An Interacting Gauge Field Theoretic Model for the Hodge Theory: Basic Canonical Brackets

TL;DR

This paper develops an alternative method for deriving canonical brackets in a 2D gauge field theory with fermions, using symmetries and generators instead of canonical momenta, linking gauge theory to Hodge theory.

Contribution

It introduces a novel approach to quantization of gauge theories based on symmetry generators, bypassing canonical conjugate momenta, and relates the structure to Hodge theory.

Findings

Derived basic brackets without canonical momenta

Utilized symmetry generators and spin-statistics theorem

Linked gauge theory structure to de Rham cohomology

Abstract

We derive the basic canonical brackets amongst the creation and annihilation operators for a two (1 + 1)-dimensional (2D) gauge field theoretic model of an interacting Hodge theory where a U(1) gauge field (A_\mu) is coupled with the fermionic Dirac fields (\psi and \bar \psi). In this derivation, we exploit the spin-statistics theorem, normal ordering and the strength of the underlying six infinitesimal continuous symmetries (and the concept of their generators) that are present in the theory. We do not use the definition of the canonical conjugate momenta (corresponding to the basic fields of the theory) anywhere in our whole discussion. Thus, we conjecture that our present approach provides an alternative to the canonical method of quantization for a class of gauge field theories that are physical examples of Hodge theory where the continuous symmetries (and corresponding generators) provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level.