Bogoliubov transformations for amplitudes in black-hole evaporation

In a previous Letter, we outlined an approach to the calculation of quantum amplitudes appropriate for studying the black-hole radiation which follows gravitational collapse. This formulation must be different from the familiar one (which is normally carried out by considering Bogoliubov transformations), since it yields quantum amplitudes relating to the final state, and not just the usual probabilities for outcomes at a late time and large radius. Our approach simply follows Feynman's +iε prescription. Suppose that, in specifying the quantum amplitude to be calculated, initial data for Einstein gravity and (say) a massless scalar field are specified on an asymptotically-flat space-like hypersurface ΣI, and final data similarly specified on a hypersurface ΣF, where both ΣI and ΣF are diffeomorphic to R3. Denote by T the (real) Lorentzian proper-time interval between ΣI and ΣF, as measured at spatial infinity. Then rotate: T→|T|exp(−iθ), 0<θ⩽π/2. The classical boundary-value problem is then expected to become well-posed on a region of topology I×R3, where I is the interval [0,|T|]. For a locally-supersymmetric theory, the quantum amplitude is expected to be dominated by the semi-classical expression exp(iSclass), where Sclass is the classical action. Hence, one can find the Lorentzian quantum amplitude from consideration of the limit θ→0+. In the usual approach, the only possible such final surfaces ΣF are in the strong-field region shortly before the curvature singularity; that is, one cannot have a Bogoliubov transformation to a smooth surface 'after the singularity'. In our complex approach, however, one can put arbitrary smooth gravitational data on ΣF, provided that it has the correct mass M; thus we do have Bogoliubov transformations to surfaces 'after the singularity in the Lorentzian-signature geometry'-the singularity is simply by-passed in the analytic continuation (see below). In this Letter, we consider Bogoliubov transformations in our approach, and their possible relation to the probability distribution and density matrix in the traditional approach. In particular, we find that our probability distribution for configurations of the final scalar field cannot be expressed in terms of the diagonal elements of some density-matrix distribution.