A Note on Functional Integral over the Local Gauge Group

TL;DR

This paper investigates functional integrals over the local gauge group using lattice discretization, revealing violations of Haar measure properties, and examines the Faddeev-Popov method through a simplified example.

Contribution

It provides a detailed analysis of functional integrals over the local gauge group and demonstrates violations of Haar measure properties in this context.

Findings

Functional integrals over the local gauge group violate Haar measure properties.

Discretization on a lattice affects measure properties in gauge integrals.

The Faddeev-Popov method's limitations are illustrated through a toy example.

Abstract

We evaluated some particular type of functional integral over the local gauge group C^{\infty}({\bf R}^n, U(1)) by going to a discretized lattice. The results explicitly violates the property of the Haar measure. We also analysed the Faddeev-Popov method through a toy example. The results also violates the property of the Haar measure.