Quantum Spin Dynamics (QSD) : VII. Symplectic Structures and Continuum Lattice Formulations of Gauge Field Theories

TL;DR

This paper develops a classical symplectic framework for gauge field theories using smeared variables and projective limits, facilitating their quantization and analysis of constraints like the Gauss law.

Contribution

It introduces a generalized projective family of symplectic manifolds for diffeomorphism invariant theories, enabling a systematic approach to quantization and semi-classical analysis.

Findings

Constructed a projective family of symplectic manifolds labeled by graphs.

Proved the existence of a limit symplectic manifold consistent with the original.

Applied the framework to the Gauss constraint in gauge theories.

Abstract

Interesting non-linear functions on the phase spaces of classical field theories can never be quantized immediately because the basic fields of the theory become operator valued distributions. Therefore, one is usually forced to find a classical substitute for such a function depending on a regulator which is expressed in terms of smeared quantities and which can be quantized in a well-defined way. Namely, the smeared functions define a new symplectic manifold of their own which is easy to quantize. Finally one must remove the regulator and establish that the final operator, if it exists, has the correct classical limit. In this paper we investigate these steps for diffeomorphism invariant quantum field theories of connections. We introduce a generalized projective family of symplectic manifolds, coordinatized by the smeared fields, which is labelled by a pair consisting of a graph and another graph dual to it. We show that there exists a generalized projective sequence of symplectic manifolds whose limit agrees with the symplectic manifold that one started from. This family of symplectic manifolds is easy to quantize and we illustrate the programme outlined above by applying it to the Gauss constraint. The framework developed here is the classical cornerstone on which the semi-classical analysis developed in a new series of papers called ``Gauge Theory Coherent States'' is based.