(−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.

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.

See also

• Parity (physics)
• Primon gas
• Möbius function

Further reading

• 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.