This article has multiple issues. Please help improve it or discuss these issues on the talk page. ( Learn how and when to remove these template messages) This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (February 2013) ( Learn how and when to remove this template message) This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (October 2020) ( Learn how and when to remove this template message) ( Learn how and when to remove this template message)

Part of a series of articles about
Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
Fundamentals Complementarity Decoherence Entanglement Energy level Measurement Nonlocality Quantum number State Superposition Symmetry Tunnelling Uncertainty Wave function Collapse
Experiments Bell's inequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett–Garg inequality Mach–Zehnder Popper Quantum eraser Delayed-choice Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral)
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Bayesian Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Local Superdeterminism Many-worlds Objective collapse Quantum logic Relational Transactional Von Neumann–Wigner
Advanced topics Relativistic quantum mechanics Quantum field theory Quantum information science Quantum computing Quantum chaos EPR paradox Density matrix Scattering theory Quantum statistical mechanics Quantum machine learning
Scientists Aharonov Bell Bethe Blackett Bloch Bohm Bohr Born Bose de Broglie Compton Dirac Davisson Debye Ehrenfest Einstein Everett Fock Fermi Feynman Glauber Gutzwiller Heisenberg Hilbert Jordan Kramers Lamb Landau Laue Moseley Millikan Onnes Pauli Planck Rabi Raman Rydberg Schrödinger Simmons Sommerfeld von Neumann Weyl Wien Wigner Zeeman Zeilinger
v t e

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.

See also

• Parity (physics)
• Primon gas
• Möbius function

References

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.