Supertrace

A supertrace is the graded analogue of the ordinary matrix trace. If a vector space has the -grading and 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.