as p . This product,

being a function of the commuting operators q qK ** t

possesses definite values in the null eigenvalue

states,

Accordingly, in considering

the factor p that occurs in q can be replacedK.

by the number (~l)n and the result expressed by

iwhere the final q is entirely an element of the

external algebra. In a similar way

and

and the ad^oxnt statements are


although the element of the external algebra that

appears in the latter equation lacks the numerical

factor (-l)n and therefore differs in sign from

that of (4.126) if n is odd. Hence it is only

for even n that complete symmetry between left

and right eigenvectors exists, which invites the

aesthetic judgment that no system described by an

odd number (of pairs) of dynamical variables of the

second kind exists in nature.

4.14 TRANSFORMATION FUNCTIONS

In constructing transformation functions, one

must eliminate explicit reference to the operators

of the physical system and express the transformation

function in terms of the eigenvalues which, for

variables of the second kind, are the elements of

the external algebra. The replacement of eigenvalues

that anticommute with the dynamical variables by

purely external quantities is accomplished generally

by equations of the type

and occurs automatically for products of eigenvalues.

With variables of the second kind, the transformation

function can be expressed more specifically as


The transformation function <p'j«ll> is character-

ized for all variables by the differential expression

and differs in its integral form only by the numeri-

cal factors that express the normalization conven-

tions for the particular type of variable. To

achieve the universal form

we must remove the factor (2IT) that appears

for Hermitian variables of the first kind. This

will be done if all integrations are performed with

the differentials

ana delta functions correspondingly redefined:


which is expressed symbolically by

Thus, for variables of the second kind, the scalar

product of two vectors can be computed from the

representative wave functions

by

since the square of any eigenvalue vanishes. The

<p'|q'> transformation function possesses the gen-

eral differential composition property

THE SPECIAL CANONCIAL GROUP

If non-Hermitian variables of the first kind

are excepted, one can construct the transformation

function

which, for variables of the second kind, becomes

We also have as the analogues of (4.137) and (4.138),

the wave functions

and the scalar product evaluation

159


With the same exception, the transformation function

<q'|q"> and <p'|p"> are meaningful without quali-

fication. The differential equation

indicates that <q'|q"> is a function of the eigen-

value differences that vanishes on multiplication

with any of its variables. (For non-hermitian

variables of the first kind, the latter equation

would read

which is solved by <y'|y"> = ty'-y"5 .) With

Herraitian variables of the first kind these properties

define the delta function,

combined with

-1

THE SPECIAL CANONICAL GROUP

and we shall retain this notation for the correspond-

ing function referring to the second class of variable,

in which the product of n antieommuting factors

is arranged in some standard order, say 1,.. , ,n , as

read from left to right. For the similar transfor-

mation function referring to the variables q we

write

with the reversed sense of multiplication. The

consistency of these definitions follows from the

composition property

161


for the opposite sense of multiplication in the two

products permits their combination without the inter-

vention of the sign changes that accompany the anti-

comrautativity of the eigenvalues.4. I f"

The transformation function <q |q > pro~

vides the connection between the wave functionst'

^Cq't) and i/i(q t) * ^e composition property

yields

and similarly

supplies the inverse relation


We also have

and

To obtain an expression of completeness re-

ferring entirely to q - eigenvectors, we observe

that

If this symbolic form is realized by < wave

functions that are commutative with the eigenvalues,

or, if n is even, the scalar product evaluation

can be presented as

The integral notation is designed to evoke an analogy

and has no significance apart from its differential

definition. In a similar way we have

and

According to the definitions adopted for variables

of the second kind,

and



Thus the extension of the delta function notation to

tho variables of the second kind is not without

justification. Another related example of formulae

that are applicable to both Hermitian variables of

the first kind and the variables of the second kind

is obtained_from (4.153), written as

rith the variables q replaced by p = iq . On

inserting the integral representation (4.164) this

Futhermore, the composition property

appears as

and similarly


becomes

For Hermitian variables of the first kind, these are

the reciprocal Fourier transformations stated in,

(4.22) .

4 .15 INTEGRATION

Although the integral notation is effective in

unifying some of the formal properties of the two

classes of variables, the nature of the operations

is quite distinct. Indeed, the symbol J has the

significance of differentiation for variables of the

second kind, and the inverse of differentiation for

variables of the first kind. This is emphasized by

the effect of subjecting the eigenvalues to a linear

transformation , q' -*• Xq' . For the Hermitian

variables, the differential element of volume in the

q' - space changes in accordance with

and (4.151) supplies the inverse formula

The latter result Is also expressed formally by

With the particular choice A=-l , we learn that

<5[g'] is an even function of the Hermitian variables,

but it possesses this property for variables of the

second kind only if n~ is even. An interesting

formal difference also appears on considering the

evaluation, by integration, of the trace of an oper-

ator. From the expression of completeness for

Hermitian variables of the first kind, (4.23), we

derive the integral formulae


which implies that

But the delta function of the variables of the second

kind is defined as a product of anticommutative

factors, and therefore


To obtain the analogues referring to variables of

the second kind, we first note the generally valid

differentiation formula, derived from (4.136), in

terms of the matrix representation <p'|x|q'> ,

and

Then, for variables of the second kind, we deduce

which restores somewhat the uniformity of the two

classes for it is the unit operator, commuting with

all variables, that plays the role of p for varia-

bles of the first kind. Since the trace is unaltered

on replacing X with pXp , one can aso write

pX in place of Xp , which is to say that odd

functions of the variables have vanishing trace.

The simplest example of a trace evaluation is


With even n_ we have, for example

and these trace formulae can also be.expressed as

Of course, with a system requiring both types of

dynamical variables for'its description, the trace

operations referring to the two classes must be

superimposed. A formula of suitable generality for


even n, , derived from (4.172), (4.174), and

(4.166) , is

4.16 DIFFERENTIAL REALIZATIONS

Finally, we note the universality of the

differential operator realizations

with their algebraic generalizations

which imply the Schrodinger differential equations

and

The <q't| and |p't> states do not permit such

general assertions, for, in addition to the asymmetry

characteristic of the non-Hermitian variables of the

first kind, we must distinguish between the two

classes of variables in the differential operator

realizations

as contrasted with


where the upper sign refers to variables of the first

kind. This sign distinction originates from the

necessary change of multiplication order in the

differential expression


Thus the following Schrodinger equations are appli-

cable to Hermitian variables of the first kind and

to the variables of the second kind,

CHAPTER FIVECANONICAL TRANSFORMATIONS

5.1 Group Properties and Superfluous 175Variables

5.2 Infinitesimal Canonical Transformations 1785.3 Rotations. Angular Momentum 1825.4 Translations. Linear Momentum 1855.5 Transformation Parameters 1875.6 Hamilton-Jacobi Transformation 1905.7 Path Dependence 1915.8 Path Independence 1945.9 Linear Transformations 195

yields the corresponding canonical form of the

generator describing infinitesimal transformations

at a specified time t ,

173

The use of a canonical version of the Lagran-

gian operator, such as

which implies the canonical commutation relations

and equations of motion. From a given generator

G other generators G can be obtained, in accor-

dance with

appears to be the result of subjecting the variables

in the action operator W to infinitesimal varia-

tions, which for q and 3 , are the special

operators variations. However, the action opera-

tor can also contain other, superfluous, canonical

variables v , and we infer the differential

equations


and we now ask whether G can also be exhibited in

a canonical form referring to new dynamical variables

q(t) , pCt) , and a new Hamiltonian operator

H (qpt) . Such new variables would then obey

canonical commutation relations and equations of

motion, characterizing the transformation of dyna-

mical variables at the time t as a canonical

transformation. The differential form

CWQMIC&L TRANSFORMATIONS 175

A canonical transformation is obtained should these

implicit operator equations possess a solution for

q and p

5.1 GROUP PROPERTIES AND SUPERFLUOUS VARIABLES

Canonical transformations form a group. The

action operator that describes the transformation

inverse to qp -*• qp is

The latter form illustrates the concept of super-

fluous variable. If the individual operators

w(q t g) and w(q , q) contain just the indicated

variables, the sum (5.7) involves q, q and q

while, for two successive transformations,

qp ~** QP "* QP » ^ne generating action operator of

the composite transformation is

.


But, according to the statements of the individual

canonical transformations,

and it must be possible to exhibit W(q , q) as a

function of the variables q , q only. It is not

always desirable, however, to eliminate the super-

fluous variables. This is particularly true when

the canonical transformation involves algebraic rela-

tions between the variables q and q , which

inhibit their independent variation. By retaining

the variables of a suitable intermediate transfor-

mation, one can deiive the desired transformation

by independent differentiation. An important ex-

ample is provided by the identity transformation.

Let us remark first on the now familiar transforma-

tion that interchanges the roles of the complementary

q and p variables, as described by

where

CANONICAL TRANSFORMATIONS

Since

which contains the p as superfluous variables.

To eliminate the latter we must impose the transfor-

mation equations q=q , which yields W=0 for the

identity transformation. But, with the superfluous

variables retained, the differential equations (5.5)

are applicable and generate the transformation.

A given canonical transformation can be derived

by differentiation with respect to either set of

complementary variables. Thus, from the action

operator W(qqt) , obeying the differential equations

this is the canonical transformation

On adding the action ope,ators for the transformation

q -* p and its inverse p •*• q = q , we obtain the

following characterization of the identity transfor-

mation ,

177


{5.5}, we obtain

and these equations are equally suitable for des-

cribing the transformation qp •*• qp . The identity

transformation, for example, is derived from the

action operator - p q

5.2 INFINITESIMAL CANONICAL TRANSFORMATIONS

Transformations in the infinitesimal neighbor-

hood of the identity - infinitesimal canonical

transformations - must be described by an action

operator that differs infinitesimally from the one

producing the identity transformation. The appro-

priate form for the variables q , q is

in which the q appear as superfluous variables.

The differential properties of the new action opera-

tor W(pqt) are now deduced to be

.

where G is an infinitesimal function of the in-

dicated variables that should have an even depen-,

dence upon, the second class of dynamical variable,

but is otherwise arbitrary. On applying (5.5),

with the p as superfluous variables, we obtain-

the explicit equations of an infinitesimal canonical

transformation,

Now, according to (3.100) and (3.123), which applies

to even functions of the variables of the second

kind, we have

or, without specialization to canonical variables,

with

CANONICAL TRANSFORMATIONS 179

and if G is a Hermitian operator this is a unitary

transformation. The subgroup of canonical transfor-

mations that preserve the Hermiticity of dynamical

variables is equivalent to the group of unitary

transformations. Without reference to Hermitian

properties the transformation (5.21) maintains all

algebraic relations, and therefore

which makes explicit the functional form of the new

Hamiltonian. Infinitesimal canonical transformations

that do not change the form of the Hamiltonian

operator.have infinitesimal generators that are

constants of the motion.


Hence

whence


We have already encountered examples of in-

finitesimal canonical transformations. The tran-

formation generated by G, = - H <5t ,

is one in which the dynamical variables at time t

are replaced by those at time t + <St ,

and the energy operator H is a constant of the

motifcn when H is not an explicit function of t ,

being the condition for the maintenance of the func-

tional form of the Hamiltonian operator under time

translation. The special operator variations, which

are distinguished by their elementary commutation

properties, appear as the canonical transformations

The new form of the Hamiltonian operator is


and

in Which we have introduced a notational distinction

between the general infinitesimal canonical trans-

formations and those of the special canonical group.

For variables of the second kind, the consideration

of the special canonical group within the framework

of general canonical transformations implies a for-

mal extension of the latter through the introduc-

tion of the elements of the external algebra.

5.3 ROTATIONS. ANGULAR MOMENTUM

The change in description that accompies a

rotation of the spatial coordinate system is a

canonical transformation, with the infinitesimal

generator

but the form of this canonical transformation is not

yet known. Differently oriented coordinate systems

are intrinsically equivalent and we should expect

that the kinematical term in the Lagrangian operator

presents the same appearance in terms of the varia-

bles appropriate to any coordinate system. This

comment also applies to the dynamical term - the

Hamiltonian operator - of a physically isolated

system, for which the total angular momentum, opera-

tor J , as the generator of a transformation

that leaves the form of H invariant, is a constant

of the motion. In view of the bilinear structure

of the kinematical term, p f -T*. , the change in-volved in an arbitrary rotation of the spatial

coordinate system will be a linear transformation

among suitably chosen q variables, combined with

the contragredient transformation of the complemen-

tary variables. For an infinitesimal rotation,

then,


in which the components of the vector j are

matrices. For Hermitain variables j is an imag-

inary matrix, while with non Hermitian variables

in the relation p = iq , the matrix j is

Hermitian. The general form of the Hermitian angu-

lar momentum operator is thus obtained as

to which the two kinds of dynamical variables make

additive contributions. The symmetrization or anti-

symmetrization indicated here is actually unnecessary,

On applying the infinitesimal transformation

generated by G^ to the operator J we findw

according to the commutation relations (2.79). But

we also have

the matrices j obey the same commutation relations

as the angular momentum operator J . It is a

consequence of these commutation relations that the

trace of the matrix j vanishes and thus the

and therefore


explicit symmetrization or antisyiraaetrization of

factors in J is unnecessary. The decompositon

of the matrix j into irreducible submatrices pro-

duces a partitioning of the dynamical variables in-

to kinematically independent sets that appear add-

itively in the structure of J . Each such set

defines a dynamical variable of several components,

the rotational transformation properties of which

are fixed by the number of components, for this

integer, the dimensionality of the corresponding

submatrix of j , essentially determines the

structure of these matrix representations of J

An irreducible set of three variables, for example,

necessarily has the rotational transformation proper-

ties of a three-dimensional space vector. The num-

ber of components possessed by a dynamical variable

of the second kind is presumably even, according to

a comment of the preceding section.

5.4 TRANSLATIONS. LINEAR MOMENTUM

The remarks concerning invariance with re-

spect to rotations of the coordinate system appJ._

equally to coordinate system translations, which

have the infinitesimal generator


QUANTUM KINEMATICS AND DYNAMICS

For systems described by a finite number of dynami-

cal variables, the appropriate transformation that

leaves the kinematical part of L invariant is the

addition of constants to suitably chosen canonical

variables q . Thus G has the structure of G

and different generators of this type do commute,

as the commutation properties of the total linear

momentum require. If we apply an infinitesimal

rotation to the operator

we recognize that the members of the class of dynam-

ical variables p that make a contribution to P

have the rotational transformation properties of

space vectors. Hence, only three-component variables

of the first kind can be affected by a translation

of the coordinate system. If the latter set of

variables is presented as the Hermitian vectors

in accordance with

186

r, » ... t T , with the complementary variables

p ,.. ., p , a suitable adjustment of the rela-

tive eigenvalue scales will guarantee that

and therefore

One exhibiting the contribution of the vector varia-

bles to the total angular momentum, we have

where the latter term contains all variables that

are uninfluenced by translations. These are evident-

ly the internal variables for a system of n part-

icles that are localized spatially by the position

vectors r,

5.5 TRANSFORMATION PARAMETERS

It is useful to regard a general infinitesimal

canonical transformation as the result of subjecting

certain parameters T , s = l,...,v , to


infinitesimal changes, say -dT , so that thes

infinitesimal generator has the form

in which the G, . may depend explicitly upon thevs /

parameters. This interpretation of the transforma-

tion is expressed by

or

Accordingly, the canonical variables obey equation

of motion,

which govern the evolution of the canonical trans-

formation. By repeated application of such infini-

tesimal transformations, a finite transformation is

generated in which the parameters t are altered


from T, to T. along a definite path. The

action operator characterizing the finite transfor-

mation is the sum of those for the individual in-

finitesimal transformations

is the action operator generating the finite canon-

ical transformation, with the operators referring

to all values of T intermediate between t. and

i appearing as superfluous variables. Since

W,„ must be independent of these intermediate

variables, it is stationary with respect to infini-

tesimal special variations of all dynamical quanti-

ties that do not refer to the terminal values of

where, according to (5.16),


Hence

190

and thus the stationary requirement, applied to a

given parameter path (61 = 0) , again yields the

differential equations (5.44), and

5.6 HAMILTON - JACOBI TRANSFORMATION

With a single parameter t and generator

-H , we regain the original action principle, now

appearing as the characterization of a canonical

transformation - the Hamiltonian - Jacobi transfor-


which in turn, equals

Now

mation - from a description at time t (=t.) to

the analogous one referring to another time

tfl (=t?) . With t_ held fixed, the action opera-tor W (=W,2) obeys

or

On comparison with {5.5} we recognize that the Ham-

ilton-Jacobi transformation is such that 5 = 0 ,

which expresses the lack of dependence on t of

the new dynamical variables x(t } , The new

Hamiltonian at time t thus differs from H eval-

uated at time t~ , which governs the dependence

of W upon the parameter t_ ,

5.7 PATH DEPENDENCE

For a. canonical transformation involving

several parameters, the last term of (5.49) displays


the effect of altering the path along which the

parameters evolve. According to the significance

of G, , as a. generator, we have\s i

is antisymraetrical with respect to the indices r

and s ,

The complete variation of W.„ is thus

With fixed terminal conditions, the consideration

of two independent path variations in the combinat-

ion

and therefore



leads to the integrability condition of the differ-

ential form (5.5?) in its dependence upon the para-

meter path,

which is indeed satisfied by virtue of the operator

identity (2.73), for

and

5.8 PATH INDEPENDENCE

If the canonical transformation is to be in-

dependent of the integration path, it is necessary

that

As an example of this situation, consider a canoni-

cal transformation with two sets of parameters and

generators: t , -H ; X , G,,, . On referringI A;

to (5.55) we see that the condition for path inde-

pendence can be presented as

Hence the Hamiltonian operator must be an explicit

function of the parameter X , which is to say that

it changes its functional form under the infinitesimal

transformation G,,v dX « Since this change isI A;

identical with (5.23), characteristic of an infini-

tesimal canonical transformation, we learn that the

same resultant canonical transformation is obtained

whether the system evolves in time and a canonical

transformation is performed at the terminal time,


or if the canonical transformation is applied

continuously in time, subject only to the fixed

endpoint. With the latter viewpoint, the superposi-

tion of the continuous change in description on the

dynamical development of the system is described

by the effective Hamiltonian operator

and the principle of stationary action includes the

numerical variable X(t) . It should be noted that

when G,,v is not an explicit function of time,(A I

the X dependence of H is such that dH/dX = 0.

5.9 LINEAR TRANSFORMATIONS

under some circumstances, this extension of

the action principle can be expressed as a widening

of the class of variations, without alteration of

the Hamiltonian. Thus, let


which produces the linear transformation described

by

the action operator is thereby expressed as that of

a pure Hamilton-Jacob! transformation. But the X

transformation can now be introduced by remarking

196 QUANTUM KINEMATICS AND DYNAMIC:

If g is a constant matrix, the explicit X trans-

formation, for constant t , is

If these linear operator relations are substituted

into the Lagrangian operator, we obtain

from which all reference to the X transformation

has disappeared. We have used the properties

for fixed X , and

that a. special variation of q(A5 and an infinit-

esimal change of A , in

implies a special variation of q , combined with

a linear transformation:

Together with the similar properties of p , this

yields an extended class of variations for the

action principle. To verify directly the correct-

ness of this extension, we observe that the latter

induces the following variation in L »

in which 3H/3A has been introduced to measure the

lack of invariance displayed by H under the A

transformation. The application of the stationary

action principle now properly yields (5.63) and

confirms the interpretation of Gi\\ ^A as ^e

generator of the infinitesimal A transformation.


