paper

Coherent and squeezed states in black-hole evaporation

arXiv:gr-qc/0603092 · doi:10.1016/j.physletb.2006.01.020

Abstract

In earlier Letters, we adopted a complex approach to quantum processes in the formation and evaporation of black holes. Taking Feynman's prescription, rather than than one of the more usual approaches, we calculated the quantum amplitude (not just the probability density) for final weak-field configurations following gravitational collapse to a black hole with subsequent evaporation. What we have done is to find quantum amplitudes relating to a pure state at late times following black-hole matter collapse. Such pure states are then shown to be susceptible to a description in terms of coherent and squeezed states - in practice, this description is not very different from that for the well-known highly-squeezed final state of the relic radiation background in inflationary cosmology. The simplest such collapse model involves Einstein gravity with a massless scalar field. The Feynman approach involves making the boundary-value problem for gravity and a massless scalar field well-posed. To define this, let T be the proper-time separation, measured at spatial infinity, between two space-like hypersurfaces on which initial (collapse) and final (evaporation) data are posed. Then, in this approach, one rotates T by a complex phase exp(-iδ) into the lower half-plane. In an adiabatic approximation, the resulting quantum amplitude may be expressed in terms of generalised coherent states of the quantum oscillator, and a physical interpretation is given. A squeezed-state representation, as above, then follows.

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