Spin-$1\over 2$ amplitudes in black-hole evaporation

We extend to the fermionic spin-1/2 case earlier work on quantum amplitudes arising from gravitational collapse to a black hole. Boundary data are specified on initial and final asymptotically-flat space-like hypersurfaces $\Sigma_{I,F}$, separated by a Lorentzian proper-time interval $T$, measured at spatial infinity. Following Feynman's $+i\epsilon$ prescription, one makes the problem well-posed by rotating $T$ into the complex: $T\to{\mid}T{\mid} \exp(-i\theta)$, with $0<\theta\leq\pi/2$. After calculating the amplitude for $\theta>0$, one takes the 'Lorentzian limit' $\theta\to 0_+$. In this paper, we treat quantum amplitudes for the case of fermionic massless spin-1/2 (neutrino) final boundary data; working in the holomorphic representation, we take these boundary data to be odd elements of a Grassmann algebra. Making use of boundary conditions originally developed for local supersymmetry, we find that this fermionic case can be treated in a way which parallels the bosonic case. With these boundary conditions, for $\theta > 0$, one obtains a unique fermionic classical solution, and we calculate its classical action as a functional of the fermionic data on the late-time surface $\Sigma_F$; the quantum amplitude follows straightforwardly from this.