The symmetry, inferable from Bogoliubov transformation, between the processes induced by the mirror in two-dimentional and the charge in four-dimentional space-time

TL;DR

This paper explores a deep symmetry between processes involving accelerated mirrors in 1+1 dimensions and charges in 3+1 dimensions, linking Bogoliubov transformations, field interactions, and fundamental invariants like spin and mass.

Contribution

It establishes a novel symmetry connecting pair creation by mirrors and charge emission, extending to spacelike field interactions and fixing the bare fine structure constant.

Findings

Bogoliubov coefficients relate to Fourier components of current and charge density.

Self-action changes and vacuum amplitudes coincide for mirrors and charges.

Trace of Bogoliubov matrices describes interactions with spacelike momentum fields.

Abstract

The symmetry between the creation of pairs of massless bosons or fermions by accelerated mirror in 1+1 space and the emission of single photons or scalar quanta by electric or scalar charge in 3+1 space is deepened in this paper. The relation of Bogoliubov coefficients with Fourier's components of current or charge density leads to the coicidence of the spin of any disturbances bilinear in scalar or spinor field with the spin of quanta emitted by the electric or scalar charge. The mass and invariant momentum transfer of these disturbances are essential for the relation of Bogoliubov coefficients with Green's functions of wave equations both for 1+1 and 3+1 spaces. Namely the relation (20) leads to the coincidence of the self-action changes and vacuum-vacuum amplitudes for the accelerated mirror in 1+1 space and charge in 3+1 space. Thus, both invariants of the Lorentz group, spin and mass, perform intrinsic role in established symmetry. The symmetry embraces not only the processes of real quanta radiation. It extends also to the processes of the mirror and the charge interactions with the fields carring spacelike momenta. These fields accompany their sources and define the Bogoliubov matrix coefficients \alpha^{B,F}. It is shown that the traces of \alpha^{B,F} describe the vector and scalar interactions of accelerated mirror with uniformly moving detector. This interpretation rests essentially on the relation (100) between the propagators of the waves with spacelike momenta in 2- and 4-dimentional spaces. The traces of \alpha^{B,F} coincide actually with the products of the mass shift \Delta m_{1,0} of accelerated electric or scalar charge and the proper time of the shift formation. The symmetry fixes the value of the bare fine structure constant \alpha_0=1/4\pi.