(−1)F

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.

[1] This operator really shows its utility in supersymmetric theories.

[1] Its trace is the spectral asymmetry of the fermion spectrum, and can be understood physically as the Casimir effect.