(−1)^F

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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.^[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.

• Parity (physics)
• Primon gas
• Möbius function

• Shifman, Mikhail A. (2012). Advanced Topics in Quantum Field Theory: A Lecture Course. Cambridge: Cambridge University Press. ISBN 978-0-521-19084-8.
• Ibáñez, Luis E.; Uranga, Angel M. (2012). String Theory and Particle Physics: An Introduction to String Phenomenology. Cambridge: Cambridge University Press. ISBN 978-0-521-51752-2.
• Bastianelli, Fiorenzo (2006). Path Integrals and Anomalies in Curved Space. Cambridge: Cambridge University Press. ISBN 978-0-521-84761-2.