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 +i epsilon 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 Sigma(I), and final data similarly specified on a hypersurface Sigma(F), where both Sigma(I) and Sigma(F) are diffeomorphic to R-3. Denote by T the (real) Lorentzian proper-time interval between Sigma(I) and Sigma(F), as measured at spatial infinity. Then rotate: T -> vertical bar T vertical bar exp(-i theta), 0 < theta <= pi/2. The classical boundary-value problem is then expected to become well-posed on a region of topology I x R-3, where I is the interval [0, vertical bar T vertical bar]. For a locally-supersymmetric theory, the quantum amplitude is expected to be dominated by the semi-classical expression exp(iS(class)), where S-class is the classical action. Hence, one can find the Lorentzian quantum amplitude from consideration of the limit theta -> 0(+). In the usual approach, the only possible such final surface Sigma(F) are in the strontg-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 Sigma(F), 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. (c) 2005 Elsevier B.V. All rights reserved.
