Vacuum Expectation Values of Products of Chiral Currents in $3+1$ Dimensions

TL;DR

This paper develops an algebraic method to compute vacuum expectation values of products of nonabelian charge operators in 3+1 dimensions, linking quantum field theory with noncommutative geometry.

Contribution

It introduces a new algebraic rule for expectation values and connects these to cyclic cocycles in noncommutative geometry, using Kirillov's orbit method.

Findings

Derived an algebraic rule for expectation values of charge operators

Connected four-operator expectation values to cyclic cocycles

Utilized Kirillov's method for current representation

Abstract

An algebraic rule is presented for computing expectation values of products of local nonabelian charge operators for fermions coupled to an external vector potential in $3+1$ space-time dimensions. The vacuum expectation value of a product of four operators is closely related to a cyclic cocycle in noncommutative geometry of Alain Connes. The relevant representation of the current is constructed using Kirillov's method of coadjoint orbits.