Quantum Spin Dynamics (QSD) : VII. Symplectic Structures and Continuum Lattice Formulations of Gauge Field Theories
T. Thiemann
TL;DR
This work develops a regulator-based, graph-labeled regularization of gauge field theories by introducing a generalized projective family of symplectic manifolds $(M_ extgamma,oldOmega_ extgamma)$ labeled by graphs and dual faces. It defines holonomy and flux variables through dual polyhedral decompositions and proves a closed, anomaly-free Poisson/commutator structure for the Gauss constraint, both at the classical level and in quantized form on graph Hilbert spaces $oldH_ extgamma$. The continuum phase space $(M,oldOmega)$ is shown to arise as a limit of the graph-based spaces, enabling a clear separation of regularization and quantization and providing a robust route to semi-classical analysis via coherent states. The framework resolves non-commutativity ambiguities inherent in unsmeared variables by promoting gauge-covariant, regulator-dependent observables and demonstrates consistent classical limits, laying groundwork for a background-independent quantum gauge theory compatible with loop quantum gravity approaches.
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.