  In earlier Letters, we adopted a complex approach to quantum processes in the
formation and evaporation of black holes. Taking Feynman's $+i\epsilon$
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(-i\delta) 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.
