Physics:(−1)F
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| [math]\displaystyle{ i \hbar \frac{\partial}{\partial t} | \psi (t) \rangle = \hat{H} | \psi (t) \rangle }[/math] |
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
- 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. https://books.google.com/books?id=zeQuWycXV3oC&q=fermion+%22%28-1%29F%22&pg=PA581.
- 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. https://books.google.com/books?id=vAUUu6DpVkUC&q=fermion+%22%28-1%29F%22&pg=PA111.
- Bastianelli, Fiorenzo (2006). Path Integrals and Anomalies in Curved Space. Cambridge: Cambridge University Press. ISBN 978-0-521-84761-2. https://books.google.com/books?id=HxpBObJ8roEC&q=%22%28-1%29F%22&pg=PA278.
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