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Crossing symmetry : ウィキペディア英語版
Crossing (physics)

In quantum field theory, a branch of theoretical physics, crossing is the property of scattering amplitudes 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 \mathrm to scatter with \mathrm and produce particle \mathrm and \mathrm will also give the scattering amplitude for \scriptstyle \mathrm+\bar to go into \mathrm, or for \scriptstyle \bar to produce \scriptstyle \mathrm+\bar{\mathrm{A}}. 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.
Crossing was already implicit in the work of Feynman, but came to its own in the 1950s and 1960s as part of the analytic S-matrix program.
==General overview==
Consider an amplitude \mathcal( \phi (p) + ... \ \rightarrow ...) . We concentrate our attention on one of the incoming particles with momentum p. The quantum field \phi (p) , 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 \bar (-p) replacing the incoming particle \phi (p) : \mathcal( \phi (p) + ... \rightarrow ...)=\mathcal( ... \rightarrow ... + \bar (-p) ) .
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 k_, k_, ... , k_ via a vertex. Conservation of momentum implies \sum_^ q_=p . In case of an outgoing particle, conservation of momentum reads as \sum_^q_=-p . Thus, replacing an incoming boson with an outgoing antiboson with opposite momentum yields the same S-matrix element.

In fermionic case, one can apply the same argument but now the relative phase convention for the external spinors must be taken into account.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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