Quantization of gauge theory for gauge dependent operators

TL;DR

This paper explores how gauge dependence affects the quantization of gauge theories, revealing that matrix elements involve gauge-transformed operators and questioning the reliability of the path integral formalism for gauge-dependent quantities.

Contribution

It introduces a canonical path integral approach showing the differences in perturbative calculations for gauge dependent operators and highlights issues with the formal path integral method.

Findings

Gauge-dependent operator matrix elements involve gauge-transformed operators.

Perturbative calculations differ for gauge-dependent and gauge-invariant operators.

The formal path integral approach may be unreliable for gauge-dependent quantities.

Abstract

Based on a canonically derived path integral formalism, we demonstrate that the perturbative calculation of the matrix element for gauge dependent operators has crucial difference from that for gauge invariant ones. For a gauge dependent operator ${\cal O}(\phi)$ what appears in the Feynman diagrams is not ${\cal O} (\phi)$ itself, but the gauge-transformed one ${\cal O}(^\omega \phi)$, where $\omega$ characterizes the specific gauge transformation which brings any field variable into the particular gauge which we have adopted to quantize the gauge theory using the canonical method. The study of the matrix element of gauge dependent operators also reveals that the formal path integral formalism for gauge theory is not always reliable.