We may well note here the special situation

of the linear Hamilton-Jacobi transformation,

corresponding to the bilinear Hamilton!an operator

and the equations of motion

Since

in virtue of the equations of motion, the action

operator W(qq0t} is identically zero, which in-

dicates the existence of algebraic relations between

the variables q and qQ . The transformation is

more conveniently described with the aid of

WCpq/vt) . According to (5,14), we must eliminate

the variables q , which is accomplished by the

explicit solution of the equations of motion,

(h constant) , and thus

198 QUANT0M KINEMATICS AND DYNAMICS

CHAPTER SIX

GROUPS OF TRANSFORMATIONS

6.1 Integrability Conditions 2026.2 Finite Matrix Representation 2046.3 Subgroups 2076.4 Differential Forms and Composition 209

Properties6.5 Canonical Parameters 2116.6 An Example, Special Canonical Group 2166.7 Other Parameters, Rotation Group 2196.8 Differential Operator Realizations 2266.9 Group Volume 2286.10 Compact Groups 2316.11 Projection Operators and Invariants 2336.12 Differential Operators and the 238

Rotation Group6.13 Non-Compact Group Integration 2436.14 Variables of the Second Kind 2476.15 Reflection Operator 2496.16 Finite Operator Basis 2506.17 Addendum: Derivation of the Action 254

Principle6.18 Addendum Concerning the Special 259

Canonical Group6.19 Addendum: Quantum Variables and the 275

Action Principle

201

6.1 INTEGRABILITY CONDITIONS

We will now examine the construction from its

infinitesimal elements of a. continuous group of

canonical transformations, where a transformation

must be completely specified by the values of the

parameters and thus is independent of the integra-

tion path. Apart from the elementary situation of

a completely commutative (Abelian) group, the gene-

rators G, . must be explicit funtions of the

parameters if the operators R are to be zero.JL 5

The group property is exploited in exhibiting the

operators G, , (x , t) , assumed finite in number,\ s}

as a linear combination of an equal number of

operators that do not depend explicity upon the

parameters,


for the condition of path independence demands that

the commutators of the operators G, » (x) be line-\3.)

arly related to the same set. On writing

we obtain the following differential equations for

the functions C c(t)O.S

The numbers g , are antisymmetrical in the last

two indices, and they are imaginary if the operators

G, v are Hermitian. Other algebraic propertiesv3/

can be conveniently presented by introducing a ma-

trix notation for the array with fixed second index,

The latter establishes a correspondence between the

operator G,, , and the finite matrix g, . This

correspondence maintains commutation properties

according to the identity (2.73),

while writing the commutation relations (6.2) in

the form

GROUPS OF TRANSFORMATIONS 203


and thus the g matrices also obey the commutation

relations (6.2). A second set of matrices withm

that property, -g , follows from the correspon-

dence

The quadratic connections among the g coefficients,

comprised in the commutation relations, are identi-

cal with the conditions of integral-ility for the

differential equations (6.3), which verifies the

consistency of the operator presentation (6.1).

6.2 FINITE MATRIX REPRESENTATION

The correspondence between the operators G, ,

and the matrices g persists under a change ofa

operator basis, to within the freedom of matrix

transformations that preserve algebraic relations;

the non-singular transformation

induces

204


An important application of this ability to change

the operator basis occurs when the operators G

are Hermitian and possess a non-zero, linearly in-

dependent, finite-dimensional, Hermitian matrix re-

presentation. Then, with the trace computed from

the bounded matrix representation,

is a real, symmetric, positive-dafinite matrix.

Accordingly, there is a choice of Hermitian basis

for which Y is a multiple of the unit matrix, and

g . is completely antisyinmetrical. Relative to

this basis, which still has the freedom of orthogo-

nal transformations, the g matrices are antisyramet:

rical,

is completely antisymmetrical in a , b , and c ,

where

205


and, being imaginary, are Hermitian matrices. Thus

the g matrices qualify as a finite-dimensional

matrix representation provided they are linearly

independent. A linear relation among the g matri-

ces will occur only if a linear combination of the

operators G, v , a = l,...,v, commutes with every\3.l

G . Should such a linear combination exist, let

it be labelled G, , by an appropriate orthogonal

basis transformation. Then

and g . = 0 , which states that G, , will never

appear in the expression for any commutator. This

procedure can be continued if there are several such

linear combinations and we reach the conclusion

that the group can be factored into an Abelian group,

and a non-commutative group with its structure

characterized by the property that the g matrices

constitute a finite dimensional representation of

the generating operators.

206

6.3 SUBGROUPS

Groups of the latter type are necessarily

semi-simple, by which is meant that they possess no

Abelian invariant subgroups (a. simple group has no

invariant subgroup). The significance of these

terms can be given within the framework of infini-

tesimal transformations. Let the generators be

divided into two sets, designated as 1 and 2 ,

of which the first refers to the subgroup. Then,

as the condition for the formation of commutators

to be closed within the subgroup, we have

The subgroup is invariant if the commutator of any

subgroup element with an outside operator is still

within the subgroup,

and, if the subgroup is Abelian,

invariant subgroup:

subgroup:

GROUPS OF TRANSFORMATIONS 20?

Abelian subgroup

Then, if the group possesses an Abelian invariant

subgroup, the only non-zero elements of a matrix

g, are of the form g , , and the matrixb l alblC2

g, cannot be antisymmetrical. Alternatively, weDlconclude from these attributes of an Abelian invar-

iant subgroup that

in contradiction with the positive-definiteness

this array of numbers should exhibit if the matrices

g constitute a finite-dimensional representation.alEvidently a group that contains an Abelian invariant

subgroup cannot possess finite dimensional matrix

representations. A fundamental example of this

situation is provided by the group of translations

and rotations in three-dimensional space. On re-

ferring to the commutation properties (2.80), we

recognize that translations form an Abelian invari-

ant subgroup, and the mathematical impossibility of

a finite dimensional representation corresponds to

the physical existence of an infinite number of

states that are connected by the operation of tran-

slation. In contrast, the subgroup of rotations,

considered by itself, is a simple group and every


matrix representation, labelled by the value of

the total angular momentum, is of finite dimension-

ality,

6.4 DIFFERENTIAL FORMS AND COMPOSITION PROPERTIES

The transformation described by the infinitesi-

mal changes dT of the parameters is produced by

the operator

where the quantities

form a set of inexact differentials (Pfaffians).

The subscript A(eft) refers to the manner in which

this operator is combined with the operator U(T)

that produces the finite transformation from the

standard zero values of the parameters, namely


when the dynamical variables are referred to the

standard values of the parameters with the aid of

the transformation

This illustrates the general composition property

of the group,

for infinitesimal t, . In addition to the infin-

itesimal transformation U(T+dT) U(t) one can

—Iconsider U(T) u(t+dt) r and there must exist a

second set of inexact differentials, <5 T suchXT ct

that

An infinitesimal change of T.. in the general

multiplication property induces a corresponding

change of t and,



serve to determine the composition properties of

the group parameters. The same function, as per-

formed by the second set of differentials, is ex-

pressed by the differential equations

6.5 CANONICAL PARAMETERS

The choice of parameters is arbitrary to

within non-singular transformations, t -* T ' ,

and the initial conditions

211

or

The ensuing differential equations (Maurer-Cartan)

together with the initial conditions

which do not affect the inexact differentials,

The differential equations (6.3) maintain their

form under parameter transformation. Through the

freedom of parameter and basis changes one could

require identity of the basis operators G, v with\a,l

the generators G, ,(T) , for T = 0 , This would

be expressed by adding the initial condition

to the 5 differential equations, A special set

of parameters, termed canonical, is defined as

follows. As the number X varies from 0 to 1

let a point in the T-parameter space move out from

the origin along the curve described by

and thus


where the t are arbitrary constants. The pointa

in the t-space that is reached at X = 1 is deter-

mined by the numbers t , which constitute thect

new set of parameters. According to the invariance

of the differential forms the same path is described

in the t-parameter space by

Now if t is replaced by Yt t Y < 1 > thea a

point reached for X = 1 is identical with the

point attained at X = y along the curve character-

izing the point t , Hence the path appears in

the t-space as

a straight line, and (6,34) asserts that

or


and thus the finite transformations of the group

have the exponential form

and, on applying this equation at the point At ,

we obtain

becomes

in terms of the canonical parameters.

The.differential equations for the functions

C »_(t) can be simplified with the aid of the

property (6.37). Indeed,


The operator producing the infinitesimal transfor-

mation characterized by dA is


and the formal solution is

Thus the differential forms 6,t are obtained

explicitly as

The substitution t •*• ~t~cft , dt -*• dt , converts

U(t+dt) U(t)"1 into U(t)~ U(t-t-dt) and therefore

In a matrix notation the latter reads

215

in which we have used an evident notation. The

structure of 6 t can now be inferred from ther

following property of the canonical parameters,


The general expression of this connection between

the two differential forms, characteristic of the

canonical parameters, is the group composition

property

6.6 AN EXAMPLE. SPECIAL CANONICAL GROUP

A simple illustration of these considerations

is provided by a group of three parameters defined

by the commutation relations

The matrices g are conveniently presented in thecl

linear combination

which asymmetrical form shows that

216


Thus the matrices g do not furnish a finite-di-a

mensional Hermitian matrix representation, which is

related to the existence of the one-parameter in-

variant subgroup generated by £*»,» . In virtue

of the algebraic property (6.51), we have

and the differential equations (6.26) read (primes

are now used to distinguish the various parameter

sets)

The solution of these equations subject to the

initial condition t' = 0 : t = t" , and the

canonical parameter composition law of the group,

is


and specializations of this result, in which the

parameters are combined with the generators, can be

presented as

It will not have escaped attention that the

commutation properties (6.49) are realized by the

special canonical group. Accordingly, the operators

expressing the finite transformations of this group

possess the multiplication property

which illustrates the reflection property (6.48).

The operator statement contained here is


6.7 OTHER PARAMETERS . ROTATION GROUP

Canonical parameters are not always the most

useful parameter choice. This can be illustrated

by the three-dimensional rotation group. We combine

the three canonical parameters into the vector w ,

and observe that the three-dimensional Hermitian

matrices j(=g } , defined by the commutation re~a

lations (2.81), can also be presented as a vector

operation:

Hence

which is applicable to all types of dynamical

variables. Specializations analogous to (6.56) are

219


and

The latter result shows that the eigenvalues of any

component of j are 1 , 0 , -1 , and therefore

On applying this result to the explicit construction

of $£**/ as given in (6.45) , we obtain

or, with a simple rearrangement.


Thus, the new parameters

are such that

The substitution u> -* -w converts u into -u

while leaving un unaltered, and therefore the

analogous expression for the differential form 6 u

is

Although u is not an independent parameter,

it can largely be treated as such. This is indicated

by the structure of the differentials du , du«

that a given 60w or & u imply:J6 3T


and

Thus, there is a unit vector in a four-dimensional

Euclidean space associated with every three-dimen-

sional rotation, and, to the composition of two three-

dimensional rotations there is associated a four-

dimensional rotation. The algebraic simplification

achieved by the u-parameters appears in the group

composition law. The invariance of the bilinear

form for the differentials 6ew, expressed by theX*

differential equations (6,26), implies a linear

relation between the parameters of the individual

transformations and those of the product transfor-

mation. One easily verifies that

for these changes maintain the normalization (6.67),

The u-parameters appear in another way on

recalling the existence of automorphisms - unitary

transformations - of the algebra defined by 2n

Hermitian canonical variables of the second kind,

£ , that have the structure of the Euclidean ro-K

tation group in 2n + 1 dimensions. Hence, a

representation of the three-dimensional rotation

group is generated by such unitary transformations

of the three anticommutative Hermitian operators

where e is the completely antisymmetrical func-

tion of its indices specified by £,„- = +1

The generators of the three independent infinitesimal

which have the following multiplication characteris

tics:

ar, of the operators



rotations are the operators %0 for, according to

(4.96) ,

and

Explicit matrix representations are obtained on

relating the operator basis 1 , a to measurement

symbols. The measurement symbols of the 0,,

representation are presented in the following array,

and (Pauli)

for example. Theus, the simplest not-trivial measur-

ment algebra, of dimensionality 2 2, provides an

angular momentum operator matrix representation

Now, apart from the freedom of multiplication by a

numerical phase factor, any unitary operator of

this algebra has the form

where the four numbers comprised in un , u are

real, and obey

The latter condition also states that U is unimo-

dular,

With any such unitary operator there is associated

a three-dimensional proper- rotation



which correspondence is 2:1 since U and -U

produces the same rotation, and successive unitary

transformations generate successive rotations. The

parameter composition law that emerges from the

product

where j signifies the three-dimensional angular

momentum matrix representation displayed in (6.60).

6.8 DIFFERENTIAL OPERATOR REALIZATIONS

The differentiable manifold of the group

parameters enables differential operator realiza-

tions of the infinitesimal generators of a group

to be constructed. Let us define for this purpose

two sets of functions rj (-T) , t, (T) , accordingas as

to

is just (6.73). Incidentally, the explicit form

of the three-dimensional rotation matrix r is


and, using canonical parameters

Now, the infinitesimal composition properties

stated in (6.20) and (6.23) can be presented as

or, on writing

as

The two sets of differential operators defined here

are commutative,

Thus


These are intrinsic properties of the differential

operators. Thus, one can verify that the differ-

ential equations implied for n(t) ,

are a direct consequence of (6.3) and the relation

(6.88).

6.9 GROUP VOLUME

An infinitesimal element of volume can be

defined on the group manifold with the aid of the

and each set obeys the G commutation relations

This volume element is independent of the choice of

parameters, and is unchanged by the parameter trans-

formation, T- •+ i , that expresses group multipli-

cation, T = T(T. , T2) • An alternative definition

of volume accompanies the differentials 6 i , and

that volume element is invariant under the parameter

transformation T, -»• T of group multiplication.£*

The relation

shows that the two definitions or volume are iden-

tical if

The latter property certainly holds if the group

possesses a finite-dimensional matrix representation,

for then all the g matrices are antisymmetrical

and its consequence


inexact differentials (6.19)


(or zero) to within the latitude of matrix transfor-

mations that do not alter the trace. But, as the

example of (6.50) indicates, this is by no means a

necessary condition. The invariance of the volume

element, stated by

is given a differential form on choosing the para-

meters T to be infinitesimal, in which circum-

stance the explicit transformation is

and we infer that

As an application of this result, let us observe that


which transformation maintains the commutation

properties of the differential operators, and yields

a. formally Hermitian differential operator to repre-

sent a Hermitian generating operator Gf \ . The(a)

study of a v-parameter group of unitary transforma-

tions can thus be performed with the aid of an equi-

valent dynamical system described by v pairs of

complementary variables of the first kind which are

generally quasi-canonical for, unless the range of

the individual parameters is -°° to » , these

variables do not possess all the attributes of

canonical variables.

6.10 COMPACT GROUPS

The ability to integrate over the group mani-

fold is particularly valuable when the group is

compact, which is to say that any infinite sequence

of group elements possesses a limit point belonging

to the group manifold. Thus the manifold of a

compact group is bounded, and its volume can be

chosen as unity by including a suitable scale factor

in the volume element. We first notice that the

matrices g for a compact group are necessarilya

232 QUANTUM. KINEMATICS AND DYNAMICS

traceless and, accordingly, the two definitions of

volume element are identical. To prove this consi-

der a particular transformation U , as character-

ized by canonical parameters t , and the corres-

ponding finite matrix

of which (6.5) is the infinitesimal transformation

form. If the imaginary matrices g are not anti-

symmetrical,

may differ from unity. Then, to the sequence of

koperators U , k = ± l , ± 2 , . . . , there corresponds

ka sequence of matrices U , for which

increases without limit as k -»• +°° , if det U > 1 ,

or, as k -»• -•» , if det U < 1 . This contradicts

the requirement that an infinite sequence of group

which appears in the general relation


elements possess a limit point on the group manifold

with its associated finite U matrix. Hence the

matrix U roust be unimodular, and every gQ has acL

vanishing trace. It may be noted here that a group

for which the g matrices supply a representation

of its infinitesimal Hermitian generators has a

bounded manifold. According to the explicit con-

struction of the volume element in terms of the

canonical parameters,

and the boundaries of the manifold are reached when

the weight factor in the element of volume vanishes.

The statement that the g matrices are Hermitian and

linearly independent implies that, for every t , gt

possesses non-zero, real eigenvalues. The numer-

ically largest of these eigenvalues equated to 2ir

then determines the finite points where the boundary

of the group manifold intercepts the ray, directed

from the origin of the parameter space, which is

specified by the relative values of the parameters

*a '

6.11 PROJECTION OPERATORS AND INVARIANTS

since intergration with respect to t , or -t ,

covers the group manifold and the volume element is

invariant under the transformation t ->• -t . Thus,

the Hermitian operator P_ is a measurement symbol

or, in geometrical language, a projection opera-

tor, for the subspace of states that are invari-

ant under all the transformations of the group.

These are also the states for which all the

generating operators can be simultaneously


The group property and the invariance aspects

of the volume element for a compact group assert

that the operator

has the following characteristics

and, using the canonical parameters,

describes the construction of the subalgebra of

operators that are invariant under all transforma-

tions of the group. One can also apply a slight

modification of the latter procedure to the finite,

real matrices U{T} , with the result

This is a real symmetrical positive definite matrix

and therefore it can be expressed as the square of

a matrix of the same type, say X . Thus the con-

tent of (6.113) is

whish is to say that a basis for the Hermitian

generating operators of a compact group can be

found that implies real orthogonal, or unitary,

matrices U(T) , ?nd antisyiranetrical matrices


assigned the value zero. In a similar way


If the operator X in (6.112) is chosen as

an algebraic function of the generating operators

G , the integration process will produce those

algebraic functions that commute with every G and

therefore serve to classify, by their eigenvalues

the various matrix representations of the infini-

tesimal generators of the group. Now

in which it is supposed that the basis is suitable

to produce a unitary matrix U . The effect of

the integration is achieved by requiring that

and, since f(G) can be chosen as a symmetrical

homogeneous function of the various G , the opera-cl

tor nature of the latter is not relevant which per-

mits (6.116) to be replaced by the numerical invar-

iance requirement

referring to functions of a vector y in a v-dimen-

sional space. The infinitesimal version of this


which are analogous to, but are less general than

the differential operators Q. and Q

of invariant functions can be constructed directly

if a finite dimensional matrix representation is

known. Let G and U symbolize a K-dimensional

matrix representation of the corresponding Hermitian

and unitary operators, Then

and

invariance property can be expressed as

where the differential operators

are realizations of the generating operators (for

a semi-simple group) f

supply invariant symmetrical functions of y, and

thereby of the operators G

6.12 DIFFERENTIAL OPERATORS AND THE ROTATION GROUP

The differential and integral group properties

we have been discussing can be illustrated with the

three-dimensional rotation group. On referring to

(6.70-1) we see that the two sets of differential

operators that realize the abstract angular momen-

tum operators are

is an invariant function. The coefficients of the

powers of X , or equivalently, the traces


these differential operators appear as

together with the results of cyclically permuting

the indices 123 , The differential operators

§. , are such that

and are evidently associated with infinitesimal

rotations in a four-dimensional Euclidean space.

(They obey the four-dimensional extension of the

angular momentum commutation relations given in

(2.80). ) Of course, the group manifold is three-di-

mensional and, on restoring n as a function of

the independent parameters u , the term contain-

ing V3uQ is omitted in (6.124). The three-di-

mensional element of volume contains

(du) = du, du2 du., , together with the factor

or, if we introduce the notation

239GROUPS OF TRANSFORMATIONS


|det 5 I which can be evaluated directly or inferred

from the relevant form of the differential equations

(6.102)

The unique solution of this equation states the

constancy of un det £ . Alternatively, an

employ the four variables u = u~ , u , and a

volume element proportional to (du) = duQ (du)

with the restriction on the variables u_ enforced*7 O

by a delta function factor 6 (u 4- u -l) . On

integrating over u» , one regains the factor

|un which, in the previous method, is supplie

by |det C| • Thus, the group manifold is the

three-dimensional surface of a four-dimensional

Euclidean unit sphere, and the volume element can

be described intrinsically, or in terms of the space

in which the manifold is imbedded,

Here the proper constants have been supplied to nor-

malize the total volume to unity, although in the

three-dimensional form one must also sum over the


two pieces of the group that correspond to

u-. SB ± (l-u ) . The advantage of the four-dimen-2

sional form stems from the commutativity of 6(u -I)

with the differential operators J , which permits

the latter to be defined in the unbounded four-di-

mensional Euclidean space, thereby identifying the

four variables u with canonical variables of thea

first kind. Thus the general properties of a three-

dimensional angular momentum can be studied in terms

of an equivalent system consisting of a particle

in a four-dimensional Euclidean space, with the

correspondence between its orbital angular momentum

and the general three-dimensional angular momentum

described by (6.126).

