View source for Physics:(−1)F

{{Short description|Term in quantum field theory}}
{{DISPLAYTITLE:(−1)<sup>''F''</sup>}}
{{Quantum mechanics}}
In a [[Physics:Quantum field theory|quantum field theory]] with [[Physics:Fermion|fermion]]s, '''(−1)<sup>''F''</sup>''' is a [[Unitary operator|unitary]], Hermitian, [[Involution (mathematics)|involutive]] [[Operator (mathematics)|operator]] where ''F'' is the [[Physics:Fermion|fermion]] number operator. For the example of particles in the [[Physics:Standard Model|Standard Model]], it is equal to the sum of the lepton number plus the baryon number, {{nowrap|1=''F'' = ''B'' + ''L''}}.  The action of this operator is to multiply [[Physics:Boson|boson]]ic states by 1 and [[Physics:Fermion|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 sectors. Bosonic operators commute with (−1)<sup>''F''</sup> whereas fermionic operators anticommute with it.<ref name="terning">{{cite book | last = Terning | first = John| title = Modern Supersymmetry:Dynamics and Duality: Dynamics and Duality | publisher = [[Organization:Oxford University Press|Oxford University Press]] | date = 2006 | location = New York | url = https://books.google.com/books?id=1JMf-fcnOHYC&q=fermion+%22%28-1%29F%22&pg=PA5 | isbn = 0-19-856763-4}}</ref>

This operator really shows its utility in [[Physics:Supersymmetry|supersymmetric]] theories.<ref name="terning"/> [[Physics:Witten index|Its trace]] is the [[Spectral asymmetry|spectral asymmetry]] of the fermion spectrum, and can be understood physically as the [[Physics:Casimir effect|Casimir effect]].

==See also==
*[[Physics:Parity|Parity]]
*[[Physics:Primon gas|Primon gas]]
*Möbius function

==References==
<references/>

==Further reading==
* {{cite book | last = Shifman | first = Mikhail A. |  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&q=fermion+%22%28-1%29F%22&pg=PA581 | isbn = 978-0-521-19084-8}}
* {{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&q=fermion+%22%28-1%29F%22&pg=PA111 | isbn = 978-0-521-51752-2}}
* {{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&q=%22%28-1%29F%22&pg=PA278 | isbn = 978-0-521-84761-2}}

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[[Category:Quantum field theory]]
[[Category:Supersymmetric quantum field theory]]

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