(−1)F

Term in quantum field theory February 2013 {{DISPLAYTITLE:(−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. Terning John Modern Supersymmetry:Dynamics and Duality: Dynamics and Duality Oxford University Press 2006 New York https://books.google.com/books?id=1JMf-fcnOHYC&q=fermion+%22%28-1%29F%22&pg=PA5 0-19-856763-4 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) Witten index Primon gas Möbius function References Further reading Shifman Mikhail A. Mikhail A. Shifman Advanced Topics in Quantum Field Theory: A Lecture Course Cambridge University Press 2012 Cambridge https://books.google.com/books?id=zeQuWycXV3oC&q=fermion+%22%28-1%29F%22&pg=PA581 978-0-521-19084-8 Ibáñez Luis E. Angel M. Uranga String Theory and Particle Physics: An Introduction to String Phenomenology Cambridge University Press 2012 Cambridge https://books.google.com/books?id=vAUUu6DpVkUC&q=fermion+%22%28-1%29F%22&pg=PA111 978-0-521-51752-2 Bastianelli Fiorenzo Path Integrals and Anomalies in Curved Space Cambridge University Press 2006 Cambridge https://books.google.com/books?id=HxpBObJ8roEC&q=%22%28-1%29F%22&pg=PA278 978-0-521-84761-2 Category:Quantum field theory Category:Supersymmetric quantum field theory Category:Fermions