There is only one independent operator that2

commutes with every component of J , namely, J ,

for there is only one independent rotationally in-

variant function of a three-dimensional vector. We

observe from (6.124-6) that


where the square of the three-dimensional angular

momentum vector is represented by the four-dimen-

sional differential operator

Thus eigenfunctions of the total angular momentum

are obtained from four-dimensional spherical har-

monics ~ solutions of the four-dimensional Laplace

equation that are homogeneous of integral degree

n = 0 , 1, 2,..., - and the eigenvalues are

2Another procedure uses complex combinations

of the parameters, as comprised in the pair of com-plex numbers that form any row or column of the ma-trix

imply

Let y indicate either two compo row vector,with y1 similarly designating a column vector.The differential composition properties.


The completeness of these eigenfunctions, and of

the three-dimensional angular momentum spectrum can

be inferred from the structure of the fundamental

solution, of degree -2 , which refers to the

inhomogeneous equation

6.13 NON-COMPACT GROUP INTEGRATION

The technique of group integration can also

be effective for groups that are not compact. We

shall illustrate this with the special canonical

group referring to Herinitian variables of the first

kind. As the analogue of the operator appearing in

(6.112), we consider

The squared angular momentum is thereby representedas, for example,

The corresponding differential operator realizationsof an angular momentum vector are:

(The transformations described by the parameter X

are without effect here), The integrations are ex-

tended over the infinite spectral range of Heraitian

canonical variables. As we can recognize directly

from the group multiplication law (6,58), the opera-/%

tor X commutes with every unitary operator U(q'p')

Hence it commutes with both sets of canonical vari-

ables , q and p , and for a system described byA

variables of the first kind, X must be a multiple

244

where, in the notation of (6.57)


~ •

and any function of the y that is homogeneousof integral degree n provides an eigenfunction,which is associated with the eigenvaluej(j+1) , j = n/2 . One recognizes that the differ-ential operators refer to an equivalent system des-cribed by two complementary pairs of non-Hermitianvariables of the first kind, and

This equivalence was used for a systematic develop-ment of the theory of angular momentum in a paper

GROUPS OP TRANSFORMATIONS 245

of the unit operator. This result can also be de-

rived through an explicit construction of the ma-^

trix representing X in some canonical representa-

tion. We first observe that

where the right side is a multiple of the unit opera-

tor. If X is chosen as a Measurement symbol,

M(a') , this equation reads

that was written in 1951, but remained unpublished.It is now available in the collection "QuantumTheory of Angular Momentum" edited by L.C. Biedenharnand H. Van Dam, Academic Press, 1965.

We have thus shown that

and therefore

The statements contained in (6.140) and (6,143) can

be regarded as assertions of the completeness and

orthonormality of the operator basis provided by


and multiplication on the right by XM(a') ,

followed by summation over a1 , converts our re-

sult into

which is the explicit exhibition of any operator as

a function of the fundamental dynamical variables

of the first kind. If the operator X of (6.138)

is replaced by such a function, F(q , p) , that

formula becomes

or, according to the multiplication property

As an application of the latter form, we have


the continuous set of unitary operator functions of

the variables of the first kindf U(q'p')

6.14 VARIABLES OF THE SECOND KIND

The general formal analogy between the two

types of dynamical variables, together with the

specific difference associated with the evaluation

of traces, correctly suggests that the statements

of Eqs. (6,138), (6.140-3) are applicable to vari-

ables of the second kind if the operation tr ...

is replaced by tr p.., . But a word of caution is

needed. The eigenvalues that appear in U(q'p')

are anticoiranutative with the variables of the sec-

ond kind and thus contain the factor p , whereas

the eigenvalues involved in the integrations and

in the delta functions are entirely elements of the

external algebra. Incidentally, the general trace

formula (4.179) emerges from (6.138) and its analogue

for variables of the second kind on forming the

<p'=0| q'=0> matrix element. It is also worth

noting that our various results could be freed from

explicit reference to complementary variables. We

shall make limited use of this possibility to trans-


form the statement for variables of the second kind

from their non-Hermitian canonical versions to forms

appropriate for Hermitian canonical variables. This

will be accomplished by using the transformation

(3.116) for operators, together with the analogous

one for eigenvalues, which yields, for example,

or

Similarly

and

When using this notation one roust not confuse the


I i n

symbol 5 (6 -O) with an eigenvalue of theEv Kl

Hermitian operator £

and

which is such that

6.15 REFLECTION OPERATOR

If the integration process (6,109) is applied

•bo the operators U(q'p'X) , referring to variables

of the first kind, we obtain zero as a result of

the X integration. Hence there is no state in-

variant under all the operations of the special

canonical group. But, if the X integration is

omitted, we are led to consider the Hermitian opera-

tor


Thus, R anticoiramutes with each variable of the first

kind.

The now familiar operator possessing these proper-

ties for the variables of the second kind is simi-

larly produced by

6.16 FINITE OPERATOR BASIS

The formal expressions of completeness and

orthogonality for the operator basis of the variables

of the second kind, which are comprised in the

various aspects of the U(lp ,

and correspondigly

can be freed of explicit reference to the special

canonical group. We first recognize that U(C')

is the generating function for the 2 n distinct

elements of the operator algebra. These we define

more precisely as

If we use the notation «'{v} to designate the

similar products formed from the eigenvalues £" iK,

we have


where each v assumes the value 0 or 1 , andis

With this definition the Hermitian operators a{v)

possess unit squares and, in particular,


The product of two operators, with indices {v}

and (1 - v} equals the operator p , to within

a phase factor,

The integration symbol stands for differentia-

tion with respect to each C * * and thus the terms

that contribute to the integrals are of the form

But one must also recall that the anticommutative

operator p is to be separated from the eigenvalues

prior to integration. This has no explicit effect


if the number of eigenvalues in the individual terms

is even, corresponding to v = £ v an even integer,H»

For odd v , however, the additional factor of

p multiplying a{v} induces

The net result has the same form in either circum-

stance, leading to the following expression of

completeness for the 4 dimensional operator basis

a{v}

The orthonormality property

can be inferred from the completeness and linear

independence of the a{v} or obtained directly

from (6.154) on remarking that


t6.17 ADDENDUM: DERIVATION OP THE ACTION PRINCIPLE


Academy of Sciences, Vol. 46, pp. 893 - 897 (1960).

VOL. 46, I960 PHYSICS: J. SCHWINGEM 893

We are now going to examine the construction of finite unitary transformationsfrom infinitesimal ones for a physical system of n continuous degrees of freedom.Thus, all operators are functions of the n pairs of complementary variables qk, pt,which we deaote collectively by x. Let us consider a continuous set of unitaryoperators labeled by a single parameter, U(r). The change from r to r + rfr is theinfinitesimal transformation

which includes a possible explicit T dependence of the generator, and

where

are the fundamental quantum variables of the system for the description producedby the transformation V(r). The accompanying state transformations are indi-cated by

A useful representation of the unitary transformation is given by the transformationfunction

where the omission of the labels a', 6' emphasizes the absence of explicit referenceto these states. Yet some variation of the states must be introduced if a sufficientlycomplete characterization of the transformation function is to be obtained. Forthis purpose we use the infinitesimal transformations of the special canonical group,performed independently on the states associated with parameters r and T + <J-r[S'l.nnu,,^.

894 PHYSICS: J. SCHWINGEB PBOC. N. A. S.

which includes

the matrix of V(r) in the arbitrary ab representation. The relation between in-finitesimally neighboring values of r is indicated by

The general discussion of transformation functions indicates that the most com-pact characterization is a differential one. Accordingly, we replace this explicitstatement of the transformation function (a'-r + dr\b'r} by a differential descrip-tion in which the guiding principle will be the maintenance of generality by avoidingconsiderations that refer to specific choices of the states o' and &', Wenote first that the transformation function depends upon the parameters T, T +AT and upon the form of the generator 0(x, T), Infinitesimal changes in theseaspects [8*] induce the alteration

in which the infinitesimal generators are constructed from the operators appropriateto the description employed for the corresponding vectors, namely x(r + dr) andX(T). It is convenient to use the symmetrical generator <?,, „ which produceschanges of the variables x by '/jte. Then

which, with the similar expression for 0,{r + dr), gives

wfoer**

and the Sx(r), &x(r -f dr) are independent arbitrary infinitesimal numbers uponwhich we impose the requirement of continuity in r.

The infinitesimal unitary transformation that relates X(T) and x(r + dr) is ob-tained from

Our result is a specialization or tne general diflerential characterization 01 trans-formation functions whereby, for a class of alterations, the infinitesimal operatorSW is derived as the variation of a single operator W. This is a quantum actionprinciple8 and W is the action operator associated with the transformation.

We can now proceed directly to the action principle that describes a finite unl-tarv transformation.

for multiplicative composition of the individual infinitesimal transformation func-tions is expressed by addition of the corresponding action operators

As written, this action operator depends upon all operators X(T) in the r intervalbetween r\ and r». But the transformations of the special canonical group, appliedto (n | ra), give

which is to say that d'Wi» does not contain operators referring to values of r in theopen interval between r\ and n, or that W-a, is stationary with respect to the specialvariations of x(r) in that interval. Indeed, this principle of stationary action,the condition that a finite unitary transformation emerge from the infinitesimalones, asserts of q(r), p(r) that

where

m which & is used here to describe the change of q, p by Sq and Sp, occurring inde-pendently but continuously at r and r + dr. The two species of variation cannow be united: J = &' + &*, and

You 46, 1960

as

Accordingly, one can write

PHYSICS: J, SCHWINGBR, 895

or

89« PHYSICS: J. SCMWINOBK PBOC. N, A. S.

which are immediate implications of the various infinitesimal generators.The use of a single parameter in this discussion is not restrictive. We have only

to write

where

The vanishing of each of these operators is demanded if the transformation ia to beindependent of path. When the operators Gt(x, r) can be expressed as a linear com-bination of an equal number of operators that are not explicit functions of the param-eters, Oa(x)f the requirement of path independence yields the previously con-sidered conditions for the formation of a group.

We now have the foundations for a general theory of quantum dynamics andcanonical transformations, at least for systems with continuous degrees of freedom.The question is thus posed whether other types of quantum variables can also beemployed in a quantum action principle.

* Publication assisted by the Office of Scientific Besearch, United States Air Force, ynder con-tract number AF49(638)-589.

« These PBOCBBMNGS, 45,1S42 (1959); 48, 257 (1960); and 46, 570 (1960),* The group can be obtained directly as the limit of the finite order group associated with each

r. That group, of order »•*, is generated by U, ¥, and the »th root of unity given by WV~lU~l.Some aspects of the latter group are worthy of note. There are »' + r — 1 classes, the commu-tator subgroup is of order e, and the order of the corresponding quotient group, F', is the numberof inequivatent one-dimensional representations. The remaining » — 1 matrix representationsmust be of dimensionality », «•' ~ »* + (» ~ I)"1) and differ only in the choice of the generating»th root of unity. That choice is already made in the statement of operator properties for U andF and there can be only one irreducible matrix representation of these operators, to within thefreedom of unitary transformation.

* In earlier work of the author, for example Phyt, Km., 91, 713 (1953), the quantum actionprinciple has been postulated rather than derived.

with each drK/dr given as an arbitrary function of T, and then regard the transforma-tion as one with p parameters, conducted along a particular path in the parameterspace that is specified by the p functions of a path parameter, T>(T). Now

is the action operator lor a traasiormation referring to a prescribed path and gener-ally depends upon that path. If we consider an infinitesimal path variation withfixed end points we find that


6.18 ADDENDUM CONCERNING THE SPECIAL CANONICAL

GROUP1"


Academy of Sciences, Vol. 46, pp. 1401-1415 (I960)

Reprinted from the Proceedings of the NATIONAL ACABEMTT op SCIENCESVol. 48, No. JO, pp. 1401-1416. October, IBM.

THE SPECIAL CANONICAL GROUP*

BY JULIAN SCHWINGER

HABVABD uNivxnuurr

Communicated August SO, 1960

This note is concerned with the further development and application of anoperator group described ia a previous paper,' It is associated with the quantumdegree of freedom labeled v = » which is characterized by the complementarypair of operators q, p with continuous spectra.8 The properties of such a degreeof freedom are obtained as the limit of one with a finite number of states, spe-cifically given by a prime integer v. We recall that unitary operators U and

define two orthonormal coordinate systems (w*| and (0*1, where

and

For any prime v > 2 we can choose the integers k and I to range from — l/t(v — 1)to l/t(v — I ) , rather than from 0 to v — J. An arbitrary state * can be representedalternatively by the wave functions

where

and the two wave functions are reciprocally related by

We now shift our attention to the Hermitian operators g, p defined by

and the spectra

Furthermore, we redefine the wave functions so that

where e = Ag' = Ap', the interval between adjacent eigenvalues. Then we have

V obeying

1402

and

PHYSICS: J, SCHWINGSB Pmoc. N. A. S.

and

This is what is implied by the symbolic notation (Dirac)

We shall also use the notation that is modeled on the discrete situation, withintegrals reolacine summations, as in

with

which must be an identity for wave functions of the physical class when theoperations are performed as indicated. There will also be a class of functionsK(p', «), such that

We shall not attempt here to delimit more precisely the physical class of states.Note however that the reciprocal relation between wave functions can be combinedinto

and

As v increases without limit the spectra of q and p become arbitrarily dense andthe eigenvalues of largest magnitude increase indefinitely. Accordingly we mustrestrict all further considerations to that physical class of states, or physical sub-space of vectors, for which the wave functions $(q') and ${p') are sufficiently wellbehaved with regard to continuity and the approach of the variable to infinitythat a uniform transition to the limit v = » can be performed, with the result

In the limit i» = oo there is a class of functions f(q'p') such, that

Vou 46, I960 PHYSICS: J. SCHWINGER 1403

There are other applications of the limit v —* <*>. The reciprocal property of theoperators U and F is expressed by

or

with an exception when q' or p' is the greatest eigenvalue, for then q' + t or p' + eis identified with the least eigenvalue, —q' or —p'. We write these relations aswave function statements, of the form

and

In the transition to the limit v = «, the subspaee of physical vectors * is dis-tinguished by such properties of continuity and behavior at infinity of the wavefunctions that the left-hand limits. « -+• 0, exist as derivatives of the correspondingwave function. We conclude, for the physical class of states, that

and

It will also be evident from this application of the limiting process to the unitaryoperators exp(feg), exp(icp) that a restriction to a physical subspaee is needed forthe validity of the commutation relation

Fhe elements of an orthonormal operator basis are given bj

or v-^Uia'-D'), with

Thus, since »-' = Ag'Ap'/2-r, we have, for an.arbitrary function f(q'p') of thediscrete variables q', p',

1404 PHYSICS: J. SCHWINGER PBOC. N. A. S.

which we express symbolically by

In particular,

where g and p on the right-hand side are numerical integration variables.The completeness of the operator basis U(g'p') is expressed by

The properties of the U(g'p') basis are also described, with respect to an arbitradiscrete operator basis X(d), by

and

When X is given as F(q, p), the operations of the special canonical group can beutilized to bring the completeness expression into the form

This operator relation implies a numerical one if it is possible to order F(q, p),so that all g operators stand to the left, for example, of the operator p: F(§; p).Then the evaluation of the (g' = 0 p' = 0} matrix element gives

which result also applies to a system with « continuous degrees of freedom if it isunderstood that

As an example of this ordering process other than the one already given by{r U(g'p'), we remark that

where

VOL. 46, I960 PHYSICS; J. SCSWINGER 1405

and thwefori

which reproduces the well-known non-degenerate spectrum of the operatorw + p5)-

Now we shall consider the construction of finite special canonical transformationsfrom a succession of infinitesimal ones, as represented by the variation of a pa-rameter T. Let the generator of the transformation associated with r -*• r + dr be

where Q(T) and P(r) are arbitrary numerical functions of r. That is, the in-finitesimal transformation is

which implies the finite transformation

Some associated transformation functions are easily constructed. We have

or

and therefore

which, in conjunction with the initial condition

gives

with

Accordingly, if we also multiply by e x p ( — i p ' f f ) and Integrate with respect todp'/Zr, what emerges is {q'n \ q"Tt)gr, as we can verify directly.

and then indicate the effect of the additional localized transformations by theequivalent unitary operators, which gives

It is important to recognize that the trace, which is much more symmetricalthan any individual transformation function, also implies specific transformationfunctions. Thus, let us make the substitutions

where, in view of the delta function factors, f+(r — T') can be replaced by otherequivalent functions, such as q+ — !/j = */»«, or */je(r ~ T'^ — (T — r')/Tt withf = T, — Tj. The latter choice has the property of giving a zero value to thedouble integral whenever Q(r) or P(r) is a constant. As an operator statement,the trace formula is the known result

We can compute the trace of a transformation function, regarded as a matrix, andthis will equal the trace of the associated unitary operator provided the otherwisearbitrary representation is not an explicit function of r. Thus

These transformation functions can also be viewed as matrix elements of theunitary operator, an element of the special canonical group, that produces thecomplete transformation. That operator, incidentally, is

where «(r — r') is the odd step function

1406 PHYSICS: J. SCHWINOEB PBOO, N. A, S.

From this result we derive

or, alternatively

on using tee tact tnat

You 46, I960 PHYSICS; J. SCHWINGSR 1407

We shall find it useful to give an altogether different derivation of the traceformula. First note that

and similarly

which is a property of periodicity over the interval T — r\ — rt. Let us, therefore,represent the operators g(r), J>(T) by the Fourier series

where the coefficients are so chosen that the action operator for an arbitrary specialcanonical transformation

acquires the form

Here the dash indicates that the term n = 0 is omitted, and

The action principle for the trace is

and the principle of stationary action asserts that

together with

The first of these results implies that the trace contains the factors &(Qo) andJ(Po). The dependence upon Qn, Pn, n 4= 0, is then given by the action principle as

First, let us observe how an associated transformation function {n| r^)0^p depends

upon the arbitrary functions Q(r), P(r). The action principle asserts that

1408 PHYSICS: J, 8CHWINOER PBOC. N. A. S.

Therefore

where the factor of 2-jr is supplied by reference to the elementary situation withconstant Q(r) and P(r). We note, for comparison with previous results, that

and

The new formula for the trace can be given a uniform integral expression by usingthe representation

for now we can write

where

and W[q, pi is the numerical function formed in the same way as the action operator,

Alternatively, we can use the Fourier series to define the numerical functionsq(r), pM. Then

f T*

and d[q, p] appears as a measure in the quantum phase space of the functions«W, P(T).

It is the great advantage of the special canonical group that these considerationscan be fully utilized in discussing arbitrary additional unitary transformations,as described by the action operator

Here ( )0 is an ordered product that corresponds to the sense of progression fromTJ to n. If T follows T' in sequence, the operator function of r stands to the left,while if r precedes T' the associated operator appears on the right. This descriptioncovers the two algebraic situations: n > r», where we call the ordering positive,( )+, and n < TI, which produces negative ordering, ( )_. For the momenttake TI > rj and compare

with

PHYSICS: J. SCHWINGER 1409

which we express by the notation

More generally, if F(T') is an operator function of x(r') but not of Q, P we have

where [ r') X (T' symbolizes the summation over a complete set of states, andtherefore

The difference of these expressions,

refers on one side to the noncommutativity of the complementary variables q andp, and on the other to the "equation of motion" of the operator g(r). Accordingto the action principle

and therefore

which yields the expected result,

Vol. 46,1960

1410 PHYSICS: J. SCHWINGER PBOO, N. A. S.

Thus, through the application of the special 'canonical group, we obtain functionaldifferential operator representations for all the dynamical variables. The generalstatement is

where F(q, p)a is an ordered function of the q(r), p(r) throughout the interval be-tween TS and n, and, as the simple example of q(T)p(r) and p(r)q(r) indicates,the particular order of multiplication for operators with a common value of tmust be reproduced by a suitable limiting process from different r values.

