标题： 量子辐射中的振幅 I. 复数方法 显示英文标题

标题： Quantum Amplitudes in Black-Hole Evaporation I. Complex Approach

摘要： 此处我们研究由引力坍缩形成的类施瓦茨希尔德黑洞的量子力学衰变过程，在非常晚的时间进入几乎平坦的时空和弱辐射，以评估最终状态的量子幅度（而不仅仅是概率）。 在坍缩成黑洞过程中没有信息丢失。 初始和最终超曲面上指定了边界数据 $\Sigma_{I, F}$，它们之间由一个洛伦兹固有时间隔 $T$分隔，该间隔在空间无穷远处测量。 为了简化，考虑爱因斯坦引力与无质量标量场 $\phi$的最小耦合。 在洛伦兹符号下，对应的经典狄利克雷边值问题，即在边界面上指定内在空间度规 $h_{ij} (i,j =1,2,3)$和 $\phi$，是不适定的，这是一个波动类似 （双曲型）方程组的边值问题。 按照费曼的$+i\epsilon$预定，通过将渐近时间区间$T$旋转到复数中：$T\to{\mid} T{\mid}\exp(-i\theta)$，其中$0<\theta\leq\pi/2$。在计算了$\theta>0$的振幅之后，取 '洛伦兹极限'$\theta\to 0_+$来得到洛伦兹量子振幅。 显示英文摘要

摘要： Here we examine the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time, in order to evaluate quantum amplitudes (not just probabilities) for final states. No information is lost in collapse to a black hole. Boundary data are specified on initial and final hypersurfaces $\Sigma_{I, F}$, separated by a Lorentzian proper-time interval $T$, as measured at spatial infinity. For simplicity, consider Einstein gravity coupled minimally to a massless scalar field $\phi$. In Lorentzian signature, the classical Dirichlet boundary-value problem, corresponding to specification of the intrinsic spatial metric $h_{ij} (i,j =1,2,3)$ and $\phi$ on the bounding surfaces, is badly posed, being a boundary-value problem for a wave-like (hyperbolic) set of equations. Following Feynman's $+i\epsilon$ prescription, the problem is made well-posed by rotating the asymptotic time interval $T$ 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' $\theta\to 0_+$ to obtain the Lorentzian quantum amplitude.