Schwinger Terms and Cohomology of Pseudodifferential Operators

TL;DR

This paper investigates the cohomology of the Schwinger term in second quantization for pseudodifferential operators, revealing its equivalence to a twisted Radul cocycle in 3+1 dimensions and relating it to phase space integrals.

Contribution

It establishes the equivalence of the Schwinger term with a twisted Radul cocycle for pseudodifferential operators in 3+1 dimensions and connects Radul cocycles to phase space integrals.

Findings

Schwinger term is equivalent to twisted Radul cocycle in 3+1 dimensions

Radul cocycle can be expressed as phase space integral of star commutator

Provides a cohomological understanding of observables in second quantization

Abstract

We study the cohomology of the Schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra. We prove that, for all pseudodifferential operators in 3+1 dimensions of this type, the Schwinger term is equivalent to the ``twisted'' Radul cocycle, a modified version of the Radul cocycle arising in non-commutative differential geometry. In the process we also show how the ordinary Radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component.