Minkowski Bessel modes

TL;DR

This paper introduces Minkowski Bessel modes and their Mellin variants, describing their mathematical properties, relation to Minkowski plane waves, and behavior in accelerated frames, with implications for quantum field theory in curved spacetime.

Contribution

It presents the explicit form and properties of Minkowski Bessel modes, including their relation to known modes and their behavior in accelerated frames, advancing understanding of quantum fields in such settings.

Findings

Minkowski Bessel modes are unitarily related to Minkowski plane waves.

They form a unitary representation of translation group in 2D Minkowski space.

Mode sum approaches a mode integral as the accelerated frame's bottom approaches the horizon.

Abstract

The global Minkowski Bessel (M-B) modes, whose explicit form allows the identification and description of the condensed vacuum state resulting from the operation of a pair of accelerated refrigerators, are introduced. They span the representation space of the unitary representation of the Poincare group on 2-D Lorentz space-time. Their three essential properties are: (1) they are unitarily related to the familiar Minkowski plane waves; (2) they form a unitary representation of the translation group on two dimensional Minkowski spacetime. (3) they are eigenfunctions of Lorentz boosts around a given reference event. In addition the global Minkowski Mellin modes are introduced. They are the singular limit of the M-B modes. This limit corresponds to the zero transverse momentum solutions to the zero rest mass wave equation. Also introduced are the four Rindler coordinate representatives of each global mode. Their normalization and density of states are exhibited in a (semi-infinite) accelerated frame with a finite bottom. In addition we exhibit the asymptotic limit as this bottom approaches the event horizon and thereby show how a mode sum approaches a mode integral as the frame becomes bottomless.