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{{Short description|Property of scattering amplitudes}}
In [[theoretical physics]], '''crossing symmetry''' is a relation between the [[S-matrix]] that describe processes that can be obtained from each other by replacing incoming particles with outgoing antiparticles after taking the [[analytic continuation]]. For example, the [[annihilation]] of an [[electron]] with a [[positron]] into two [[photon]]s is related to an [[elastic scattering]] of an electron with a photon by crossing symmetry. This relation allows to calculate the [[scattering amplitude]] of one process from the amplitude for the other process if negative values of [[energy]] of some particles are substituted.▼
{{Quantum field theory|cTopic=Crossing symmetry}}
In [[quantum field theory]], a branch of [[theoretical physics]], '''crossing''' is the property of [[scattering amplitude]]s that allows antiparticles to be interpreted as particles going backwards in time.
Crossing states that the same formula that determines the [[S-matrix]] elements and scattering amplitudes for particle <math>\mathrm{A}</math> to scatter with <math>\mathrm{X}</math> and produce particle <math>\mathrm{B}</math> and <math>\mathrm{Y}</math> will also give the scattering amplitude for <math>\scriptstyle \mathrm{A}+\bar{\mathrm{B}}+\mathrm{X}</math> to go into <math>\mathrm{Y}</math>, or for <math>\scriptstyle \bar{\mathrm{B}}</math> to scatter with <math>\scriptstyle \mathrm{X}</math> to produce <math>\scriptstyle \mathrm{Y}+\bar{\mathrm{A}}</math>. The only difference is that the value of the energy is negative for the antiparticle.
The formal way to state this property is that the antiparticle scattering amplitudes are the [[analytic continuation]] of particle scattering amplitudes to negative energies. The interpretation of this statement is that the antiparticle is in every way a particle going backwards in time.
==History==
[[Murray Gell-Mann]] and [[Marvin Leonard Goldberger]] introduced crossing symmetry in 1954.<ref>{{cite journal | last1=Gell-Mann | first1=M. | last2=Goldberger | first2=M. L. | title=Scattering of Low-Energy Photons by Particles of Spin ½ | journal=Physical Review | publisher=American Physical Society (APS) | volume=96 | issue=5 | date=1 November 1954 | issn=0031-899X | doi=10.1103/physrev.96.1433 | pages=1433–1438| bibcode=1954PhRv...96.1433G | url=https://authors.library.caltech.edu/60413/1/PhysRev.96.1433.pdf }}</ref> Crossing had already been implicit in the work of [[Richard Feynman]], but came to its own in the 1950s and 1960s as part of the [[Bootstrap model|analytic S-matrix]] program.
==Overview==
Consider an amplitude <math>\mathcal{M}( \phi (p) + ... \ \rightarrow ...) </math>. We concentrate our attention on one of the incoming particles with momentum p. The quantum field <math> \phi (p) </math>, corresponding to the particle is allowed to be either bosonic or fermionic. Crossing symmetry states that we can relate the amplitude of this process to the amplitude of a similar process with an outgoing antiparticle <math> \bar{\phi} (-p) </math> replacing the incoming particle <math> \phi (p) </math>: <math>\mathcal{M}( \phi (p) + ... \rightarrow ...)=\mathcal{M}( ... \rightarrow ... + \bar{\phi} (-p) ) </math>.
In the bosonic case, the idea behind crossing symmetry can be understood intuitively using [[Feynman diagrams]]. Consider any process involving an incoming particle with momentum p. For the particle to give a measurable contribution to the amplitude, it has to interact with a number of different particles with momenta <math> q_{1}, q_{2}, ... , q_{n} </math> via a vertex. Conservation of momentum implies <math> \sum_{k=1}^{n} q_{k}=p </math>. In case of an outgoing particle, conservation of momentum reads as <math> \sum_{k=1}^{n}q_{k}=-p </math>. Thus, replacing an incoming boson with an outgoing antiboson with opposite momentum yields the same S-matrix element.
<!-- Commented out: [[Image:Crossing_symmetry.jpg]] -->
In fermionic case, one can apply the same argument but now the relative phase convention for the external spinors must be taken into account.
==Example==
▲
==See also==
*[[Feynman–Stueckelberg interpretation]]
*[[Feynman diagram]]
*[[Regge theory]]
*[[Detailed balance]]
==References==
{{reflist}}
==Further reading ==
*{{cite book
| last1 = Peskin
| first1 = M.
| authorlink1 = Michael Peskin
| last2 = Schroeder
| first2 = D.
| title = An Introduction to Quantum Field Theory
| publisher = Westview Press
| year = 1995
| isbn = 0-201-50397-2
| page = [https://archive.org/details/introductiontoqu0000pesk/page/155 155]
| url = https://archive.org/details/introductiontoqu0000pesk/page/155
}}
*{{cite book
| last = Griffiths
| first= David
| authorlink = David Griffiths (physicist)
| title = An Introduction to Elementary Particles
| edition = 1st
| publisher = John Wiley & Sons
| year = 1987
| isbn = 0-471-60386-4
| page = 21
}}
[[Category:Quantum field theory]]
[[Category:Scattering theory]]
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