The connection with the previous considerations emerges on supplying G witha variable factor X. For states at n and n that do not depend explicitly upon Xwe use the Action principle to evaluate

The formal expression that gives the result of integrating this differential equationfrom X = 0 to X = 1 is

where the latter transformation function is that for X = 0 and therefore refersonly to the special canonical group. An intermediate formula, corresponding to0 = Gi + Gt, contains the functional differential operator constructed from G\acting on the transformation function associated with (?j. The same structureapplies to the traces of the transformation functions. If we use the integralrepresentation for the trace of the special canonical transformation function, andperform the differentiations under the integration sign, we obtain the generalintegral formula'

Here the action functional W[q, p] is

which is formed in essentially the same way as the action operator PFM, the multi-plication order of noncommutative operator factors in G being replaced by suitableinfinitesimal displacements of the parameter r. The Hermitian operator G canalways be constructed from symmetrized products of Hermitian functions of qand of p, and the corresponding numerical function is real Thus the operatorVtf/ifato, MvM] is represented by 'AC/jCfCr + «)) + /,(<?(r - t)))MpM),for example. One will expect to find that this averaged limit, * -*• 0, implies nomore than the direct use of /I(#(T)/J(P(T)), although the same statement is certainlynot true of either term containing e. Incidentally, it is quite sufficient to construct

Another property of the example is that the contributions to the trace of allFourier coefficients except n = 0 tend to unity for sufficiently small T at ft. Thisis also true for a class of operators of the form G = !/»pz + /(<?)• By a suitabletranslation of the Fourier coefficients for p(X) we can write w[q, p] as

This example also illustrates the class of Hermitian operators with spectra thatare bounded below and for which the trace of exp (iTG) continues to exist on givingT a positive imaginary component, including the substitution T -*• iff, ft > 0.The trace formula can be reatated for the latter situation on remarking that theFourier series depend only upon the variable (r — T$)/T = A, which varies from0 to 1, and therefore


the action from pdq, for example, rather than the more symmetrical version, invirtue of the periodicity,

As a specialization of this trace formula, we place Q — P = 0 and consider theclass of operators G that do not depend explicitly upon T, Now we are computing

in which we have used the possibility of setting ra = 0. A simple example isprovided by G = '/»(?* + <?*)» where

&nfL

For sufficiently small B, the trem involing dq/dy to which all the Fourier coffi-

cients of q(y) contribute except q0, will effectively suppress these Fouier coefficientPovides appropiate restric are imposed concering sigular pointes in the neighborhood of which f(q) acquires arge negative values. Then f(q (y)f (q0))and we can reduce the integrations to just the contribution of qo and po, as experssed

by

1412 PHYSICS: J. SCBWINGSR PEOC. N. A. S.

Comparison with the previously obtained trace formula involving the ordering ofoperators shows that the noneommutativity of g and p is not significant in this limit,Thus we have entered the classical domain, where the incompatibility of physicalproperties at the microscopic level is ao longer detectible. Incidentally, a, firstcorrection to the classical trace evaluation, stated explicitly for one degree offreedom, is

Now introduce the functional differential operator that is derived from thenumerical action function W0[q, p] which refers only to the transformationsrcnerated bv G. namelv

These equations are valid for any such transformation function. The trace isspecifically distinguished by the property of periodicity,

which gives the exact value when /(g) is a positive multiple of q2.Another treatment of the general problem can be given on remarking that the

equations of motion implied by the stationary action principle,

can be represented by functional differential equations*

which asserts that the trace depends upon Q(r), P(r) only through the Fouriercoefficients Qn, Pn, and that the functional derivatives can be interpreted by meansof ordinary derivatives:

and observe that the differential equations are given by

VOL. 46, 1960 PHYSICS; J. SCHWINGER 1413

>r by

The latter form follows from the general expansion

on noting that [Wo, Q(r)], for example, is constructed entirely from differentialoperators and is commutative with the differential operator WG- Accordingly,

which asserts that exp(—iW0)(tr) vanishes when multiplied by any of the Fouriercoefficients Qn, Pn and therefore contains a delta function factor for each of thesevariables. We conclude that

where, anticipating the proper normalization constants,

A verification of these factors can be given by placing G = 0, which returns usto the consideration of the special canonical transformations. In this procedurewe encounter the typical term

the proof of which follows from the remarks that

and

The result is just the known form of the transformation function trace

When integral representations are inserted for each of the delta function factorsin 8[Q, P], we obtain

and the consequence of performing the differentiations in Wa under the integrationsigns is*

where the action function W[q, p] now includes the special canonical transformationdescribed by Q and P.

1414 PHYSICS: J. SCHWINOER PBOC. N. A, 8.

We shall also write this general integral formula as

in order to emphasize the reciprocity between the trace, as a function of Qn, Pn orfunctional of Q(r), P(r) and exp(«fPG[g, p]) as a function of qn, pn or functional of?(T), p(r). Indeed,

where

is such that

A verification of the reciprocal formula follows from the latter property on insertingfor the trace the formal differential operator construction involving 5[Q, F]. Thereality of Wg[q, p] now implies that

f d[Q, P]d[Q', F']eVTW(p-F')-«0-e')) (tr)0e* (&•)«'/•' = \

or, equivalently,

f d[Q, P](fr)x+*,*(fr)*+*i - «[Qi - Qt, Pi - Pt]

where X combines Q and P.

The trace possesses the composition property

in view of the completeness of the operator basis formed from f(-ri), p(n),No special relation has been assumed among TO, n and T». If n and TS are

equated, one transformation function in the product is the complex conjugate ofthe other. We must do more than this, however, to get a useful result. The mostgeneral procedure would be to choose the special canonical transformation inft" {raj Ti)e«p = tr (n\ Tt)g<>r* arbitrarily different from that in tr (n TJ)OW. Thecalculations! advantages that appear in this way will be explored elsewhere.

The operation involved is the replacement of Q(r), P(r) in the respective factorsby Q(r) =F q'S(r — n), F(r) =F p'S(r — n) followed by integration with respectto dq'dj>'/2ir. The 'explicit form of the left side is therefore

VOL. 46, 1960 PHYSICS: /. SCHWINOER 1415

Here we shall be content to make the special canonical transformations differonly at r». The corresponding theorem is

where

and similarly for Q"{r), P*(r). This statement follows immediately from theorthonormality of the U(q'p') operator basis on evaluating the left-hand side as

* Supported by the Air Force Office of Scientific Research (ARDC).1 These PBOOBBOIMGS, 46, 883 (1960).' These PEOCBBDINOS, 46, 570 (1960),' This formulation is closely related to the algorithms of Feynman, Pkys. Bee., 84, 108 (1951),

Res. Mod, Phys., 20, 36 (1948), It differs from the latter in the absence of ambiguity associatedwith noncommutative factors, but primarily in the measure that is used. See Footnote 5.

4 These are directly useful as differential equations only when 0(fp) is a sufficiently simplealgebraic function of q and p. The kinematical, group foundation for the representation ofequations of motion by functional differential equations is to be contrasted with the dynamicallanguage used in these PmocmEDisas, 37, 4S2 (1851).

* In this procedure, q(r) and p(T) are continuous functions of the parameter r and the Fouriercoefficients that represent them are a denumerably infinite set of integration variables. Analternative approach is the replacement of the continuous parameter r by a discrete index whileinterpreting the derivative with respect to r as a finite difference and constructing S[Q, PI as sproduct of delta functions for each discrete r value. With the latter, essentially the Peynman-Wiener formulation, the measure d[q, p] is the product of dg(r)<ip(r)/2*r for each value of T,periodicity is explicitly imposed at the boundaries, and the limit is eventually taken of an infinitelyfine partitioning of the interval T = n — TJ. The second method is doubtless more intuitive,since it is also the result of directly compounding successive infinitesimal transformations but it igmore awkward as a mathematical technique.


6.19 ADDENDUM: QUANTUM VARIABLES AND THE ACTIONPRINCIPLE


Academy of Sciences, Vol. 47, pp. 1075 -1083 (1961)

Hpprhited from the Proeeetlmgs of the NATSOKAZ, ACAOJKMY or SCIENCESVol. ff, No. 7, pp. 1078-1083, July, W6I.

QUANTUM VARIABLES AND THE ACTION PRINCIPLE

BY JULIAN SCHWIJJGER

HiBVARD UNIVERSITY AND I'XIVEHSITY OF CALIFORNIA AT I,OH ANGBIJU5

Communicated Alaij 89, lUfii

In previous communications, a classification of quantum degrees of freedom by aprime integer v has been given,1 and a quantum action principle has been con-structed2 for v=<*>. Can a quantum action principle be devised for other types ofquantum variables? We shall examine this question for the simplest quantum de-gree of freedom, »• = 2.

Let us consider first a single degree of freedom of this type. The operator basisis generated by the complementary pair of Hermitian operators

is the three-dimensional infinitesimal rotation

Accordingly, an infinitesimal transformation that varies £» and not & can only be arotation about the second axis,

The corresponding infinitesimal generator is

Jimilarly,

is generated by

and the generator for the combination of these elementary transformations is (k =1,2)

which obey

The basis is completed by the unit operator and the product

We have remarked1 upon the well-known connection between the 0%, k = 1,2, 3,and three-dimensional rotations. In particular, the most general unitary operatorthat differs inftnitesimally from unity is, apart from a phase factor, of the form

and the cosTesponding operator transformation

1075

1078 PHYSICS: J. SCUWINOER PBOC. N. A. S.

It is evident that G» and ft must be supplemented by

to form the infinitesimal generators of a unitary group, which is isomorphie to thethree-dimensional rotation group. The transformation induced by 63 is

and

Thus, the concentration on the complementary pair of operators fi and fe does notgive a symmetrical expression to the underlying three-dimensional rotation group.This is rectified somewhat by using, for those special transformations in which |jand fe are changed independently, the generator of the variations VtSft, k =» 1,2,

n>

An arbitrary infinitesimal unitary transformation is described by the transforma-tion function

Infinitesimal variations in r, r + dr, and the structure of G induce

To this we add S', the transformations generated by Gt, performed independentlybut continuously in r, on the states (r + dr\ and I r).

where S', in its effect upon operators, refers to the special variations 8ft, k = 1, 2,performed independently but continuously at r and r + dr.

It is only if the last term is zero that one obtains the quantum action principleis = r + «1

with

Since the special variation is such that 8|i and Jfe are arbitrary multiples of &, it

Here

or

VOL. 47, 1961 PHYSICS: J. SCMWINGKS 1077

is necessary that [6, |3] commute with & and fe. Hence this commutator must bea multiple of the unit operator, which multiple can only be zero, since the trace ofthe commutator vanishes or, alternately, as required by [G, Is2] = 0. For an actionprinciple formulation to be feasible, it is thus necessary and sufficient that

Terms in & and S-t are thereby excluded from G, which restriction is also conveyedby the statement that a permissible <? must be an even function of the {», k = 1, 2.Apart from multiples of the unit operator, generating phase transformations, theonly allowed generator is fife, which geometrically is a rotation about the thirdaxis.

It should be noted that the class of variations &' can be extended to include theone generated by (?», without reference to the structure of G. Thus,

where the latter term equals

and therefore, for this kind of $' variation,

The action operator for a finite unitary transformation is

or, more symmetrically,

since Wn is only defined to within an additive constant, and

The principle of stationary action

which refers to a fixed form of the operator G({(r), r), expresses the requirementthat a finite transformation emerge from the succession of infinitesimal transforma-tions. It will be instructive to see how the properties of the quantum variablesare conversely implied by this principle. The discussion will be given withoutexplicit reference to the single pair of variables associated with one degree of free-dom, since it is of greater generality.

The bilinear concomitant

The generator term G&T evidently restates the transformation significance of theoperator G, The effect upon operators is conveyed by the infinitesimal unitarytransformation

which is the general equation of motion,

Let us take Sf* to be a special variation, which we characterize by the followingproperties: (I) Each 8& anticommutes with every &,

(2) the &^(T) have no implicit T dependence; and (3) every Sft is an arbitrary in-finitesimal numerical multiple of a common nonsingular operator, which does notvary with r. The second basic property asserts that

which restricts 0 to be an even function of the &, in virtue of the anticommutativityof the special variations with each member of this set. Furthermore, the r deriva-tives of the special variations are also antieommutative with the {», according toproperty (3), and therefore,

Then, if we write

which defines the left and right derivatives of G with respect to the f»,- we get

which is consistent with the general equation of motion., and

1078

shows that

PHYSICS: J,

and gives

PRoc. N. A. S.

Vol.. 47, 1981 PHYSICS: J. SCHWINOBR 1079

The nonsingular operator contained in every special variation can be cancelledfrom the latter equation, and the arbitrary numerical factor in each SfctM impliesthat

In the opposite signs of left and right derivatives, we recognize the even property ofthe function G.

On comparing the two forms of the equation of motion for f* we see that

If we reintroduce the special variations, this reads

The left side is just the change induced in (? by (?{, the generator of the special vari-ations, which also appears in {?», t, while the right side gives the result in terms ofchanges of the {» by Vifi£». Since (? is an arbitrary even function of the f*, we acceptthis as the general interpretation of the transformation generated by (?$, whichthen asserts of any operator F that

The implication for an odd function of the ft is

and the particular choice F = f j gives the basic operator properties of these quantumvariables

In this way, we verify that the quantum action principle gives a consistent accountof all the characteristics of the given type of quantum variable.

The operator basis of a single degree of freedom is used in a different mannerwhen the object of study is the three-dimensional rotation group rather than trans-foimations of the pair of complementary physical properties. The generator ofthe infinitesimal rotation Sir = Su X e is

On applying this transformation independently but continuously to the states ofthe transformation function (r 4- drl T). we encounter

The identity,

1080 PHYSICS: J. SCHWINGIBR Pnoc. N. A, S.

The most general form for G, describing phase transformations and rotations, is

The action principle asserts that

The extension to n degrees of freedom of type v = 2 requires some discussion.At first sight, the procedure would seem to be straightforward. Operators asso-ciated with different degrees of freedom are commutative, and the infinitesimalgenerators of independent transformations are additive, which implies an actionoperator of the previous form with the summation extended over the n pairs ofcomplementary variables. But we should also conclude that G must be an even

which appear in the anticipated form on remarking that the left-hand side of thelatter equation equals

where the operator variations are arbitrary infinitesimal rotations, So- = fo X <r.Hence the equations of motion are

We shall be content to verify that the principle of stationary action reproduces "theequations of motion that also follow directly from the significance of G,

combined with the equivalence of the two left-hand terms, both of which equal

then gives

where 8' describes the independent operator variations S«r(r) = &i(r) X a(r) atr and T + dr. If we add the effect of independent variations of r, T + dr, and thestructure of (7 (8"], we obtain an action principle3 without restrictions on the formof G.

The action operator for a finite transformation is

Vol. 47, 1961 PHYSICS; J. SCHWINGSR 1081.

function of the complementary variables associated with each degree of freedomseparately, and this is an unnecessarily strong restriction.

To loosen the stringency of the condition for the validity of the action principle,we replace the relationship of commutativity between different degrees of freedomby one of anticommutativity. Let {<a>i,»,i be the operators associated with the adegree of freedom. We define

and this set of 2n Hermitian operators obeys

The inverse construction is

and ia

we have an operator that extends by one the set of anticommuting Hermitianoperators with squares equal to Vs. In particular this operator anticommuteswith every {*, k = 1 ... 2n.

An infinitesimal transformation that alters only f* must be such that

which identifies SJ» as an infinitesimal numerical multiple of {»„+.). We shall write

and the generator of all these special variations is

The latter can also be written as

On forming the commutator of two such generators, we get

where the Hermitian operators

for the condition that permits an action principle formulation. Each special varia-tion is proportional to the single operator $s*+i. It is necessary, therefore, that[G, £**+i] commute with every £*, k = 1 . . . 2«, or equivalently, with each comple-mentary pair of operators fi, j(a>, a — I ... n. Such an operator can only be a mul-tiple of the unit operator and that multiple must be zero. Hence the infinitesimalgenerators of transformations that can be described by an action principle mustcommute with f»«+i. Considered as a function of the anticomrnutative operatorset Jt, k = 1 . . . In which generates the 22" dimensional operator basis, an admis-sible operator (?({) must be even. This single condition replaces the set of n condi-tions that appear when commutativity is the relationship between different degreesof fieedom. Of course, if the class of transformations under consideration is suchthat Q is also an even function of some even-dimensional subsets of the {*, one can

1082 PHYSICS: J. SCHWINOSR Pnoc. N. A. S.

are n(2n — 1} in number, and obey (k ?£ I)

The generators G( and their commutators can thus be constructed from the basisprovided by the n(2n + 1) operators {*;, with k and I ranging from 1 to 2n + 1.And, since

these operators are the generators of a unitary transformation group, which has thestructure of the Euclidean rotation group in 2n + 1 dimensions. For n > 1, theoperators {«, k, I = 1 .. 2n + 1 can also be combined with the linearly independentset

to form the (n + 1) (2n + 1) generators of a similar group associated with rotationsin 2n -f- 2 dimensions.

It should also be Doted that

induces the linear transformation

and one can writ

These generators have the structure of the rotation group in 2n dimensions.The discussion of the change induced in a transformation function (r -j- dr\ T) by

the special vanations, applied independently but continuously at r and T + dr,proceeds as in the special example n = 1, and leads to

and the earlier discussion can be transferred intact. It may be useful, however, toemphasize that the special variations anticommute, not only with each &>(r), butalso with dit/dr, since this property is independent of r. Then it follows directlyfrom the implication of the principle of stationary action,

The action principle is still severely restricted in a practical sense, foi the genera-tors of the special variations cannot be included in G since the operators &fc»+i areodd functions of the 2w fundamental variables. It is for the purpose of circum-venting this difficulty, and thereby of converting the action principle into an ef-fective computation device, that we shall extend the number system by adjoiningan exterior or Grassmann algebra,

1 These PROCBSDINGS, 46, 570 (1960),'Ibid., 46, 883(1960).* The possibility of using the components of an angular momentum vector as variables in an

action principle was pointed out to me by I), Volkov during the 1959 Conference on High EnergyPhysics held at Kiev, U.8.S.R.

that the equations of motion an

for this type of 6' variation.The action operator associated with a finite transformation generalizes the form

already encountered for n = 1,

which gives


consistently adopt eommutativity as the relationship between the various subsets.Incidentally, the eommutativity of G with &»+I(T) asserts that the latter operatordoes not vary with T. Accordingly, the special variations S&(T) have no implicitoperator dependence upon T and one concludes that the special variations are anti-commutative with the 2w fundamental variables {», without reference to the as-sociated r values.

