In a quantum field theory with fermions, (−1) 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, . 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) whereas fermionic operators anticommute with it.
This operator really shows its utility in supersymmetric theories.<ref name="terning"/> Its trace is the spectral asymmetry of the fermion spectrum, and can be understood physically as the Casimir effect.
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