(−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.

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${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }$ Schrödinger equation
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This operator really shows its utility in supersymmetric theories. Its trace is the spectral asymmetry of the fermion spectrum, and can be understood physically as the Casimir effect.