Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism

TL;DR

The paper develops an operator-formalism proof of gauge invariance and Ward identities for non-Abelian gauge theories, including cases with spontaneous symmetry breaking. By introducing a gauge-invariant Lagrangian with fictitious Fermionic scalars and a generalized statistical averaging that yields even-frequency Fermionic Green's functions, Tyutin provides a framework for gauge-invariant statistical averages and a gauge-independent partition function. It demonstrates how Ward identities arise from equations of motion and canonical relations, and shows how Green's functions transform under changes of gauge parameters, ensuring gauge-invariant physical quantities. The approach also reconciles finite-temperature formalism with gauge symmetry through carefully chosen statistics of the auxiliary fields, yielding a consistent, gauge-invariant description of both dynamical and statistical aspects of gauge theories.

Abstract

We obtain the Ward identities and the gauge-dependence of Green's functions in non-Abelian gauge theories by using only the canonical commutation relations and the equations of motion for the Heisenberg operators. The consideration is applicable to theories both with and without spontaneous symmetry breaking. We present a definition of a generalized statistical average which ensures that the Fourier images of temperature Green's functions of the Fermionic fields have only even-valued frequencies. This makes it possible to set up a procedure of gauge-invariant statistical averaging in terms of the Hamiltonian and the field operators.