(−1)F

In a quantum field theory with fermions, (−1)^F is a unitary, Hermitian, involutive operator where F is the fermion number operator. For the example of particles in the Standard Model, it is equal to the sum of the lepton number plus the baryon number, F=B+L. The action of this operator is to multiply bosonic states by 1 and fermionic states by −1. This is always a global internal symmetry of any quantum field theory with fermions and corresponds to a rotation by 2π. This splits the Hilbert space into two superselection sectors. Bosonic operators commute with (−1)^F whereas fermionic operators anticommute with it.

This operator really shows its utility in supersymmetric theories. Its trace is often a useful computation.