Quantum amplitudes in black-hole evaporation: coherent and squeezed states

The quantum amplitude for processes involving the formation and evaporation of black holes was previously calculated by means of a complex-time approach. In that treatment, we followed Feynman's $+i\epsilon$ approach in quantum field theory. The Lorentzian time interval $T$, measured at spatial infinity between a pair of asymptotically flat spacelike hypersurfaces $\Sigma_I$ and $\Sigma_F$ carrying initial and final boundary data for the gravitational and other fields, is rotated: $T\to{\mid}T{\mid}\exp(-i\delta)$, where $0< \delta \leq{\pi}/2$. Classically and quantum mechanically, this procedure is expected to lead to a well-posed boundary-value problem. Thus, we have found quantum amplitudes (not just probability densities) relating to a pure state at late times following gravitational collapse of matter to a black hole. Such pure states, arising from gravitational collapse, admit a description in terms of coherent and squeezed states. Indeed, this description is not so different from that arising in a well-known context, namely, the highly-squeezed final state of the relic radiation background in inflationary cosmology. For definiteness, we study the simplest model of collapse, based on Einstein gravity with a massless scalar field. Following the complex rotation above, one finds that, in an adiabatic approximation, the resulting quantum amplitude may be expressed in terms of generalized coherent states of the harmonic oscillator. A physical interpretation is given; further, a squeezed-state representation follows.