The action principle is also valid for the linear variations induced by (?„, withoutregard to the structure of G. Thus,

and the last term equals

CHAPTERCANONICAL TRANSFORMATION FUNCTIONS

7.1 Ordered Action Operator 2857.2 Infinitesimal Canonical Transformation 287

Functions7.3 Finite Canonical Transformation 293

Functions7.4 Ordered Operators. The Use of 297

Canonical Transformation Functions7.5 An Example 2997.6 Ordered Operators and Pertubation 302

Theory7.7 Use of The Special Canonical Group 3067.8 Variational Derivatives 3097.9 Interaction of Two Sub-Systems 3177.10 Addendum: Exterior Algebra and the 321

Action Principle

7.1 ORDERED ACTION OPERATOR

According to the significance of the action

operator W(q , q , t) that defines a canonical

transformation at time t , infinitesimal altera-

tions of the eigenvalues and of t produce a change

in the canonical transformation function <q't|q't>


given by

For some transformations, the commutation proper-

ties of the q and q variables can be used to

rearrange the operator SW so that the q's every-

where stand to the left of the q's . This ordered

differential expression will be denoted by

6W(q ; q , t) and

From the manner of construction of the latter opera-

tor, the variables q and q act directly on

their eigenvectors in (7.1) and this equation be-

comes

Hence 8W must be an exact differential, and in-

tegration yields

in which a multiplicative integration constant is

incorporated additively in W . This constant is

CANONICAL TRANSFORMATION FUNCTIONS 287

fixed, in part, by the composition properties of

the transformation function. It is to be emphasized

that the ordered operator W does not equal W

and, indeed, is not Hermitian, should W possess

this property. We have in effect already illustrat-

ed the ordering method by the construction of the

transformation function <q'jp'> . For Hermitian

variables of the first kind, for example,

7.2 INFINITESIMAL CANONICAL TRANSFORMATIONFUNCTIONS

The transformation function for an arbitrary

infinitesimal canonical transformation is easily

constructed in this way, if one uses the action

operator

whereas

for

is brought into the ordered form by performing the

required operation on the infinitesimal quantity

SG with the aid of the known commutation relations

between q = q and p . It is convenient to

supplement G with a numerical factor X so that

we obtain a one-parameter family of transformations

that includes the desired one (A = 1} and the

identity transformation (X=0) . Then

and the ordering operation is to be applied to G

as well as to 5G . From the former we obtain an

equivalent operator which we call G(p ; q) and thus



The integrability of the ordered operator <5W now

demands that the ordered version of 6G be simply

the variation of G(p;g) and thus (A=l)

in which the additive constant is that appropriate

to the identity transformation, W(p;q) , and de-

pends upon the kinds of variables employed. Hence

or

which simply repeats the significance of G as the

generator of the infinitesimal transformation. When

the transformation corresponds to an infinitesimal

change of parameters, (7.12) reads

and in view of the infinitesimal nature of G


The composition properties of transformation

functions can now be used to introduce other choices

of canonical variables. For example, with Hermitian

variables of the first kind,

according to {4.21} and- (7.12). The integrations

are easily performed if G is a linear function of

the p variables, or a non-singular quadratic

function. In the first situation

and, in particular,


and

For the quadratic functions or p , it is conven-

ient to move the origin of the p' variables to

the point defined by

Then

where

On eliminating p with the aid of (7.20) , we find


2in which M(q") is the matrix inverse to 3 G/3p3p

In the limit as the latter approaches zero the delta

function form (7.19) is obtained. Both results are

also easily derived directly, without reference to

the intermediate p representation. For the example

of an infinitesimal Hamiltonian-Jacobi transformation

in which H meets the requirements of a non-singu-

lar quadratic dependence on p , we have

2where rn is the matrix inverse to 3 H/3p3p and

An equivalent form for the latter is


in which pn is defined by

7.3 FINITE CANONICAL TRANSFORMATION FUNCTIONS

The transformation function for a finite

canonical transformation can be constructed by the

repeated composition of that for an infinitesimal

transformation. To evaluate <T,JT2> along a

particular path in the parameter space, we choose

N intermediate points on the path, and compute

in which x signifies the application of a method

of composition appropiate to the canonical variables

employed. In the limit N -»• « the required path

is traced as an infinite sequence of infinitesimal

transformations. With Hermitian variables of the

first kind and composition by integration, for

example, we begin with


and obtain

in which

and the delta function factor enforces the restric-

tion q - q" . For the Hamacobi

transformation with a Hamiltonian that depends

quadratically on p, this general form reduces to

in which w is given by (7.25) or (7.26). The

generally applicable technique of composition by

differentiation can be applied directly to the

infinitesimal transformation function (7.15) and we

obtain



apart from numerical factors of (2-n) for Hermi-

tian variables of the first kind. The boundary

values are

and the sense of multiplication is that of (7.28).


7.4 ORDERED OPERATORS. THE USE OF CANONICALTRANSFORMATION FUNCTIONS

the transformation function is obtained as a matrix

in which the states and canonical variables do not

depend on T . The result of composing successive

transformations then appears as the matrix of the

product of the associated operators and thus

which defines the ordered exponential operator. The

+ subscript refers to the manner of multiplication,

in which the positions of the operators correspond

to the path in parameter space as deformed into a

On writing


straight line, with T- on the right and T, on

the left. We speak of negative rather than positive

ordering if the sense of multiplication is reversed,

so that

in analogy with the more specific group structure

(6,39). Should the operators G, , commute, thus

generating an Abelian group, the exponential form

(7.38) is valid without reference to the integration

path. Let us suppose that the generators constitute,

or can be extended to be a complete set of commuting

Hermitian operators. Then a G representation exists

and a canonical transformation function describing

When the generators are independent of T , and

the path is a straight line, the ordering is no

longer significant and


the transformations of the group, say <p'T1|q*T?> ,

can be exhibited as

which shows how the canonical transformation function

serves to determine the eigenvalues of the operators

G as well as wave functions representing the G

states.

7.5 AN EXAMPLE

An elementary example for n = 1 is provided

by the Hermitian operator

in which the non-Hermitian canonical variables may

be of either type. A direct construction of the

canonical transformation function can be obtained

from (7.33) by repeated application of


namely

The eigenvalues thus obtained are

and wave functions of these states are given by

The derivation applies generally to both types of

variables. However, for variables of the second


*5 «t» | o

kind (q1) = (q ) = 0 , and the summation in

(7.42) terminates after n = 0 , 1 , which are in-

deed the only eigenvalues of q, The result(7.42) is reached more quickly, however, along the

lines of (5,78). For non-Hermitian variables

syranetrization or antisymmetrization (,) is un-

necessary and it is omitted in (7.40). No ordering

is required to obtain

which immediately gives (7.42). An alternative

procedure is

according to the solution of the equations of motion

which gives (7.42) on integration when combined with

the initial condition provided by


7.6 ORDERED OPERATORS AND PERTURBATION THEORY

A more general ordered operator form appears

on decomposing the generators into two additive parts

in which the subscripts indicate that the infinites-

imal parameter change is governed by the generators

G or G , respectively. The resulting finite

transformation function is

Then (7,15) can be written


Another derivation of the latter result pro-

.ceeds directly from the fundamental dynamical prin-

ciple, now applied to a change in the dynamical

characteristics of the system. Thus, for a system

with the Hamiltonian

an infintesimal change of the parameter X produces

a change in Hamiltonian, and hence of a canonical

transformation function <tii't2> ' *ascri^e<^ y

in which the dynamical variables and states that

appear on the right vary along the integration path

in accordance with the generatorexample of the Hamilton-Jacobi transformation we

havfi


In the latter form, the dependence upon X appears

in the two transformation functions <t,|t> and

<t|t2> « A second differentiation yields

in which the two terms are equal, the limits of

integration being alternative ways of expressing

the ordered nature of the operator product. One

can also write


and the general statement is

Hence the transformation function for the system

0 iwith Barailtonian H = H + H (X=l) can be obtained

formally as a power series expansion about A = 0 ,

and is thus expressed in terms of properties of the

The identification of this result with (7.52)

supplies the expansion of the ordered exponential

0system with Hamiltonian H


The transformation function construction given in

(7.52) is the foundation of perturbation theory,

whereby the properties of a dynamical system are

inferred from the known characteristics of another

system. The expansion (7.58) is the basis of re-

lated approximation procedures.

7.7 USE OF THE SPECIAL CANONICAL GROUP

The properties of the special canonical group

can be exploited as the foundation of a technique

for obtaining canonical transformation functions.

If we are interested in the Kami1ton-Jacobi transfor-

mation, with infinitesimal generator - H fit we


consider the extended transformation characterized

by

which includes infinitesimal displacements of the

canonical variables. On supposing such displacements

to be performed independently within each infinitesi-

mal time interval, as described by

and the extended transformation appears as a Hamil-

ton-Jacobi transformation with an effective Hamil-

tonian. The corresponding equations of motions are

which exhibits the independent changes in the

canonical variables that occur in a small time

interval. If the displacements Q and P area a

localized at a time t which means that

the infinitesimal generator becomes


the equations of motion imply a finite discontinuity

in the canonical variables on passing through the

time tQ ,

As an application of the latter result, the

arbitrary eigenvalues of canonical variables that

specify states at a certain time can be replaced by

convenient standard values, provided the compensat-

ing canonical displacement is included. Thus with

tx > t2

where the displacements localized at the terminal

tiroes,

do indeed convert the standard states into the

desired ones at times t.(-0) and t2(+0) , A

proof of equivalence can also be obtained from


(7.52), applied to produce the transformation

function of the system with Hamiltonian

H + I (p Q -Pq) from that with Hamiltonian H <

With the localized displacement (7.66), the ordering

of the exponential operator is immediate and

since the two exponential operators produce the

required canonical transformations at times t, and

7.8 VARIATIGNAL DERIVATIVES

A more significant use of the special canoni-

cal group appears on considering arbitrary displace-

ments throughout the interval between t. and t_

t23


It is now convenient to unify the canonical varia-

bles and write

An infinitesimal variation of X (t) produces thect

corresponding transformation function change

which defines the left variational derivative ofi

the transformation function with respect to X (t)

The operator structure of the displacements that

is used here,


refers explicitly to variables of the second kindi

where the X are anticommuting exterior algebract

elements and p is the operator that anticommutes

with all dynamical variables, but it also covers

variables of the first kind on replacing "anticommut-

ing" by "commuting". In particular, p •> 1 . The

effect of a second variation is obtained from (7.55)

as

and


where

It follows from the properties of the 6X1 , or ex-

plicitly from (7.74), that variational derivatives

are commutative, with the exception of those referr-

ing entirely to variables of the second kind, which

are anticommutative.

In the limit as t •* t1 in a definite sense,

we obtain operator products referring to a common

time in which the order of multiplication is still

determined by the time order, Thus

whereas


in which the minus sign appears only for pairs of

second kind variables. The difference between the

two limits is related to the commutation properties

of the fundamental dynamical variables

where the ambiguous bracket implies the commutator,

for variables of the first kind or a single variable

of the second kind, and becomes the anticommutator

for variables of the second kind. Now according to

the equations of motion for the system with Hamil-

tonian

the change of the dynamical variables in a small

time interval, as determined by the displacement, is


Hence

and

which combines the commutation properties of all

the fundamental variables.

Products of three operators at a common time

are expressed, by a formula of the type (7.76), as

and more generally, if p(x(t)} is any algebraic

function of the dynamical variables at time t , re-

stricted only to be an even function of variables

of the second kind, we have


in which the variational derivatives refer to times

differing infinitesimally from t as implied by

the particular multiplication order of the operators,

If the Hamiltonian operator is an algebraic function

of the dynamical variables, we can utilize this

differential operator representation through the

device of considering a related system with the

Hamiltonian AH , and examining the effect of an

infinitesimal change in the parameter A ,

This differential equation is analogous in structure

to a Schrodinger equation, with the transformation

function, in its dependence upon the infinite num-i

ber of variables X (t) , t, > t > t- , appearinga L ~ ~ e.

as the wave function, On integrating from A = 0

to A-l , the formal solution,


exhibits the transformation function as obtained by

a process of differentiation from the elementary

transformation function that refers only to the

special canonical transformations. A more general

form emerges from the decomposition

It must be noted that, since H and H are

necessarily even functions of the variables of the

second kind, the associated differential operators

are commutative. On applying first the exponential

operator constructed from H , we obtain

which could also be derived directly by appropriate

modification of (7.52). This produces a basis for

perturbation theory, by which the desired transfer-


mation function for Hamiltonian H is obtained from

a simpler one for Hamiltonian H by

differentiation.

7.9 INTERACTION OF TWO SUBSYSTEMS

The general dynamical situation of a system

formed of two sub-systems in interaction is des-

cribed by

where x1 and x_ refer to the dynamical variables

of the respective sub-systems. Correspondingly the

displacements X decompose into two sets X, and

X~ . In the non-interacting system described by

the Hamiltonian H , the two sub-systems are

dynamically independent and, in accordance with

the additive structure of the action operator, the

transformation function appears as the product of

that for the separate sub-systems,


The transformation function for the interacting

systems is therefore given by

In this construction, both sub-systems appear quite

symmetrically. It is often convenient, however, to

introduce an asymmetry in viewpoint whereby one

part of the system is tnought of as moving under

the influence of the other. The Hamiltonian

^l^xl' "*" ^12 xl ' o(t}) describes the first system

only, as influenced by the external disturbance

originating in the second system, with its variables

regarded as prescribed but arbitrary functions of

the time. The transformation function for this

incomplete system will be given by


and (7.91) asserts that the complete transformation

function can be obtained by replacing the prescribed

variables of the second system by differential

operators that act on the transformation function

referring to the second system without interaction,

The displacements have served their purpose, after

the differentiations are performed, and will be

placed equal to zero throughout the interval

between t. and t_ . They will still be needed

at the terminal times if the transformation functions

refer to standard eigenvalues, but will be set equal


to zero everywhere if this device is not employed.

Considering the latter for simplicity of notation,

the transformation function of the system with

Hamiltonian H is obtained as

which has the form of a scalar product evaluated by

the differential composition of wave functions.

Accordingly, from the known equivalent evaluations

of such products, other forms can be given to (7.94) ,

such as

and, for variables of the first kind, one can sub-

stitute composition by integration for the differ-


ential method here employed.

7.10 ADDENDUM: INTERIOR ALGEBRA AND THE ACTION

PRINCIPLE1"


Academy of Sciences, Vol. 48 pp. 603-611 (1962).

Reprinted from the Proceedings of the NATIONAL ACA&EM* OF SCIENCESVol. 48, No. 4, pp. 803-611, April, 1962.

EXTERIOR ALGEBRA AND THE ACTION PRINCIPLE, /*

BY- JULIAN SCHWINOBB

HAKVABD UNIVEBSITY

Communicated February %7, 19&%

The quantum action principle1 that has been devised for quantum variables oftype v = 2 lacks one decisive feature that would enable it to function as an instru-ment of calculation. To overeome_ this difficulty we shall enlarge the numbersystem by adjoining an exterior or Grassmann algebra.2

An exterior algebra is generated by N elements «„ « = 1 • • .V, that obey

which includes «,J = 0.

A basis for the exterior algebra is supplied by the unit element and the homo-geneous products of degree d

for d = 1 • • JV. The total number of linearly independent elements is counted as

The algebraic properties of the generators are unaltered by arbitrary nonsingularlinear transformations.

To suggest what can be achieved in this way, we consider a new class of special

604 PHYSICS: J. SCBWINGSR PBOC. N. A, 8.

variations, constructed as the product of £2n+1 with an arbitrary real infinitesimallinear combination of the exterior algebra generators,

The members of this class antieommute with the & variables, owing to the factorf2«-f.!, and among themselves,

since they are linear combinations of the exterior algebra generators. Accordingly,the generator of a special variation,

commutes with any such variation,

If we consider the commutator of two generators we get

where the right-hand side is proportional to the unit operator, and to a bilinearfunction of the exterior algebra generating elements,

The latter structure commutes with all operators and with, all exterior algebraelements. The commutator therefore commutes with any generator 0(, and thetotality of the new special variations has a group structure which is isomorphicto that of the special canonical group for the degrees of freedom of type » » = * » .This, rather than the rotation groups previously discussed, is the special canonicalgroup for the v = 2 variables.

We are thus led to reconsider the action principle, now using the infinitesimalvariations of the special canonical group. The class of transformation generators0 that obey

includes not only all even operator functions of the In variables {», but also evenfunctions of the ft that are multiplied by even functions of the N exterior algebragenerators e,, and odd functions of the J* multiplied by odd functions of the ««.The generators of the special canonical transformations are included in the lastcategory.

The concept of a Hermitian operator requires generalization to accommodatethe noncommutative numerical elements. The reversal in sense of multiplicationthat is associated with the adjoint operation now implies that

where X is a number, and we have continued to use the same notation and language

VOL. 48,1962 PHYSICS: J. SCMWINGSR 805

despite the enlargement of the aumber system. Complex conjugation thus hasthe algebraic property

The anticonimutativity of the generating elements is maintained by this operation,and we therefore regard complex conjugation in the exterior algebra as a linearmapping of the JV-dimensional subspaee of generators,

The matrix R obeys

which is not a statement of unitarity. Nevertheless, a Cayley parameterizationexists,

provided det(l + R) * 0. Then,

obeys

which asserts that a basis can always be chosen with real generators. There stillremains the freedom of real nonsingular linear transformations.

This conclusion is not altered when R has the eigenvalue — 1. In. that circum-stance, we can construct p(R), & polynomial in R that has the properties

The matrix

also obeys

while

since the contrary would imply the existence of a nontrivial vector » such that

or, equivaleatly,

which is impossible since p(—1) = 1. Now,

and if we use the Cayley construction for R' in terms of a real matrix rft which is

a function of R with the property r'(—1) *= 0, we get

600 PHYSICS: J, SCHWINGER PBOC. N. A. S,

This establishes the generality of the representation

where the nonsingular matrices a and p* are commutative, and thereby provesthe reality of the generator set

When real generators are chosen, the other elements of a real basis are suppliedby the nonvanishing products

for

A Hermitian operator, in the extended sense, is produced by linear combinationsof conventional Hermitian operators multiplied by real elements of the exterioralgebra. The generators Gt are Hermitian, as are the commutators i[Cf^, (?'!*].

The transformation function (n | rz) associated with a generalized unitary trans-formation is an element of the exterior algebra. It possesses the properties

and

where X symbolizes the summation over a complete set of states which, for themoment at least, are to be understood in the conventional sense. These attributesare consistent with the nature of complex conjugation, since

For an infinitesimal transformation, we have

where 0 is Hermitian, which implies that its matrix array of exterior algebraelements obeys

The subsequent discussion of the action principle requires no explicit referenceto the structure of the special variations, and the action principle thus acquires adual significance, depending upon the nature of the number system.

We shall need some properties of differentiation in an exterior algebra. Since«x2 = 0, any given function of the generators can be displayed uniquely in thealternative forms

where /a, /i, /, do not contain «x- By definition, /, and ft are the right and leftderivatives, respectively, of /(«) with respect to «x,

Vou 48, 1962 PBY8ICS: J. SCHWINGMR mi

If /(«) is homogeneous of degree d, the derivatives are homogeneous of degreed — I , and left and right derivatives are equal for odd d, but of opposite sign whend is even. For the particular odd function «w we have

so that

Since a derivative is independent of the element involved, repetition of the opera-tion annihilates any function,

To define more general second derivatives we write, for X <& it,

where the coefficients are independent of sx and «„. The last term has the alterna-tive forms

with

Then,

and

which shows that different derivatives are anticommutative,

A similar statement applies to right derivatives.The definition of the derivative has been given in purely algebraic terms. We

now consider /(« + $«), where St\ signifies a linear combination of the exterioralgebra elements with arbitrary infinitesimal numerical coefficients, and concludethat

to the first order in the infinitesimal numerical parameters.If this differential

property is used to identify dervatives, it must be supplemented by the require-ment that the derivative be of lower degree than the function, for any numbericalmultple of E1E2..EN can be added to a derivative without changing the differential

608 PHYSICS: J. SCHWINGMR PBOC. N. A. S.

form. Let us also note the possibility of using arbitrary nonsingular linear combi-nations of the «x in differentiation, as expressed by the matrix formula

where the If* are variations constructed from the exterior algebra elements. Wecannot entirely conclude that

This apparent incompleteness of the action principle is removed on stating theobvious requirement that the transformation function (n| r») be an ordinary numberwhen all Xt(r) vanish. Thus, the elements of the exterior algebra enter onlythrough the products of the X»(T) with &(r), as the latter are obtained by inte-grating the equations of motion, and terms in these equations containing «!••«»are completely without effect,

Let us examine how the transformation function (rijrj}cx depends upon the

X»(r). It is well to keep in mind the two distinct factors that compose XI,(T},

which follows directly from the differential expression,We shall now apply the extended action principle to the superposition of two

transformations, one produced by a conventional Hermitian operator G, an evenfunction of the ft, while the other is a special canonical transformation performedarbitrarily, but continuously in T, The effective generator is

where —Xt(r)dr is the special variation induced in |jt(r) during the interval dr.The objects -3T*(r) are constructed by multiplying %in+i with a linear combinationof the «x containing real numerical coefficients that are arbitrary continuous func-tions of r. To use the stationary action principle, we observe that each XI(T),as a special variation, is commutative with a generator of special variations andthus

The action principle then asserts that

since there remains an arbitrariness associated with multiples of «> • • ew> as in theidentification of derivatives from a differential form. No such term appears,however, on evaluating

Here,

VOL, 48, 1962 PHYSICS: J. SCHWINGER 809

while Xt'(r) is entirely an element of the exterior algebra. Thus, it is really theJT'(r) upon which the transformation function depends. An infinitesimal changeof the latter induces

where the summation index k has been suppressed. The repetition of such varia-tions gives the ordered products

which form is specific to an even number of variations. Complex conjugation isapplied to the exterior algebra elements in order to reverse the of multi-plication. In arriving at the latter form, we have exploited the fact that specialcanonical variations are not implicit functions of r and therefore anticommutewith the £* without regard to the r values. Thus, one can bring together the1m quantities &X(rl), • -SX^™), and this product is a multiple of the unit operatorsince pl =» 1. The multiple is the corresponding product of the exterior algebraelements MT'Cr1), • •&Z'(r2at), which, as an even function, is completely commuta-tive with all elements of the exterior algebra and therefore can be withdrawn fromthe matrix element. The reversal in multiplication sense of the exterior algebraelements gives a complete account of the sign factors associated with antieom-mutativity.

