field quantization via discrete approximations: problems and perspectives

Field quantization via discrete approximations:

problems and perspectives.

Jerzy KijowskiCenter for Theoretical Physics PAN

Warsaw, Poland

2QFTG - REGENSBURGOct. 1, 2010

Perturbative QFT

Conventional (perturbative) approach to quantum field theory:

Reason (???)

1) Fundamental objects are plane waves.

2) Their (non-linear) interaction is highly non-local.

In spite of almost 80 years of (unprecedented) successes, this approach has probably attained the limits of its applicability.

: violation of locality principle.


Alternative (non-perturbative) approach, based on discrete approximations:

1) Physical system with infinite number of degrees of freedom(field) replaced by a system with finite number of (collective)degrees of freedom (its lattice approximations).

2) These collective degrees of freedom are local.They interact according to local laws.

Lattice formulation of QFT

Quantization of such an approximative theory is straightforward(up to minor technicalities) and leads to a model of quantum mechanical type.


Hopes


Quantizing sufficienly many (but always finite number) ofdegrees of freedom we will obtain a sufficiently good approximation of Quantum Field Theory by Quantum Mechanics.

Weak version:

There exists a limiting procedure which enables us toconstruct a coherent Quantum Field Theory as a limitof all these Quantum Mechanical systems.

Strong version:Spacetime structure in micro scale???


To define such a limit we must organize the family of all thesediscrete approximations of a given theory into an inductive-projective family.


Example: scalar (neutral) field.

Typically (if the kinetic part of the Lagrangian function ,,L’’is quadratic), we have: .

Field degrees of freedom described in the continuum version of the theory by two functions: – field Cauchy dataon a given Cauchy surface .


Scalar field

For any pair and of vectors tangent to the phase space we have:

Phase space of the system describes all possible Cauchy data:

Symplectic structure of the phase space:


Discretization – classical level

where are finite (relatively compact) and ``almost disjoint’’(i.e. intersection has measure zero for ).

Define the finite-dimensional algebra of local observables:

Possible manipulations of the volume factors

Choose a finite volume of the Cauchy surfaceand its finite covering (lattice) :

Symplectic form of the continuum theory generates:


There is a partial order in the set of discretizations:

Hierarchy of discretizations

Every lattice generates the Poisson algebra of classical observables, spanned by the family .

Example 1:



Example 2: (intensive instead of extensive observables)


Discretization – quantum levelQuantum version of the system can be easily constructed on the level of every finite approximation .

generate the quantum version of the observable algebra.

Quantum operators:

Schrödinger quantization: pure states described by wavefunctions form the Hilbert space .

Different functional-analytic framework might be necessaryto describe constraints (i.e. algebra of compact operators).

Simplest version:


Inductive system of quantum observables

Observable algebras form an inductive system:

Theorem:

Definition:

More precisely: there is a natural embedding .

Proof:where describes ``remaining’’ degrees of freedom(defined as the symplectic annihilator of ).



Example:

Annihilator of generated by:



Example 2: (intensive observables)

Anihilator of generated by:



Embeddings are norm-preserving and satisfy the chain rule:

Complete observable algebra can be defined as the inductivelimit of the above algebras, constructed on every level of thelattice approximation:

(Abstract algebra. No Hilbert space!)


Quantum statesQuantum states (not necessarily pure states!) are functionalson the observable algebra.

Projection mapping for states defined by duality:

On each level of lattice approximation states are represented by positive operators with unital trace:


Quantum states

„Forgetting” about the remaining degrees of freedom.

Physical state of the big system implies uniquely the state of its subsystem

EPR


Projective system of quantum states

Chain rule satisfied:

States on the complete algebra can be describedby the projective limit:


Hilbert space

Given a state on the total observable algebra (a ``vacuum state’’), one can generate the appropriateQFT sector (Hilbert space) by the GNS construction:

Hope:

The following construction shall (maybe???) lead to theconstruction of a resonable vacuum state:

1) Choose a reasonable Hamilton operator on every levelof lattice approximation (replacing derivatives by differencesand integrals by sums).


Hilbert spaceHope:

The following construction shall (maybe???) lead to theconstruction of a resonable vacuum state:

1) Choose a reasonable Hamilton operator on every levelof lattice approximation (replacing derivatives by differencesand integrals by sums).

2) Find the ground state of .3) (Hopefully) the following limit does exist:

Corollary:

is the vacuum state of the complete theory.


Special case: gauge and constraintsMixed (intensive-exstensive) representation of gauge fields:

Implementation of constraints on quantum level:Gauge-invariant wave functions (if gauge orbits compact!),otherwise: representation of observable algebras.

parallell transporter on every lattice link .


General relativity theory

Einstein theory of gravity can be formulated as a gauge theory.

Field equations:

After reduction: Lorentz groupPossible discretization with gauge group: .

Further reduction possible with respect to

What remains? Boosts!

Affine variational principle: first order Lagrangian function depending upon connection and its derivatives (curvature).


General relativity theory

This agrees with Hamiltonian formulation of general relativityin the complete, continuous version:

Cauchy data on the three-surface : three-metric and theextrinsic curvature .

Extrinsic curvature describes boost of the vector normal to when dragged parallelly along .


Loop quantum gravity

The best existing attempt to deal with quantum aspects of gravity!

In the present formulation: a lattice gauge theory with .

Why?

But:

After reduction with respect to rotations we end up with:


Loop quantum gravity

The theory is based on inductive system of quantum states!

contains more links or gives finer description of the same links:

But


Inductive system of quantum states

State of a subsystem determines state of the big system!!!

contains more links or gives finer description of the same links:

Inductive mapping of states:

is a subsystem of the ``big system’’


LQG - difficulties

Lack of any reasonable approach to constraints.

Leads to non-separable Hilbert spaces.


But the main difficulty is of physical (not mathematical) nature:

Positivity of gravitational energy not implemented.

Non-compact degrees of freedom excluded a priori.


LQG - hopes


A new discrete approximation of the 3-geometry (both intrinsicand extrinsic!), compatible with the structure of constraints.

Representation of the observable algebra on every level ofdiscrete approximations.

Positivity of gravitational energy implemented on every levelof discrete approximations of geometry.

Replacing of the inductive by a projective system of quantumstates.

State of a system determines state of the subsystem!!!


References

field quantization via discrete approximations: problems and perspectives

Documents

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