Bogoliubov transformations in black-hole evaporation

In previous papers and letters on quantum amplitudes in black-hole evaporation, a boundary-value approach was developed for calculating (for example) the quantum amplitude to have a prescribed slightly non-spherical configuration of a massless scalar field phi on a final hypersurface Sigma(F) at a very late time T, given initial almost-stationary spherically symmetric gravitational and scalar data on a space-like hypersurface Sigma(I) at time t = 0. For definiteness, we assumed that the gravitational data are also spherically symmetric on Sigma(F). Such boundary data can correspond to a classical solution for the Einstein/scalar system, describing gravitational collapse from an early low-density con. guration to a nearly Schwarzschild black hole. This approach provides the quantum amplitude (not just the probability) for a transition from an initial to a final state. For a real Lorentzian time-interval T, the classical boundary-value problem refers to a set of hyperbolic equations (modulo gauge), and is badly posed. Instead, the boundary-value approach of the previous letters and papers requires (following Feynman) a rotation into the complex: T --> \ T\ exp(- i theta), for 0 < theta <= pi/2, of the time-separation-at-infinity T. The classical boundary-value problem, for a complex solution of the coupled nonlinear classical field equations, is expected to be well-posed for 0 < theta <= pi/2. For a locally supersymmetric Lagrangian, containing supergravity coupled to supermatter, the classical Lorentzian action S-class, a functional of the boundary data (which include the complexified T), yields a quantum amplitude proportional to exp(iS(class)), apart from possible loop corrections which are negligible for boundary data with frequencies below the Planck scale. Finally (still following Feynman), one computes the Lorentzian quantum amplitude by taking the limit of exp(iS(class)) as theta --> 0(+). In the present paper, a connection is made between the above boundary-value approach and the original approach to quantum evaporation in gravitational collapse to a black hole, via Bogoliubov coefficients. This connection is developed through consideration of the radial equation obeyed by the (adiabatic) non-spherical classical perturbations. When one studies the probability distribution for configurations of the final scalar field, based on our quantum amplitudes above, one finds that this distribution can also be interpreted in terms of the Wigner quasi-probability distribution for a harmonic oscillator.