Transition amplitudes of interacting charged particles in an electric field and the Unruh effect

We compute the transition amplitudes between charged particles of mass $M$ and $m$ accelerated by a constant electric field and interacting by the exchange of quanta of a third field. We work in second quantization in order to take into account both recoil effects induced by transitions and the vacuum instability of the charged fields. In spite of both effects, when the exchanged particle is neutral, the equilibrium ratio of the populations is simply $\mathrm{exp}[\ensuremath{\pi}{(M}^{2}\ensuremath{-}{m}^{2})/eE].$ Thus, in the limit $(M\ensuremath{-}m)/\stackrel{\ensuremath{\rightarrow}}{M}0,$ one recovers Unruh's result characterized by the temperature $a/2\ensuremath{\pi}$ where $a$ is the acceleration. When the exchanged particle is charged, its vacuum instability prevents a simple description of the equilibrium state. However, in the limit wherein the charge of the exchanged particle tends to zero, the equilibrium distribution is once more Boltzmannian, but characterized not only by a temperature but also by the electric potential felt by the exchanged particle. This work therefore confirms that thermodynamics in the presence of horizons does not rely on a semiclassical treatment. The relationship with horizon thermodynamics and the role of the horizon area as an entropy are stressed.