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In a [[quantum field theory]] with [[fermion]]s, '''(−1)<sup>''F''</sup>''' is a [[unitary operator|unitary]], [[Hermitian operator|Hermitian]], [[Involution (mathematics)|involutive]] [[Operator (mathematics)|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 [[boson]]ic states by 1 and [[fermion]]ic 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 sector]]s. Bosonic operators [[Commutativity|commute]] with (−1)<sup>''F''</sup> whereas fermionic operators [[anticommute]] with it. |
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This operator really shows its utility in [[supersymmetry|supersymmetric]] theories.<ref>{{cite book | last = Terning | first = John| title = Modern Supersymmetry:Dynamics and Duality: Dynamics and Duality | publisher = [[Oxford University Press]] | date = 2006 | location = New York | url = https://books.google.com/books?id=1JMf-fcnOHYC&pg=PA5&dq=fermion+%22(-1)F%22&redir_esc=y#v=onepage&q=fermion%20%22(-1)F%22&f=false | isbn = 0-19-856763-4}}</ref> [[Witten index|Its trace]] is the [[spectral asymmetry]] of the fermion spectrum, and can be understood physically as the [[Casimir effect]]. |
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==See also== |
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*[[Parity (physics)]] |
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*[[Primon gas]] |
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*[[Möbius function]] |
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==References== |
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<references/> |
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==Further reading== |
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* {{cite book | last = Shifman | first = Mikhail A. | authorlink = Mikhail A. Shifman | title = Advanced Topics in Quantum Field Theory: A Lecture Course | publisher = Cambridge University Press | date = 2012 | location = Cambridge | url = https://books.google.com/books?id=zeQuWycXV3oC&pg=PA581&dq=fermion+%22(-1)F%22&redir_esc=y#v=onepage&q=fermion%20%22(-1)F%22&f=false | isbn = 978-0-521-19084-8}} |
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* {{cite book | last1 = Ibáñez | first1 = Luis E. | first2 = Angel M. | last2 = Uranga | title = String Theory and Particle Physics: An Introduction to String Phenomenology | publisher = Cambridge University Press | date = 2012 | location = Cambridge | url = https://books.google.com/books?id=vAUUu6DpVkUC&pg=PA111&dq=fermion+%22(-1)F%22&redir_esc=y#v=onepage&q=fermion%20%22(-1)F%22&f=false | isbn = 978-0-521-51752-2}} |
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* {{cite book | last = Bastianelli | first = Fiorenzo | title = Path Integrals and Anomalies in Curved Space | publisher = Cambridge University Press | date = 2006 | location = Cambridge | url = https://books.google.com/books?id=HxpBObJ8roEC&pg=PA278&dq=fermion+%22(-1)F%22&redir_esc=y#v=onepage&q=%22(-1)F%22&f=false | isbn = 978-0-521-84761-2}} |
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{{DEFAULTSORT:-1F}} |
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[[Category:Quantum field theory]] |
[[Category:Quantum field theory]] |
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[[Category:Supersymmetry]] |
[[Category:Supersymmetry]] |
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{{quantum-stub}} |
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Revision as of 09:05, 16 December 2017
 | This article includes a list of general references, but it lacks sufficient corresponding inline citations. (February 2013) |
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.[1] Its trace is the spectral asymmetry of the fermion spectrum, and can be understood physically as the Casimir effect.
See also
References
- ^ Terning, John (2006). Modern Supersymmetry:Dynamics and Duality: Dynamics and Duality. New York: Oxford University Press. ISBN 0-19-856763-4.
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.
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