Transient evolution of a photon gas in the nonlinear QED vacuum Page: 6 of 21
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The technical steps of the derivation start from the photon-photon differential scattering rate.
The total scattering rate is expressed in terms of manifestly covariant quantities in the S-matrix
formulation. Next, using the symmetries of the problem, the differential rates of production of
scattered photons and loss of incident photons are derived for arbitrary isotropic spectral
distributions. Finally, the integro-differential evolution equation thus obtained is applied to
specific cases of interest. Explicit calculations are carried out showing transient evolution of a
photon gas under photon-photon scattering.
It is also important to emphasize the assumptions under which the derivation is performed and
our conclusions are valid. First and foremost, the Heisenberg-Euler theory must hold - the
photon energy is small compared to the electron rest energy and fields weak relative to the
Schwinger critical field. Second, the mean free path between photon-photon collisions must
remain much longer than the wavelengths involved, so that the photon energy is well defined.
Another key point is that the photon number density and energy density conservation are
conserved. Interaction with matter can change one or both quantities, hence, for this assumption
to be valid, there must be negligible matter content in the region of interest. While 4-momentum
conservation and photon number conservation are evident requirements for a given interaction,
volumetric density conservation implies that the system is homogeneous, isotropic and
unbounded, and that the interaction time scale is short compared to all other time scales. In
reality, at any given time, part of the energy-momentum content of the system appears in the
form of electron-positron pairs, thus diminishing the electromagnetic component of the 4-
momentum and introducing more scattering mechanisms; this is neglected in the present
analysis.
The transition rate density dW of a general scattering process in which two incident particles
collide and are transformed into two outgoing particles, h+k ->h'+k', is [6]:
dW = (2t4 4 (ph + pk -- ph -- , d3ph (3 )
4EnE' (2w)32Eh' (2w)32Ek
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Wu, S Q & Hartemann, F V. Transient evolution of a photon gas in the nonlinear QED vacuum, report, October 4, 2011; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc834710/m1/6/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.