(−1)^F

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In a quantum field theory with fermions, (−1)^F is a unitary, Hermitian, involutive operator which multiplies 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.