The notation of functional differentiation can be used to express the result.With left derivatives, we have

where the antieontmutativity of exterior algebra derivatives implies that

Here,

according to whether an even or odd permutation is required to bring r1, • • r2"1

into the ordered sequence. This notation ignores one vital point, however. Inan exterior algebra with N generating elements, no derivative higher than theNth exists. If we wish to evaluate unlimited numbers of derivatives, in order toconstruct correspondingly general functions of the dynamical variables, we mustchoose N = <*>; the exterior algebra is of infinite dimensionality. Then we canassert of arbitrary even ordered functions of the |»(T) that

In the alternative right derivative form, &T signifies that successive differentiationsare performed from left to right rather than in the conventional sense. We alsonote that the particular order of multiplication for operator products at a commontime is to be produced by limiting processes from unequal r values. As an appliea-

610 PHYSICS: 3. SCHWISGER Pnoc, N. A. S,

tion to the transformation function (n \ Tt)gx, we supply the even operator function

0 with a variable factor X and compute

which gives the formal construction

where the latter transformation function refers entirely to the special canonicalgroup.

The corresponding theorems for odd ordered functions of the f*(r) are

and

provided the states are conventional ones. Since the variations SX(r) have noimplicit T dependence, the factors pe can be referred to any r value. If this ischosen as n or T% there will remain one p operator while the odd product of exterioralgebra elements can be withdrawn from the matrix element if, as we assume, theproduct commutes with the states (n| and TJ). The sign difference between theleft and right derivative forms stems directly from the property {AST = —iXf.

As an example, let F(Q be the odd function that appears on the left-hand sidein the equation of motion

and

Some properties that distinguish the trace of the transformation function canalso be derived from the statement about odd functions. Thus,

The corresponding functional derivatives are evaluated as

since the X(r) are special canonical displacements, and this yields the functionaldifferential equations obeyed by a transformation function {n | Tt)0

x, namely,

since either side is evaluated as

YOL. 48, 1962 PHYSICS: J. SCHWINGMB 611

Accordingly,

which shows that the trace is an even function of the X'(r). The nature of thetrace is involved again in the statement

which is an assertion of effective antiperiodicity for the operators {(T) over theinterval T = n — T*. The equivalent restriction on the trace of the transforma-tion function is

or

* Publication assisted by the Air Force Office of Scientific Research.» These PBQCBKBINQS, 47, 1075 (1961).* A brief mathematical description can be found in the publication The Construction, and Study

o/ Certain Important Algebras, by C. Cfaevalley (1955, the Mathematical Society of Japan).Although such an extension of the number system has long been employed in quantum fieldtheory (see, for example, these PROCEEDINGS, 37, 482 (1951)}, there has been an obvious needfor an exposition of the general algebraic and group theoretical basis of the device.

and, in particular,

CHAPTER EIGHTGREEN'S FUNCTIONS

8.1 Incorporation of Initial Conditions 3318.2 Conservative Systems. Transforms 3358.3 Operator Function of a Complex 337

Variable8.4 Singularities 3408.5 An Example 3418.6 Partial Green's Function 343

8.1 INCORPORATION OF INITIAL CONDITIONS

The most elementary method for the construc-

tion of canonical transformation functions associat-

ed with parameterized transformations is the direct

solution of the differential equations that govern

the dependence upon the parameters. For time de-

velopment these are the Schrodinger equations


where H here refers to the differential operator

representatives of the Hamiltonian» at times

t or t0 , which depend upon the particularJ~ <&

choice of canonical representation. The desired

transformation function is distinguished among the

solutions of these Schrodinger equations by the

initial condition referring to equal times,

which means that the canonical transformation func-

tion is independent of the common time and is deter-

mined only by the relation between the descriptions.

This formalism is given an operator basis on writing

where the unitary time development operator

is to be constructed as a function of dynamical

variables that do not depend upon time, by solving

GREEN'S FUNCTIONS 333

the differential equations

-it is usetux to incorporate tne initial con-*

ditions that characteri-ze transformation functions,

or the time development operator, into the differ-

ential equations. This is accomplished by intro-

ducing related discontinuous functions of time -

the Green's functions or operators. The retarded

and advanced Green's functions are the matrices, in

some representation, of the retarded and advanced

Green's operators defined, respectively, by

where

together with the initial condition

The discontinuities of the functions n and n,

are expressed in differential form by

334

md


and therefor, in consequence of the differtial

equations (8.5) and the initial conditions (8.6)

both the retarded and advanced green,s operator

obey the inhomogeneous equations

The two Green's soperators are distingushed as the

solutions of these equations that by

GREENS'S FUNCTIONS 335

8.2 CONSERVATIVE SYSTEMS . TRANSFORMS

For a conservative system, in which t does

not appear in the Hamiltonian operator, the time

development operator and the Green's operators can

depend only upon the relative time,

and the defining properties of the Green's operators

appear as

which is evidently consistent with the adjoint rela-

tion

From these operators the unitary time developmen-

operator is constructed as


It is now possible to eliminate the time dependence

in the Green's operators by defining the transform

operators

As we have already indicated, since the time

integrations are extended only over semi-infinite

intervals these operators exist for complex values

of the energy parameter E , when restricted, to

the appropriate half-plane. The application of the

transformation to the differential equations (8.16)

yields


for both Green's operators which, as functions of the

complex variable E, are now defined by the respective

domains of regularity indicated in (8.17). Corres-

pondingly the adjoint connection now appears as

where the integration path, extended parallel to

the real axis, is drawn in the domain of regularity

appropriate to the Green's operator under considera-

tion.

8.3 OPERATOR FUNCTION OF A COMPLEX VARIABLE

Both Green's operators are given formally by

and therefore form together a single operator fun-

ction of the complex variable E , defined every-

where except perhaps on the common boundary of the

The inversions of (8.17) are comprised in


two half-planes, the real axis. Indeed, the form

(8.21), expressed in terms of the eigenvectors of

the operator H and a supplementary set of constants

of the motion y , shows that the singularities of

G(E) are simple poles on the real E axis, coin-

ciding with the spectrum of energy values for the

system. The construction of the Green's function

in some convenient representation, and an investi-

gation of its singularities will thus supply the

entire energy spectrum of the system together with

automatically normalized and complete sets of wave

functions for the energy states. For a system with

the Hamiltonian H(q , q) , for example, the

Green's function G(q , q' , E) could be obtained

by solving the inhomogeneous differential equation

On exhibiting the solution as

+

+


the desired information concerning energy values

and wavefunctions is disclosed. We should also

note the possibility of a partial Green's function

construction, which supplies information about a

selected group of states. Thus if we place the

eigenvalues q' equal to zero in (8.22), the differ-

ential equation reads

and the solution of this equation will yield the

energy values only for those states with

<E"Y*|0> 0 * &H these states are still repre-

sented in the further specialized Green's function

O

where the coefficients |<E'Y*|O>| obey

and give the probabilities for realizing the various

energy states in a measurement on the zero eigen-

value state 0> .


8.4 SINGULARITIES

The singularities of a Green's function can

be determined from the discontinuities encountered

on crossing the real E axis. Thus, for real E ,

according to the delta function construction

Note that this is also a measure of the non-Hermi-

tian part of G (E) for real E ,

There is no discontinuity and G is Herraitian at

any point on the real axis that does not belong to

the energy spectrum of the system, whereas a dis-

crete energy value is recognized by the correspond-

ing localized discontinuity. If the energy spectrum


forms a continuum beginning at EQ » the discontin-

uity for E > EO is

The existence of such a finite discontinuity for

every E > EQ implies that G(E) possesses a

branch point singularity at E = EQ , the contin-

uous line of poles extending from En to infinity

supplying the cut in the E plane. Thus the

precise nature of the energy spectrum for a system

is implied by the character of the singularities

exhibited by G(E) as a function of a complex

variable.

8.5 AN EXAMPLE

The elementary example of a single free

particle in space is described non-relativistically

in the r representation by the Green's function

equation


The solution

J«involves the double-valued function E of the

complex energy parameter, which must be interpreted

kas +i|E| for E < 0 in order that the Green's

function remain bounded as |r-r'| -> «• . Accor-ie i-

dingly, for E > 0 we must have E = -f-|E| 2 on

the upper half of the real axis and E = -|E|

on the lower half. Hence there is a discontinuity

across the real axis for E > 0 which constitutes

the entire energy spectrum, since the Green's func-

tion is always bounded as a function of E . The

discontinuity is given by

GREEN'S FPNCTIONS 343

where the integral in the latter form is extended

over all directions of the vector p = p n . On

comparison with (8.30) we see that the various states

of a given energy can be labelled by the unit vector

n , specified within an infinitesimal solid angle

dw . The corresponding wave functions of the r

representation are

where (dp) is the element of volume in the p space.

8.6 PARTIAL GREEN'S FUNCTION

The utility of a partial Green's function

construction appears in two general situations,

which can overlap. One or more compatible constants

of the motion may be apparent from symmetry conside-

rations and it is desired to investigate only states

with specific values of these quantities, or, one

may be interested for classification purposes in

constructing the states of a perturbed system which

correspond most closely to certain states of a re-

lated unperturbed system. Both situations can be


characterized as a decomposition of the complete

set of states into two parts» or subspaces» as

symbolized by

in which the measurement symbols , or projection

operators, obey

and where one seeks to construct the projected

Green's operator referring only to the subspace

M, ,

From the equation

one obtains

and


and we obtain as the determining equation for

where

If the two subspaces refer to distinct values of

constants of the motion for the complete Hamiltonian,

there will be no matrix elements connecting the sub-

spaces, , H_ = ?H, = 0 » and (8.42) reduces to the

fundamental form of the Green's operator equation,

which we write as

The second equation is formally solved, within the

subspace ML , by


(8.18), now defined entirely within the space ML

CHAPTER NINESOME APPLICATIONS AND FURTHER DEVELOPMENTS

9.1 Brownian Motion of a Quantum Oscillator 3479.2 Coulomb Green's Function 374

9.1 BROWNJAN MOTION OF A QUANTUM OSCILLATORt

tReproduced from the Journal of MathematicalPhysics, Vol. 2, pp. 407 -432 (1961).

347

Reprinted from the JOURNAL OF MATHEMATICAL Pavsicss Vol. 2, Mo. 3, 407-432, Jtinted S» V- S, A,

lay-June, 1961

Browaian Motion of a Quantum Oscillator

JCUAS SdffiWIUQES*Elt&wfd Uniserstfy, Csm^fi4ge9 Massachuseiis

(Received November 28,I960)

Aa action principle tec&nkjue for the direct coasputatioR of esEp^etatioti values is described aad illustratedm detail by a special physical example, tfe« effect on an oscillator of another physics! system. This simpleproblem has the advaatage of orobiaiiisg immediate physical applicability (e,f,, resistive damping ormaser amplification of a single electromagnetic cavity mode) with a slgniieaat idealisation of the complexproblems encountered in many-particle and relativiatic fieM titeory. Successive sections eemtain discussionsof the oscillator subjected to external forces, the oscillator loosely coupled io the external systemt aaimproved treatment of this problem and, finally, there is a brief account of a general fonrnsktkta.

IfTOGOTCTIG!?

HE title of this paper refers to an elementaryphysical example that we shall as© to illustrate,

at some length, a solution of the loHowimg methodologicalproblem, The quantum action principle1 is a differentialcharacterization of transformation functions, (o'fc 1 &%}»and thus is ideally suited to the practical computationof transition probabilities (which includes the deter-mination o£ stationary states). Many physical questionsdo not pertain to individual transition probabilities,however^ btit rather to expectation vaiyes of a physicalproperty for a specified initial state,

or, more generally, a mixture of states. Can one deviseais aciiSoa principle technique that is adapted to thedirect computation of such expeefcati&B values, withoutrequiring knowledge of the individual transformationfunctions?

The action principle asserts that {^^ 1}>

which expresses the fked numerical value of

But now, imagine that the positive and negative sensesof time development are governed by differtot dy-namics. Then the transformation function for the closedcircuit will be described by the action principle

ami

in which we shall take ^>la. These mutually complex-conjugate forms correspond to the two viewpointswhereby states at different times can be compared,either by progressing forward from the earlier time* orbackward from the later time. The relation betweenthe pair of transformation functions is such that

in which abbreviated notation the multiplication signsymbolizes the composition of transformation functionsby summation over a complete set of states, Ilf inparticular, the Lagraagiaa operators Z± contain thedynamical term X±(<)^T(<), we have

and, therefore^

* Supported by the Mi Force Office of Scientific Research(ARDCV1 Some reierenees are: Julias Scfewinger, Fhys. Rev. , 9M(!»51}j 91, ?J3 (!9»)j Phil Mag. 44, H?J (WSJ). The first twopapers also appear io Selected Papers OK Quantum Ele£tf®dyn®mfo$(Dover Pttbffcallons, New fork, »S8), A recent discussion iscontained in Julian Scbwinger, Proc. JMatl. Acad. Sd. U. S. 46,883 (I960).

where Xj. can BOW be identified. Accordingly, if asystem is suitably perturbed2 in a manner that dependsupon the time sense, a knowledge of the transformationfunction referring to a dosed time path determines theexpectation value of any desired physical quantity fora specified initial state or state mixture.

OSCILLATOR

To illustrate this remark we first consider an oscillatorsubjected to an arbitrary external force, as described bythe Lagrangian operator

a ESespite this dynamical language, a change in the Hamiltoroan

ratot of a system can be kineiratical in character, arising fromconsideration of another transformation along with (lie

dynamical aw generated by the Hamiltonkn. See Use last paperquoted in footnote I, sod Julias Schwinger, Proc Natl Acad.Sd. U. S. «, 1401 (1«0).

407

7

408 J . S C H W I N G E R

We shall begin by constructing the transformationfunction referring to the lowest energy state of the andunperturbed oscillator, (GfelQfe)-**, This state can becharacterized by

in which the complementary pair of nor*-Hermiiian The choice of initial state implies effective boundaryoperators y, iy>, are constructed from Hermitian conditions that supplement the equations of motion,

The equations of motion implied by the action principleare

and solutions are given by

together with the adjoint equation. Since we nowdistinguish between the forces encountered, in thepositive time sense, K+(t), K+*(i), and in the reversetime direction, K~.(t), K~*{t), the Integral must betaken along the appropriate path. Thus, when I isreached first in the time evolution from 4, we have

together with the similar adjoint equations obtained byinterchanging the ± labels. For convenience, stepfunctions have been introduced:

or, equivalently, by the eigenvector equations

Since the transformation function simply equalsunity if K+***K~ and K+*~K-*, we must examine theeffect of independent changes in K+ and JC_, and ofJC+* and K-", as described by tie action principle

The requirement that the transformation functionreduce to unity on identifying X+ with £_, K+* withK~.*, is satisfied by the null sum of all elements of Ge,as assured by the property %.+>!_= 1.

An operator interpretation of Gt is given by thesecond variation

Generally, on performing two distinct variations ia thestructure of L that refer to parameters upon which

operatorss q,p, by

Hence in effect we have

while on the subseqent return to time t,

The solution of the resulting integrable differentialexpression for log(ot2/ ot2)k is given by

We shall also havcasion to use the odd function

Note

in a matrix notation with

and

B R O W N 1 A N M O T I O N O F A Q U A N T U M O S C I L L A T O R 409

If we are specifically interested in {afef^s)'8'*, whichsupplies all expectation values referring to the initialstate nt we myst extract the coefficient of (3t*'y'l)*/nlfrom aa exponential of the form

All the terms that contribute to the required coefficient

s A discussion of non-Hermitjan representations is given mLectures on Quantum Mec&snies (Les HoycheSj 1955) unpublished.

iims, unaer the influence oi these lorces, me states|%) and (0^1 become, at the time fe-f-O, the statesiy%) &&d (yf%], which are right an<l left eigenvectors!respectively, of the operators y(fe) and yf(J»). OH takiaginto account arbitrary additional forces* the transforma-tion function for the closed time path can be expressed

where the expectation values and operators refer to thelowest state and the dynamical variables of the un~perturbed oscillator. The property of Go that the sumof rows and columns vanishes is here a consequence ofthe algebraic property

The choice ol oscillator ground state Is no essentialrestriction since we eas now derive the analogousresults for any initial oscillator state. Consider, forthis purpose, the impulse forces

the effects of which are described by

Tfae eigenvectors of the Bon-Hermitiaa canonicalvariables are complete and have an intrinsic physicalInterpretation in terms of g and p measurements ofoptimum compalibility.3 For otir immediate purposes.,however, we are roore interested in the unperturbedoscillator energy states. The connection between the twodescriptions can be obtained by considering theunperturbed oscillator transformation function

in which the multiplication order follows the sense oftime development. Accordingly*

the dynamical variables at a gives time are not explicitly &nddependent, we have

We infer She norincgative integer spectrum o! yry( andthe corresponding wave functions

since

Accordingly^ a non-Hermitian canonical variable trans-formation function can serve as a generator for thetransformation (unction referring to unperturbedoscillator energy statesr

in which

Now

W h c h Y i e l s d s

a s


are contained m

and since the elements of G& are also given by un-perturbed oscillator thermal expectation values

inhere we have written

where the latter version is obtained from

andthe designation {n}$ is consistent with its identificationas (yW*

The thermal forms can also be derived directly bysolving the equations of motion, in the manner used tofind (0^ I Qi%}K*. On replacing the single diagonal element

in which the nth Lagaerre polynomial has been, intro-duced on observing that

we find the following relation

One obtains a much neater form, however, from whichthese results can be recovered, on considering anInitial mixture of oscillator energy states for whichthe nth state is assigned the probability

instead of the effective initial condition {^^O. Thisis obtained by combining

s*ith the property of the trace

and

can be interpreted as a temperature. Then, since

We also hav;

and therefore} effectively,

Hence, to the previously determined y^(t) is to beadded the term

we obtain

with

and in whichand correspondingly

which rej>reduces the earlier result.

bby the statistical avera

T h u s


As an elementary application let us evaluate the there appearsexpectation value of the oscillator energy at time t\tot a system that was in thermal equilibrium at time (5and is subsequently disturbed by an arbitrarily time-varying force. This can be computed as

It may be worth remarking, in connection with theseresults, that the attention to expectation values doesThe derivative i/SK+*(k) supplies the factor * j • * .1 ».•!• .,. ,, ,

not (Jepnve uj of ti,e ability to compute individualprobabilities. Indeed, if probabilities for specific oscil-lator energy states are of interest, we have only toexhibit, as functions of y and y1, the projection operatorsfor these states, the expectation values of which are the

the subsequent variation with respect to K.(h) gives required probabaitics. frow

is represented by the matrix

and the required enetgy expectation value equals *""> therefore,

u{n)0+u

More generally, the expectatian values of all functions of y((.) and y«(fc) are known from that of

exp{ - i[X/{h)+MyOi)]}, n whicl, we vs jntro<iuced a notation to indicate thisand this quantity is obtained on supplementing K+ ordered multiplication of operators. A convenientand K-t* by the impulsive forces (note that in this use generating function for these projection operators isof the formalism a literal eoraplex-conjugat© relationshipis not required)

K+(t)-\t(t h), g(ves tj,e probability of finding the oscillator in theK^.*(t)"iit{l—lt+0), nth energy state after an arbitrary time-varying force

Then

and we observe that

Accordingly,

which involves the special step-function value

JtsmaHvelv. if Wf* rh*msf>

412 j . S C H W 1 N G E R

has acted, il it was initially m a thermal mixture ofstates,

To evaIlia If?

rhls ferai; and the implied transition probabilities, haveilready been derived in another connection,4 and weshall only state the general resell here;

in which ra> and ^< represent the larger and smaller ofthe two integers n and nf.

