Supertrace

A supertrace is the graded analog of the ordinary matrix trace. A -grading of a vector space is a decomposition into even and odd subspaces. An endomorphism is even if it preserves these subspaces and odd if it interchanges them. If an endomorphism has block matrix form

(1)

then its supertrace is

(2)

Thus an ordinary matrix trace is taken on each graded component, with the odd component entering with a minus sign. For endomorphisms and of definite parity, the supertrace satisfies the graded cyclic identity

(3)

where denote their parities. It therefore vanishes on graded commutators.

In quantum field theory, the even and odd components correspond respectively to bosonic and fermionic degrees of freedom. A functional supertrace may also include sums or integrals over continuous field labels, as in the Wetterich equation.