Spin-$\frac{1}{2}$ amplitudes in black-hole evaporation

Abstract

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 $0<\theta \le \pi /2$. 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.

References in corpus (3)

• Bogoliubov transformations for amplitudes in black-hole evaporation
• Scalar--Field Amplitudes in Black--Hole Evaporation
• Spin-2 Amplitudes in Black-Hole Evaporation

Cited by in corpus (9)

• Vaidya Space-Time in Black-Hole Evaporation
• Black hole evaporation in a spherically symmetric non-commutative space-time
• Gravitational amplitudes in black-hole evaporation: the effect of non-commutative geometry
• Bogoliubov transformations for amplitudes in black-hole evaporation
• Coherent and squeezed states in black-hole evaporation
• Bogoliubov transformations in black-hole evaporation
• Quantum amplitudes in black-hole evaporation: Spins 1 and 2
• Quantum amplitudes in black-hole evaporation: coherent and squeezed states
• Quantum Amplitudes in Black-Hole Evaporation: Complex Approach and Spin-0 Amplitude