Another kind of probability is also easily identified,that referring to the continuous spectrum erf theMermitian operator

from which foilo^

we first remark that

If we multiply this result by exp(—»#Y) and integratewith respect to p'/ln: from — « to «, we obtain theexpectation value of Qf {**) ""* $1 which is the probabilityof realizing a value of q(t\) in a unit interval about (/i

Still another derivation ojf the lormuk giving thermalexpectation values merits attention. Now we let thereturn path terminate al a different time l/=4~"^»and on regarding the resulting transformation functionas a matrix, compute the traces or rather the tracerattn

which reduces to unity in the absence of externalforces. The action principle again describes the dqsend-enee upon JC±*(/}, K±(t) through the opera-tore y±0)»y^(f) which are related to the forces by the solutionsof the equations of motion, and, is particular*

and <m referring 10 the previously used Lagiier:polynomial sum formula, we obtain

as one shows with a similar procedure^ or by directseries expari&iors, Tfeeref&re.,

Here

or

In addition to describing the physical situation o(initial thermal equiliibrisimt this result provides agenerating function for the Individual transitionprobabilities between oscillator energy states,

Next we recognize that the structure ol the trace impliesthe effective boundary condition

w i t h

w h r e r

julan Schwinger Phy, Rev 91,728,1953

For this purpose,we place

abd obtain

B R O W N I A N M O T I O N O F A Q U A N T U M O S C I L L A T O R 413

Let us consider

ixW\y-(t*')\ti-Z« <aVb-(V)k%),where we require of the a representation only that ithave no explicit time dependence. Then

a&e

which is the slated result.T%e ffffrfiw? initial rnrtHitmn nnw aranp&rs asi

in which /,#,({ characterizes the external system andQ(i) is a Hermltlan operator of that system.

We begin our treatment with a discussion of thetransformation function {(»1*^*9*** that refers initiallyto a thermal mixture at temperature $ lor the externalsystem, and to an Independent thermal mixture attemperature$9 for the oscillator. The latter temperaturecan be interpreted literally, or as a convenient para-metric device lor obtaining expectation values referringto oscillator energy states. To study the effect of thecoupling between the oscillator and the externalsystem we supply the coupling term with a variableparameter \ and compute

and the action principle supplies the following evalua-tion of the trace ratio: where the distinction between the forward and return

paths arises only from the application of differentexternal forces &&(!} OR the two segments of the closedtime contour. The characterization of the externalsystem as essentially macroscopic now enters throughthe assumption that this large system is only slightlyaffected by the coupling to the oscillator. In a corre-sponding first approximation, we would replace theo|>erat0rs Q&(t) by the effective aumerical quantity(CCO)*- The phenomena that appear in this order ofaccuracy are comparatively trivial, however^ and weshall suppose that

where the time variable in K+ and £„ ranges from£g to It and from I/ to l\t respectively. To solve the givenphysical problem we require that &.({) vanish le theinterval between // and t$ so that all time integrations3l"P *ixt**nrfw'! HpfWfPfTi to anH f.t TThj^n einff

which forces us to proceed to the next approximation.A second differentiation with respect to X giveswfaat has been evaluated equals

aad by addmg the remark that this ratio continues Uexist on making the complex substitution

the desire<l formula emerges asThe introduction of an approximation based upon theslight disturbance at the macroscopic system convertsthis into

EXraRHAt SYSTEM

This concludes our preliminary survey of the oscil-lator and we turn to the specific physical problem &iinterest: An oscillator subjected to prescribed externalforces and loosely coupled to an essentially macroscopicexternal system. All oscillator interactions are linearin the oscillator variables, as described by the Lagran-glan operator

whre


and we have also discarded all terms containingy(l)y(t') and /(«}/(<')• The latter approximationrefers to the assumed weakness of the coupling of theoscillator to the external system, for, durmg the manyperiods that are needed tor the effect of the coupling toaccumulate, quantities with the time dependencee±«0e(H-i'> ^rjj} Ijecome suppressed in comparison withthose varying as g^***"™1'). At this point we ask whateffective term in an action operator that refers to thedosed time path of the oscillator would reproduce thisapproximate value of (9/dX)*(fs\t,} at A=>0. The

complete action that satisfies this requirement, withXs set equal to unity, is gives by

The nonlocal character of these equations is not verymarked if, for example, the correlation between Q(t)and Q(f} in the macroscopic system disappears when\t—lf\ is still small compared with the period of theoscillator, Then, since the behavior of y(t) over a shorttime interval is given approximately by «"**', the matrixA (<—;') is effectively replaced by

The application of the principle of stationary actionto this action operator yields equations of motion thatare nonlocal in time, namely, *

and the equations of motion read

The latter set is also obtained by combitning the formaladjomt operation with the interchange of the + and~ labels attached to the operators and K{t). Anothersignificant form is conveyed by the pair of equations

One consequence

and

ana

it should be noted that A+,(a) and 4_+(a) are reaKjsitive Quantities since

It also follows from

that

is real. Furthermore

where

whers

and

together with

and

B R O W N I A N M O T I O N O F A Q U A N T U M O S C I L L A T O R 415

and the required solution is given by

and, for the choice at an initial thermal mixture,

We now find that

The net result of this part of the discussion is toremove al! explicit reference to the external system asa dynamical entity. We are given elective equations ofmotion for y+ and y^ that contain the prescribedexternal forces and three parameters, the angularfrequency »(£»»»), 7, and <t, the latter pair beingrelated by the temperature of the macroscopic system.

We have not yet made direct reference to the natureof the expectation value for the macroscopic system,which is now taken as the thermal average;

where S is the energy operator of the external system.The implication for the structure of the expectationvalues is contained in

The corresponding solutions for y^(tj are obtained byinterchanging the ± labels in the formal adjointequation.

The differential dependence of the transformationfunction (fe \ ti)»t>tK* upon the external forces is describedby these results, and the explicit formula obtained onintegration is

which employs the formal property

On introducing the "time Fourier transforms, howevithis becomes the explicit relation

and we conclude that

which is a positive even function of . As a consequentwe have

which can also be written as

so that

snd « emerges as the real quantity

although the simplest description of G is supplied by

The accompanying boundary conditions areA n o t h e r w a y o f p e r e a e n t i n g t h i s r e s u l t s i s

w h i c h s u p p l i t e s t h e i n i t i a l c o n d i t i r o n f o r t h e s

e q u a t i o n o f m o t i o n


the differential equation

(where f indicates differentiation to the left) hconjunction with the initial value

We note the vanishing sum of ail G elements, and thatthe rote of complex conjugation in exchanging the twosegments of the dosed time path is expressed by

which is to say that

It will be observed that only when

is Gfg»(<— f t , f—1%) independent of k and a function of(—('. This dearly refers to the initial physical situationof thermal equilibrium between the oscillator and theexternal system at the common temperature <>»=$> 0,which equilibrium persists ia the absence of externalforces. If the initial circumstances do not constitutethermal equilibrium, that will be established in thecourse of time at the macroscopic temperature tf>0.Thus, all reference to the initial oscillator temperaturedisappears from G»e>(t— Is, (—fe) when, for fixed t—t.

from which a variety of probability distributions andexpectation values can be obtained.

The latter calculation Illustrates a general character-istic of the matrix G(t,(), which is implied by thelack of dependence on the time ti. Indeed, such aterminal time need not appear explicitly in the structureof the transformation function {k\i%)K* and all timeintegrations can range from it to +<*>. Then ti isimplicit as the time beyond which K+ and K~ areidentified, and the structure of G must be such as toremove any reference to a time greater than t\. In thepresent situation, the use of an impulsive force at hproduces, for example, the term

in which K+ and K~ are set equal. Hence it is necessarythat

which says that adding the columns of G(/,f) givesretarded functions of/ £—/, while the sum of rowssupplies a vector that is an advanced function oft—t'. In each instance, the two components must have asero sum- These statements are immediately verified forthe explicitly calculated C?^Q>?(£—la, ?—t%) and followmore generally from the operator construction

and the boundary conditions

A more symmetrical version of this differential equatio!is given by

The previously employed technique of impulsive force;applied at the time t\ gives the more general result

and is expressed by

md similarly that

The thermal relaxation of the oscillator energy is

for, as we have already noted in connection with Qproducts,

d e r v i e d f o r m

B R O W N I A H M O T I O N O F A Q U A N T U M O S C I L L A T O R

and

Our results show^ incidentally, that

This is a formula for the direct computation, of expecta-tion values of general functions of y(t) and yfO). A lessexplicit but simpler result can also be givesi by means ofexpectation values for functions of the operators

Another general property can be illustrated by ourcalculation, the posttlveness of -~iG(t/)+~t

Let us recognize at once that

and therefore that the fluctuations of y(t)t yr(/) can beascribed to the effect of: the forces Kft jf/, which appearas the qtiantam analogs of the random forces In theclassical l.angevin approach to the theory of theBrownian motion. The change in viewpoint is accom-plished by introducing

and it is clearly necessary that each term obey sep-arately the positiveness requirement. The first term is

where we assume, just for simplicity f that the functionsw(l), ?(/} vanish at the time boundaries* Then, partialtime integrations will replace the operators % y* withKt, K,t.

To carry this out, however, we need the followinglemma on time-ordered products:

tnd the required property of the second term, followsrom the formula

which involves the unessential assumption that B(i)vanishes at the time terminals; and the hypothesis that[X(0,#(0] and [dB(t}/dt, B(tYj are commutativewith all the other operators. The proof is obtained byreplacing B(t) with XJS(0 aad diiferentlatJBg withrespect to X,

All the information that has been obtained aboutthe oscillator is displayed on considering the forces

and making explicit the effects ofequivalent time-ordered operators:

Tfees, a partial integration, yields

417

418 j . S C H W I N G E R

according to the hypothesis, and the stated resultfellows &n integrating this differential equation,

Tbe structure of the lemma Is given; by the rearrange-ment

Such expectation values are to be understood as effectiveevaluations that serve to describe the properties of theoscillator under the circumstances that validate thevarious approximations that have bees used,

It will be observed that when » is sufficiently largeto permit the neglect of all other terms,

for then

and the seEse ol operator multiplication is no longersignificant. This is the classical limit, for which

wb ere we have placed % —• -«_—tt% &$, ™ tr_ ~ i f . Onintroducing real components of the raadom force

Kfiri^+iKd, ir/'-z-Htfi-flCt),the classical limiting result reads

The fluctuations at different times are independent. IIwe consider time-averaged forces,

which is the Gaussian distribution giving the probabilitythat the force averaged over a time interval &£ willhave a vatae within a small neighborhood of the pointK'. In this classical limit the fluctuation constant a isrelated to the damping or dissipation constant 7 andthe macroscopic temperature $ by

ine last term involves aiscaromg a touu time oenvauvethat will not contribute to the final result. To evaluate[4 3] we must refer to the mean-teg of K/ and K/ thatis supplied by the actual equations of motion.

which is also proportional to the unit operator. Ac-cordingly,

where

we find by Fourier transformation thatand complex conjugation yields the analogous result fornegatively time^-ordered products.

With the aid of the differential equation obeyed byG, we now get

The elements of this matrix are also expressed b;

Our simplified equations can also he applied tosituations io which the external, system is not at thermalequilibrium. To see this possibility let us return to thereal positive functions A,+(ta)t A+~(<a) that describe theexternal system arscl remark that, generally,

and we immediately find a commutator that is amultiple of the unit operator.

B R O W H 1 A H M O T I O N O F & Q U A H T 0 M O S C I L L A T O R 419

These properties can be expressed by writing

where 0(w) is a real eves function that cas range from— «* to 4" °°. When only one value of « is of interest^all conceivable situations for the external system can bedescribed by the single parameter & the reciprocal ofwhich appears as an effective temperature of themacroscopic system. A new physical domain thatappears m this way is characterised by negativetemperatarej $<G. Since & is an iatrmsicaJly positiveconstant, it is y that will reverse sign

aad the effect of the external system on the oscillatorchanges from damping to amplification,

We shall discuss the following physical sequence. Attime Ig the oscillator, in a thermal mixture of states attemperature $#, is acted on by external forces whichare present lor a time, short in comparison with 1/J7J,Altar a sufficiently extended interval "-(/i—fc) suchthat the amplification factor or gain is very large,

measurements are made in the neighborhood of timet\. A prediction of all such measurements is contained IBthe general expectation value formula. Approximationsthat convey the physical situation under considerationare given by

with

and, on defining

-exp[- ««>.,+ (l-«-i«i")->)wJ

which Implies that

From the appearance of the combinations #,f—/*_**/*,Xf.~-)u™X only, we recognize that noaoesmmtitativity ofoperator multiplication Is no longer significant, aadthus the motion of the oscillator has been amplified tothe classical level. To express the consequences mostsimply, we write

Thus, the oscillator coordinate y(l) is the amplifiedsuperposition of two harmonic terms, one of definiteamplitude and phase (signal), the other with random,amplitude and phase (noise), governed by a two-dimensional Gaussian probability distribution.

These considerations with regard to amplification canbe viewed as a primitive model of a ntaser device,6

with the oscillator corresponding to a slogte mode of aresonant electromagnetic cavity, aod the externalsystem to an atomic ensemble wherein, for a selectedpair of levels, the thermal population inequality isreversed by some meaas such as physical separation orelectromagnetic pumping,

m IMFEO?E0 TEEA1MEHT

In this section we seek to remove some of the limita-tions ot the preceding discussion. To aid in dealingsuccessfully with the Ronlocal time behavior of theoscillator, it is conveaient to replace the non-Hermitianoperator description with one employing Hermitlanoperatois. Accordingly^ we begin the developmentagain, now using the Lagrangian operator

where Q has altered its meanirsg by a omstaat factor.One could also include a& external prescribed ferce thatjs coupled to p* We repeat the previous approximateconstructioii of the transformation function (lsifeVfl*JF*which proceeds by the introduction of an effectiveaction operator that retains only the simplest correlationaspects of the external system^ as comprised in

s A sinsilar nwtei has been discussed recently by R. Ser!>erand C. E, TOWMS, Symposium an Qusntum Ele&sfmm (ColumbiaUmwts&y Press, New York, I960).

we obtain the time"-mQepeadent result

and

J . SCH W! M G E R420

The action operator, with no other approximations, is The accompanying boundary conditions are

and the implied equations of motion,, presented assecond-order differentia! equations after eliminating

or> more conveniently expressed,

which replace the nom-Herrmtiaa relations

anc

Note that il is the intrinsic oscillator frequency we thatappears here since the initial condition refers to athermal raisture oi unperturbed oscillator states,

The required solution of the equation lor q~~~g-$,eats be written as

It will be seen that the adjoint operation is equivalentto the interchange of the ±labels.

We define

Implicit is the time /i as one beyond which F~.~F+

equals zero. The initial conditions Cor the secondequation, which this solytiou supplies, are

arm

together witl

and

which enables us to present the integro-differenlialequations as

and

The Green's function that is appropriate lor theequation obeyed by q.++•§-- is defined by

&nd

are

and

B R O W N I A N M O T I O H O F A Q U A N T U M O S C I L L A T O R 4 2 1

and the two real factions are related by The latter obeys

The desired solution of the second differential equationis

and its elements are given by

whereWe note the identifications

It is also seen that the mm of the columns of G isproportional to (*>(£—/)? while the sum of the towscontains only <?«(*--0-

We shall suppose that Gr(t~~^} can have iw> more thanexponential growth, ^g«^~^^ as |—|r--» «?. Thee thecomplex Fourier transformis a real symmetrical mnctioa ol its two arguments.

The differential description of the transformatikmfunction that these solutions imply is indicated by

exists in the upper faalf-plaae

and is given explicitly by

and the result oi integration isHere

'bb can also be displayed m the matrix form sr, since (A~,+-~A^~)(&) is an odd function ol <&}

We have already remarked on the generality of therepresentation

with

and thus we shall write

and

422 J. S C H W I N G E R

axis of fs. As to the negative real axis, GKI")"1 Is araonotordcaliy decreasing function of 1"*= that beginsat -I- * for £= — <» and will therefore have no zero onthe negative real axis if it is still positive at a;=0,The corresponding condition is

which gives

Since IMS is art even function of f, it also representsthe Fourier transform of Ga in the lower half-planeIm£<-«.

If the effective temperature is positive and finiteat all frequencies, $(&?)>0, <?(!*} can have no complexpoles as a function of the variable £*. A complex poleat ft^x+iy, y+0, is a zero of G(f)~l and requires that

tinder these circumstances a—0, for G(%), qua- functionof f!, has no singularity other than the branch line onthe positive real axis, and the f singularities are thereforeconfined entirely to the real axis. This is indicated by

which is impossible since the (jusntity in bracketsexceeds unity. On letting y approach zero, we see thata pole of G(£) can occur at a point £=w'2>Q only ifs(t/)~0. If the external system responds through theoscillator coupling to any impressed frequency, <j(a»)>Gfor all a and no pole can appear on the positive real *nd B(t*>2) is the positive quantity

Some integral relations are easily obtained by of regularity. Accordingly,comparison of asymptotic forms. Thus

The integral relations mentioned previously can beexpressed in terms of these Green's functions. Thus,

while, in the limit of small positive r,

while setting {=0 yields

The Green's functions are recovered on using thiinverse Fourier transformation

which indicates the initial effect of the coupling to theexternal system.

