SCHWINGER TERMS AND COHOMOLOGY OF PSEUDODIFFERENTIAL-OPERATORS

We study the cohomology of the Schwinger term arising in second quanti zation of the class of observables belonging to the restricted general linear algebra. We prove that, for all pseudodifferential operators i n 3 + 1 dimensions of this type, the Schwinger term is equivalent to t he ''twisted'' Radul cocycle, a modified version of the Radul cocycle arising in non-commutative differential geometry. In the process we al so show how the ordinary Radul cocycle for any pair of pseudodifferent ial operators in any dimension can be written as the phase space integ ral of the star commutator of their symbols projected to the appropria te asymptotic component.