Hilbert space cocycles as representations of $(3+1)-$ D current algebras

TL;DR

This paper explores the use of Hilbert space cocycles as a novel approach to representing $(3+1)$-dimensional current algebras in gauge theories, emphasizing group extensions and regularization techniques.

Contribution

It introduces a new cocyclic representation framework for current algebras in 3+1 dimensions using Hilbert space cocycles and regularization methods.

Findings

Cocyclic representations are constructed via background-dependent regularizations.

A specific cocycle is evaluated for the group of smooth maps from space to a Lie group.

Quantization is achieved through a combination of conjugation and subtraction regularizations.

Abstract

It is proposed that instead of normal representations one should look at cocycles of group extensions valued in certain groups of unitary operators acting in a Hilbert space (e.g the Fock space of chiral fermions), when dealing with groups associated to current algebras in gauge theories in $3+1$ space-time dimensions. The appropriate cocycle is evaluated in the case of the group of smooth maps from the physical three-space to a compact Lie group. The cocyclic representation of a component $X$ of the current is obtained through two regularizations, 1) a conjugation by a background potential dependent unitary operator $h_A,$ 2) by a subtraction $-h_A^{-1}\Cal L_X h_A,$ where $\Cal L_X$ is a derivative along a gauge orbit. It is only the total operator $h_A^{-1} Xh_A-h_A^{-1}\Cal L_X h_A$ which is quantizable in the Fock space using the usual normal ordering subtraction.