Symmetries and causes of the coincidence of the emission spectra of mirrors and charges in 1+1 and 3+1 spaces

Abstract

This paper discusses the symmetry of the wave field that lies to the right and left of a two-sided accelerated mirror in 1+1 space and satisfies a single condition on it. The symmetry is embodied in the Bogolyubov matrix coefficients α and β that connect the two complete sets of solutions of the wave equations. The amplitudes of the quantum processes in the right and left half-spaces are expressed in terms of α and β and are related to each other by the transformation (12). The coefficient β ^* _ω′ω plays the role of the source amplitude of a pair of oppositely directed particles with frequencies ω and ω′ of which one is in the left half-space and the other is in the right half-space because one of them has undergone reflection. Such an interpretation makes β ^* _ω′ω observable and explains why, as shown by Eq. (1) and found earlier by Nikishov and Ritus [Zh. Éksp. Teor. Fiz. 108, 1121 (1995); transl. JETP 81, 615 (1995)] and by Ritus [Zh. Éksp. Teor. Fiz. 110, 526 (1996); transl. JETP 83, 282 (1996)], the emission spectra of a mirror in 1+1 space coincide with those of charges in 3+1 space. The reason is that the angular momentum of the pair emitted by the mirror coincides with the angular momentum of the single particle emitted by the charge.