Difficulties in a Kind of Averaging Procedure for Constructing Gauge-invariant Operators out of Gauge-variant Ones

TL;DR

The paper demonstrates that a common averaging method for creating gauge-invariant operators from gauge-variant ones is fundamentally flawed and cannot be generally applied.

Contribution

It provides a rigorous proof showing the inapplicability of a widely used averaging procedure in gauge theory constructions.

Findings

The averaging procedure is erroneous for many operators.

The method cannot be reliably used to produce gauge-invariant operators.

The proof clarifies limitations in gauge-invariant operator construction.

Abstract

We prove that a kind of averaging procedure for constructing gauge-invariant operators(or functionals) out of gauge-variant ones is erroneous and inapplicable for a large class of operators(or functionals).