Minkowski Bessel modes

The global Minkowski Bessel (MB) 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 a unitary representation of the Poincaré group on two-dimensional 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 space-time, and (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 MB 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. This is the infinite Regge-Wheeler volume limit.