Crossing symmetry
Encyclopedia
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
to scatter with 
and produce particle 
and 
will also give the scattering amplitude for 
to go into 
, or for 
to scatter with 
to produce 
. 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.
. We concentrate our attention on one of the incoming particles with momentum p. The quantum field 
, 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 
replacing the incoming particle 
: 
.
In 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
via a vertex. Conservation of momentum implies 
. In case of an outgoing particle, conservation of momentum reads as 
. 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.
of an electron
with a positron
into two photon
s is related to an elastic scattering
of an electron with a photon (Compton scattering
) 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
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, 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
to scatter with and produce particle and will also give the scattering amplitude for to go into , or for to scatter with to produce . 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
Bootstrap model
In physics, the term bootstrap model is used for a class of theories that use very general consistency criteria to determine the form of a quantum theory from some assumptions on the spectrum of particles...
program.
General overview
Consider an amplitude. We concentrate our attention on one of the incoming particles with momentum p. The quantum field , 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 replacing the incoming particle : .In 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
via a vertex. Conservation of momentum implies . In case of an outgoing particle, conservation of momentum reads as . 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.
Example
For example, the annihilationAnnihilation
Annihilation is defined as "total destruction" or "complete obliteration" of an object; having its root in the Latin nihil . A literal translation is "to make into nothing"....
of an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
with a positron
Positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...
into two photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s is related to an elastic scattering
Elastic scattering
In scattering theory and in particular in particle physics, elastic scattering is one of the specific forms of scattering. In this process, the kinetic energy of the incident particles is conserved, only their direction of propagation is modified .-Electron elastic scattering:When an alpha particle...
of an electron with a photon (Compton scattering
Compton scattering
In physics, Compton scattering is a type of scattering that X-rays and gamma rays undergo in matter. The inelastic scattering of photons in matter results in a decrease in energy of an X-ray or gamma ray photon, called the Compton effect...
) by crossing symmetry. This relation allows to calculate the scattering amplitude
Scattering amplitude
In quantum physics, the scattering amplitude is the amplitude of the outgoing spherical wave relative to the incoming plane wave in the stationary-state scattering process...
of one process from the amplitude for the other process if negative values of energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
of some particles are substituted.