The function S(u*} is bounded, and the Green'sfunctions mast therefore approach sero as 11~~ t1 \ ~~t «s.Accordingly, all reference to the initial oscillator condi-tion and to the time k must eventually disappear.For sufficiently large t—It, f—h, the function »(«— />,where the path of integration is drawn in the half-plane

and

B R O W N I A N M O T I O N OF A Q U A N T U M O S C I L L A T O R 423

with»(»)- («M«">-!)->,

which describes the oscillator in equilibrium at eachfrequency with the external system. When the tempera-ture is frequency independent, this is thermal equilib-rium. Note also that at zero temperature »(w)~0, andG(t~ <%+ is characterized by the temporal outgoingwave boundary condition—positive (negative) fre-quencies (or positive (negative) time difference. Thesituation is similar for G(t~~t'}~~ as a function of £'—t,

It can no longer be maintained that placing ft>=tfremoves all reference to the initial time. An intervalmust elapse before thermal equilibrium is establishedat the common temperature. This can be seen byevaluating the fe derivative of w(<~fe, t'—t^j:

for if this is to vanish, the integrals involving Gn say,must be expressible as linear combinations of Gf(t—t$)and its time derivative, which returns us to the approx-imate treatment of the preceding section, including theapproximate identification of o^ with the electiveoscillator frequency. Hence $s»$ does not representthe initial condition of thermal equilibrium betweenoscillator and external system. While it is perfectlyclear that the latter situation is described by the matrix

G»(<—(*)» a derivation that employs thermal equilib-rium as an initial condition would be desirable,

The required derivation is produced by the device ofcomputing the trace of the transformation function(fe'Ha)^ in which the return path terminates at thedifferent time t,'**f,~T, and the external force F_{»)is zero in the interval between (j and It. The particularsignificance of the trace appears on varying the param-eter \ that measures the coupling between oscillatorand external system:

The operators ft are needed to generate infinitesimaltransformations of the individual states at the corre-sponding times, if these states are defined by physicalquantities that depend upon X, such as the total energy.Th^re is no analogous contribution to the trace,however, for the trace is independent of the representa-tion, which is understood to be defined similarly at t>and tj, and one could use a complete set that does notrefer to X, More generally, we observe that G\{lt)bears the satn« relation to the (VI states as does 6n(<j)to the states at time % and therefore

Accordingly, the construction of an effective actionoperator can proceed as before, with appropriatelymodified ranges of time integration, and, for theexternal system, with

and, therefore,

The corresponding asymptotic form of the matrixG(t~(t, t'—tt) is given by

Ms trace structure implies that

or, since these correlation functions depend only on

Butt1-t2) reduces to

424 J , S C H W I N G E R

WMCB nas seen written tor external torees tftat areaero until the m&m&nt t% has passed.

Perhaps the simplest procedure at this point is toask for the dependence o£ the latter solution upon t$,for fixed T, We find that

time pjiiereiKesi that

ivhicfa if -"'so expressed by

The equations of motion for ^> it &ra given by

on using the relations

These are supplemented by the eciuation lor f_(/) inthe interval from I/ to fe:

Therefore,

smce, with positive time argument;

The utility of this result depends upon the approach ofthe Green's functions to sera with Increasing magnitudeol the time argume&t, which is assured, after making thesubstitution r -* ijff, under the circumstances we haveindicated. Then we can let *,-+ — » and obtain

and the effective bcnmaary condition

The equation lor $-—#$. is selved as before.

wit!whereas

as anticipated.Our results determiae the trace ratio

where H is the Hamiltonian op^aU>r of the completeSystran, and the substitution T~~»-i$ yidds the trans-formation ftiEtction

B R O W H I A N M O T I O N O F A Q U A N T U M O S C I L L A T O R 425

In addition to the trace ratio, which determines thethermal average transformation function {414}sF*with its attendant physical information, it is possibleto compute the trace

which describes the complete energy spectrum andthereby the thermostatic properties of the oscillator inequilibrium with the external system. For tMs purposewe set F±—0 lor i">t% and apply an arbitrary externalforce F-{%) in the interval from £/ to 4. Moreover, thecoupling term between oscillator and external system"m the effective action operator is supplied with thevariable factor X (formerly Xs). Then we have

ana

with

We can also write

where

What is asserted here about expectation values in thepresence of ari external field F($ becomes explicit onwriting

with the accompanying boundary condition

where ?-(£) obeys the equation of motion

and indicating the effect of j±(t} by equivalent time-ordered operators,

Thin

and the properties of $~~{g}$F, which are mdeperideatof PS are given by setting F—0 m the geaeral result.In particular, we recover the matrix identity

and the requirement of periodicity. We em now placeF~_™0 is the differential equation for the tracef andobtain

which is a statement 01 periodicity for the intervalT«l/—Ig, The solution of this equation is

The relation between w and <?.—G? can then bedisplayed as a connection between symmetrical prodactand commutator expectation values

The periodic Green's function. Is given by the Fourierseries

with

where the Green's function obeys

J . S C H W I N G E R

vanishes at infinity in this cat plane. Hence

426

and

where, it is to be recalled,

where the value

•eproduces the pole of G-'EO/t-P) at ?2=0, We also•ecogmse, on relating the two forms,

so that the iategrand has BO singularities at wT~ 2w [ n \.Now we have

that

which, together with the initial conditionThe positive value of the right-hand side as i*? —* 0shows that <f(a) approaches the zero frequency limitingvalue o! ?r from below, and the assumption that a(&s)>0for all w implies

where the lower limit is approached as &; —* .A comparison of asymptotic forms for G~l{$) shows

that

yields

We have already introduced the function

and examined some of its properties for real andpositive jl_j,(w}, A+-(a), This situation is recoveredon makine the substitution T—*i8, and thus

The introduction of the phase derivative can also beperformed directly in the structure &i G~~l({),

the existence oi which for all ft>Q requires thatG^l($) remain positive at every value comprised inj"*— — (2rn/ft)1, which is to say the entire negative faxis including the origin. The condition

and equating the two values for G~"(Q) gives

is thereby identified as a stability criterion. To evaluatethe summation over » most conveniently we shall givean alternative construction for the function lag(G~l(()/—I1), which, as a function of fl, has all its singularitieslocated on the branch line extending from 0 to <* arid

We now have the representation

We shall suppose that the stability requirement iscomfortably satisfied, so that the right-hand side ofthe equation for eot$>(&t} is an appreciable fraction ofwe2 at sufficiently low frequencies. Then tan^u is verysmall at such frequencies, or ?(iu}"v*v and this persistsuntil we teach the immediate neighborhood o! thefrequency &Ji<wt such that

That the function in question, ReG^^-HD), has azero, follows from its positive value at w^O and itsssyraptotic approach to — <# with indefinitely inereasmgfrequency. Under1 the conditions we have described.,with the major contribution to the integral comingfrom high frequencies^ the gero point is given approxi-mately as

ind somewhat more accurately byThe second factor can be ascribed to the oscillator,with its properties modified by interaction with theexternal system, The average energy of the oscillatorat temperature $~0rl is therefore gives by


aaa the summation tormula derived worn the pnxtuciform of the hyperbolic sine function,

gives us the desired result

in which the temperature dependence of the phase^(w) is not to be overlooked. In an extreme faigh-temperatyre limit, such that &i$<sCl for all significantfrecitiencks, we have

As we shall see, B Is less than, unity, but only slightlyso under the circumstances assumed,

In the neighborhood of the frequency w(, the equationthat determines ^(<y) can be approximated by

where

with the definitionand the simple classical result £—$ appears when{Q% is proportional to $„ The oscillator energy alxero temperature is given by

Hence, as w rises through the frequency &»i} ^ decreasesabruptly from a value close to w to one near zero. Thesubsequent variations of the phase are comparativelygradual, and <p eventually approaches aero as <y —* *>.A simple evaluation of the average oscillator energycan be given when the frequency range «>wi overwhich e(w) is appreciable in magnitude is such thatj3w^j, Ttiere will be n& significant temperature varia-tion in the latter domain a&d in particular &n shouldbe essentially temperature independent Thes, since— (I/-$}(&%>/$&} in the neighborhood of on closelyre^mbles ft(w—wi), we haw approxiisately

and the osctllaior coEtrtbution to the specific heatvanishes.

The following physical situation has consequencesthat resemble the simple model of the previous section.For values o? to<w0J a(«) tanh(|t^)4Cw«Si and *(w)differs significantly from zero until one attains fre-quencies that are large in comparison with &j». Themagnitudes that a{&?) can assume at frequencies greaterthan &}g is limited only by the assumed absence of rapidvariations and by the requirement of stability. Thelatter is generally assured if

which describes a simple oscillator of frequency wj,with a displaced origin of eaergy,

Note that with $>(«} wry small at a frequencyslightly greater than w and zero at infinite frequency,eye have

428

Related integrals are

J . SCH W I N G E R

The farther concentration on the immediate vicinityof &fi, 1 &f—^i j -^^ gives

and

The latter result confirms that B< i. A somewhat moreaccurate formula lor ,8 fa

If the major contributions to all these integrals comefrom the general vicinity of a frequency «»««, wecan mate toe crude estimates

which clearly identifies B<1 with the contribution tothe integral J"AaiJS(in'') that comes from the vicinity ofthis resonance of width t at frequency «i, although thesame result is obtained without the last approximation.The remainder of the integral, t—B, arises from fre-quencies considerably higher than an according to ourassumptions.

There is a similar decomposition of the expressionsfor the Green's functions. Thus, with Of,

The second high-frequency term will decrease veryquickly on the time scale set by I/MI,. Accordingly, inusing this Green's function, say in the evaluation of

Then neither the energy shift nor the deviation of thefactor B from unity are particularly significant effects.

The approximation of Re<3{aj-HQ} as B™1^!*"™^}evidently holds from zero frequency up to a frequencyconsiderably in excess of &n. Throughout this frequencyrartere we }IK\TP

or

wito

If in particular /9»i<Cl, the frequencies under considera-tion are in the classical domain and 7 is the frequencyindependent constant

for an external force that does not vary rapidly inrelation to &si, the contribution of the high-frequencyterm is essentially given by

To regard j as constant for a quantum oscillator requiresa suitable frequency restriction to the vicinity of on,The function B(ws) can be computed from

and the response to such an external force is adequatelydescribed by the low-frequency part of the Green'sfunction. We can represent this situation by an equi-valent differential equation

which needs no further qualification when the oscilla*lions are classical but implies a restriction to a frequencyinterval within which y is constant, for quantum oscilla-tions. We note the reduction in the effectiveness of theexternal force by the factor B. Under the circumstances

and accordingly is given by

But

B R O W N I A N M O T I O N O F A O U A H T U M O S C I L L A T O R 429

mtlined this effect is not important an4 we shall place5 equal to unity.

One can ntafee a general replacement of the Green'sunctions by their low "frequency parts:

which here asserts that

if one limits the time focahzability ©f measurements sothat only time averages of q{t) are of physical Interest.This is represented in the expectation value formula byconsidering only functions f±{t) that do not vary tooaukklv. The corresmwirfin? replacement iotwit—i*} is

with

and the entire matrix G&(t—tf) obtained in this wayofsevs the. differential ^filiation

The latter matrix can also be identified as

where s^o^Wi),The simplest presentation of results is again to be

found m the Laogevin. viewpoint, which directs theemphasis fom the coordinate operator g(t) to thefluctuating force defined by

and

In the classical limit

If a comparison is made with the similar results o! theprevious section it cars be appreciated that the f reqtsencyrange has been extended and the jrestriction «tscs&j»removed.

We return from these extended considerations onthermal equilibrium and consider am extreme exampleof negative temperature for the external system, itiisis described by

which Is to say

This change is introduced by the substitution

and the necessary partial integrations involve thepreviously established Jemima oa tiffie-ordered operatorsj

With the definition

and

430

we have

J . S C H W I N G E R

sfithin the range dg i, 4g% Is

where g» and «? ate the amplitude and phase of f,(i).Despite rather different assumptions about the externalsystem, these are the same conclusions as before, apartfrom a factor of f in the formula (or the gain.

6IHERAL THE0M

The whole of the preceding discussion assumes anexternal system that is only slightly influenced by thepresence of the oscillator. Now we must attempt toplace this simplification within the framework ol ageneral formulation, A more thorough treatment isalso a practical necessity in situations such as thoseproducing amplification of the oscillator motion, for asizeable reaction in the external system must eventuallyappear, unless a counter mechanism is provided,

It is useful to supplement the previous Lagrangianoperator with the term f'('X?, in which q'{i) is anarbitrary numerical function of time, and also, toimagine the coupling term gQ supplied with a variablefactor X. Then

We shall suppose, for simplicity, that wt^wa andsfXsat. Then the poles of

Its a function of £*t G(£) now has complex poles if

After the larger time intervals j»(«—i,}, f(l'—%)»!,we have

with

When I is in the vicinity of a time li, such that th<mptification factor

the oscillator is described by the classical coordins provided that the states to which the transformationfunction refers do not depend upon the couplingbetween the systems, or that the trace of the trans-formation functisn is being evaluated. A similarstatement would apply to a transformation fusctioiswith different terminal times. This differential equationimplies an integrated form, in which the transformationfunction for the fully coupled system (X= 1) is expressedin terms of the transformation function for the un-coupled system (X=0), The tatter is the product oftransformation functions for the independent escalatorand external system. The relation is

where g-t and ^ are characterised by the expectationvalue formula

Here

and

in which

AceordJKgly, the probability of observing ft and j»

are located at andAcoordingly is regulat outside a stirip of wich

The associated Greeen's Futcation are given by

and the function compoued fior

431

On moving P&($ to the left ol the exponential, thisa Such positive and negative time-ordered products occur in a

recent paper bv it. Svnmnak Q, Math. Phys* 1, 249 (1960)1which appeared after this paper had beem written and its contentsssed as a basis for lectures delivered at the Brandos SummerSefeaol, July, I960.

B E O W N I A H M O T I O N O F A Q U A N T U M O S C I L L A T O R

and we have indicated that q^ is finally set equal tozero If we are eonceraed OBly with nseasarements onthe oscillator.

Let us consider for the moment just the exteraalsystem with the perturbation gfQ, the effect 0! which Isindicated by8

become

Bui

We shall define

and similarlywhich leads us to the following functional differentialequation lor the transformation function (lali^*, inwhich a knowledge is assumed of the external systemsreaction to the perturbation ?±'(Q:

and farthermore,

When q±(l)**tf(t)t we haye

which is the expectation value ol Q{$ in the presence ofthe perturbation described by ?'(^). This is assumed tobe aero for §'00—0 and depends generally upon thehistory of /(/} between % and the given time.

The operators §&(!} are produced withlm the trans-formation function by the functional differentialoperators (±1/0 V^±(*)» &n^ ^nce the equation ofmotion for the uncoupled oscillator is

Throughout this discussion, one must distinguishbetween the ± signs attached to particular compoHeaf saud those involved in the listing of complete sets ofvariables.

The differential equations for time development aresupplemented by boundary conditions which assert, ata time h beyond which F^(I)~F_(0, that

we have

fhe previous treatment caa sow be identified as the

while, for the exai^ple of the transformation function(4f4Ve^*» ^e have the initial conditions

432 J . S C H W I M G E R

approximation of the Q±(ttq±) oy linear functions of converts the functional differential equations into

%&>

wherein the linear equations tor the operators q±(t) anatheir meaning in terms of variations of the P& havebeen united in one pair of functional differentialenuafJons. This relation becomes clearer if one writes

Tne bouiitoaiy <3>uaitiOBS EOW appear as

When the Q$, are linear functions of g±( the functionaldifferential operators disappear7 amcl we regam thelinear eolations for q±(t)t which in turn imply thequadratic form of $T{F&) that characterizes thepreceding discussion,

TThe degeneration of th« lunctioriml ^jtiation? iato or^nafyEliffeiwitial equations a^e» oeeurs when the RHJ!ion of the oscillatoris classical aad free of Iuct«atioR.

and, with the defiuition

and


9.2 COULOMB GREEN'S FUNCTION

tReproduced from the Journal of Mathematical Physics,

Vol. 5, pp. 1606-1608 (1964).

+

Reprinted liom

JOUKKAL OF M A T H E M A T I C A L PHYSICS

Printed in VSA.

N O V E M B E R I 9 6 <

r has long been known that the degenwasy ofthe bound states in the nonrektrristio Coulomb

problem can be described by a four-dimensionalEuclidean rotation group, and that the momentumrepresentation is most eonvenient for realizing theconnection. It seems not to have been recofpiraed,however, that the same approach cat* b§ used toobtain an exploit construction for the Green'sfunction of this problem. The derivation1 is givenhere.

The momentum representation equation for the* It was wolfed cm$ to preseat at a Harvard quaatum

mechanics eourae give® in ttte late 1940*8. I h&ve beenstimulated to rescue it from tfa® quiet d«ath oj lectee aotesby recent publications is this Jouro&l, which give alternativeforms of the Green's faaetwm: B. H. Wielimaaa aad C. H.Woo, J. Mutt. Phys. Z, 178 (1W1): JU Ho«tter, iW. S, S91(J964).

Coulomb Green's Function

JyiaAN ScHwxKasitffarmrd £/nwers%, Cambridge, MasmekmeM*

(Beceirad 19 June 1964)

A one-parameter istegml Eeprestutation is given for tlis momestucft 8[>a«e Green's fuoctioahe nasrelativistie Coulomb problem.

VOLOKB 5, N U M B E R 11

Sreen's function a (ft = 1)

We shall solve this equation by assuming, at first,that

ia real and negative. The general result is inferredby analytic continuation.

The parameters

define the surface of a unit four-dimensional By-

r

COBLOMB GEBBN'S FUNCTION 1607

elidean sphere,

the points of which are in, one to one eorrespondencwith the momentum spaee. The element of areon the sphere is

if one keeps JB mind that p ^. Pa corresponds to thetwo semispheres Is » ^(1 — £*)** As another form ofthis relation, we write the delta function connect-ing two points on the uait sphere as

whei

The Green's function D is exhibited as

conveys the oorroaliwtion and completeness of thesurface harmonies. One can verify that D has theradial discontinuity implied by the delta functioninhomogeneity of the differential equation,

The function D is used in the integral equation forF with p « = / » ' • » 1. The equation is solved byNext, observe thi

The singularities of this function at f-n —1,2, • • •give the expected negative energy eigenvalues.The residues of Q at the corresponding poles inthe B plane provide the normalized wavefune-tions, whieh are

Then, if we define

that function obeys a four-dimensional auelideaEsurface integral equation,

One can exhibit r(tt, 0') in essentially closedform with the end of the expansion for D. We usethe following version of this expansion:

where

and

The function D that is defined similarly through-out the Euclidean space is the Green's funetiocof the four-dimensional Poisson equation,

where { and £' are of unit length and 0 < p < I.Note, incidentally, that if we set { ** J* and inte-grate over the unit sphere, of area 2ir3, we get

It can be constructed in terms of a eomplete setof four-dimensional solid harmonics. In the sphericalcoordinates indicated by />, Q, these are where m* is the multiplicity of the quantum number

n. This eoRfirms that m* *= ns,The identitywhere the quantum numbers J, m provide a three-

dimensional harmonic classification of the four-dimensional harmonics. The largest vslue of i con-tained in the EQisogeneous polynomial fli>~tY*tm($l)is the degree of the polynomial, n — 1. Thus,

together with the integral representation

label the n distinct harmonies that have a commonvalue of n. ?alid for » < 1, gives

1608 J U t l A N S C B W I N G E R

anil

One would have found the same asymptotic formfor any potential that decreases more rapidly thaathe Coulomb potential at large distances, but with<J*(p) = (g - I*)"*. The factors GV) and G°(p)describe the propagation of the particle before andafter the collision, respectively, and / is identifiedas the scattering amplitude. The same interprets-"tion is applicable here since the modified 6° justincorporates the long-range effect of the Coulombpotential. This is most evident from the asymptoticbehavior of the corresponding spatial function,which is a distorted spherical wave.

The scattering amplitude obtained in this waycoincides with the known result,

The path C begme »t p — 1 + Qz, where the phaseof p is Bero and terminates at p « 1 — Gi, afterencircling the origin within the unit circle,

The Green's function expressions implied by (1),f2>. and C31 are

and

where

The restrietion p < 1 ean be removed by re-placing the real integrals with contour integrals,

The $eeond of the three forms given for G is mostconvenient here. The asymptotic behavior is domi-nated by small f values, and one immediatelyobtains

Equivalent forms, produced by partial integra-tions, are

U>

The Green's function is regular everywhere inthe complex E plane with the exception of thephysical energy spectrum. This consists of thenegative-energy eigenvalues already identified andthe positive-energy eontitmam. The integral repre-sentations (!'), ('2'), &sd (30 are not completelygeneral sinee it is required that

C2)As we have indicated, this restriction ean be re-moved. It is not necessary to do 80, however, ifone is interested in the limit of real k. Tliese repre-sentations can therefore be applied directly to thephysical scattering problem.

The asymptotie conditions that characterise finiteangle deflections are

am

which uses the limiting relatio

Note that i is a function of a single variable,

where

schwinger j

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