Quantum Amplitudes in Black-Hole Evaporation II. Spin-0 Amplitude
  This work on spin-0 amplitudes in black-hole evaporation is based on the underlying results and methods outlined in our first paper, "I. Complex Approach". The main result here, and the model calculation for work on all higher spins, as described in several further papers, is the computation of the quantum amplitude (rather than merely the probability) for a given slightly anisotropic configuration of a scalar field $\phi$ on a space-like hypersurface $\Sigma_F$ at a very late time $T$. For simplicity, one may take the initial data for gravity and the massless scalar field at an initial surface $\Sigma_I$ to be spherically symmetric. This applies to perturbations of spherically-symmetric collapse to a black hole, starting from a diffuse, nearly-stationary configuration, where the bosonic part of the Lagrangian consists of Einstein gravity and the massless scalar field. As in Paper I, Feynman's $+i\epsilon$ approach is taken; this involves a rotation into the complex: $T\to {\mid}T{\mid} \exp (-i\theta)$, with $0<\theta\leq\pi/2$. A complex solution of the classical boundary-value problem is expected to exist, provided $\theta>0$; although for $\theta =0$ (Lorentzian time-separation), the classical boundary-value problem is badly posed. Once the amplitude is found for $\theta>0$, one can take the limit $\theta\to 0_+$ to find the Lorentzian amplitude. The paper also includes a discussion of adiabatic solutions of the scalar wave equation, needed for the spin-0 calculation. 
A.N.St.J.Farley and P.D.D'Eath