Generalized canonical approach to deformation problem in gauge theories

TL;DR

This paper introduces a general method using the BFV formalism to construct deformations of dynamical systems with first-class constraints, enabling transformations into other systems, exemplified by a non-local deformation of gauge theories.

Contribution

It provides a novel, explicit construction of deformations via supercanonical transformations within the BFV framework, applicable to gauge theories and beyond.

Findings

Derived a general solution for deformation using supercanonical transformations.

Constructed a non-local deformation of Abelian to non-Abelian gauge theory.

Showed the deformed theory can have a local sector identical to Yang-Mills theory.

Abstract

We develop a general approach to constructing a deformation that describes the mapping of any dynamical system with irreducible first-class constraints in the phase space into another dynamical system with first-class constraints. It is shown that such a deformation problem can be efficiently explored in the framework of the Batalin-Fradkin-Vilkovisky (BFV) formalism. The basic objects of this formalism are the BRST-BFV charge and a generalized Hamiltonian that satisfy the defining equations in the extended phase space in terms of (super)Poisson brackets. General solution to the deformation problem is found in terms of a (super)canonical transformation with a special generating function which is explicitly established. It is proved that this generating function is determined by a single arbitrary function which depends only on coordinates of initial dynamical system. In principle, such a function may be non-local, but the deformed theory may have a local sector. To illustrate the developed approach, we have constructed a non-local deformation of the Abelian gauge theory into a non-local non-Abelian gauge theory whose local sector coincides with the standard Yang-Mills theory.