Cherenkov radiation in a photon gas

It is well-known that a charged particle moving with constant velocity in vacuum does not radiate. In a medium the situation can be different. If the so called Cherenkov condition is satisfied, i.e. the particle velocity exceeds the phase speed in the medium, the particle will radiate. We show that a charge moving with a constant velocity in a gas of photons emits Cherenkov radiation, even in the gamma-ray regime, due to nonlinear quantum electrodynamic effects. Our result is evaluated with respect to the radiation background in the early universe, and it is argued that the effect can be significant. PACS numbers: 41.60.Bq, 98.80.-k In 1934, Cherenkov observed the type of radiation now bearing his name [1]. His experimental result was explained by Tamm & Frank [2]. In an isotropic dielectric medium, a charged particle in rectilinear motion satisfying the so called Cherenkov condition, i.e. its velocity exceeds the (parallel) phase speed in the medium in which it moves, will thus radiate [3]. The radiation chock-front, called the Cherenkov cone, is analogous to the Mach cone formed as objects move with supersonic speeds through air. In quantum mechanical terms, the Cherenkov condition corresponds to energy and momentum conservation. Cherenkov radiation has technological uses, e.g. in determining particle velocities. Quantum electrodynamics (QED) predicts many phenomena with no classical counterparts, such as the Casimir effect and elastic photon–photon scattering [4, 5, 6]. While the former has been experimentally confirmed, the latter has still to be detected [7]. In addition, the effective field theory describing photon–photon scattering [6] has been widely used to predict possible effects in, for example, extreme magnetized objects, such as neutron stars and magnetars. One effect of a strong magnetic field is to downshift the frequency of a test photon, the so called photon splitting [8, 9]. In this Letter, we predict that a charged particle moving in an equilibrium radiation gas will emit Cherenkov radiation. The possible importance of this effect is then discussed for the cosmic radiation background. A related, but significantly different, study was presented by Dremin [10]. The dispersion relation for electromagnetic waves Cherenkov radiation in a photon gas 2 in an isotropic and homogeneous photon gas with refractive index n is ω = kc/n, where n = 1 + δ and [11, 12] δ = 2bαE 135πǫ0E2 S ≈ bE