标题： 黑洞蒸发中振幅的布洛赫-戈尔基夫变换 显示英文标题

标题： Bogoliubov transformations for amplitudes in black-hole evaporation

摘要： 遵循黑洞坍缩后的量子辐射的熟悉方法是通过Bogoliubov变换进行的，并给出了最终结果的概率。 在我们的（复数）方法中，我们通过遵循费曼的$+i\epsilon$公式来找到量子振幅，而不仅仅是概率。 爱因斯坦引力和（例如）无质量标量场的初始和最终数据是在一对渐近平坦的类空超曲面$\Sigma_I$和 $\Sigma_F$上指定的；两者都微分同胚于${\Bbb R}^3$。 将$T$表示为在空间无穷远处测量的两个曲面之间的（实数）洛伦兹固有时间隔。 然后旋转：$T\to{\mid}T{\mid}\exp(-i\theta),0<\theta\leq \pi/2$。 {\it 经典}边界值问题在拓扑为$I\times{\Bbb R}^3$的区域上预期是适定的，其中$I$是一个闭区间。 对于局部超对称理论，量子振幅应由半经典表达式$\exp(iS_{\rm class})$占主导，其中$S_{\rm class}$是经典作用量。 从极限$\theta\to 0_+$可以得到洛伦兹量子振幅。 在通常的方法中，可能的最终表面仅存在于曲率奇点之前的强场区域。 在我们的方法中，可以在$\Sigma_F$上放置任意平滑的引力数据，只要它具有正确的质量$M$—— 奇点在解析延拓中被绕过了。 这里，我们考虑Bogoliubov变换及其与传统方法中概率分布和密度矩阵的可能关系。 我们发现，最终标量场配置的概率分布无法用某种非平凡密度矩阵分布的对角元素来表示。 显示英文摘要

摘要： The familiar approach to quantum radiation following collapse to a black hole proceeds via Bogoliubov transformations, and yields probabilities for final outcomes. In our (complex) approach, we find quantum amplitudes, not just probabilities, by following Feynman's $+i\epsilon$ prescription. Initial and final data for Einstein gravity and (say) a massless scalar field are specified on a pair of asymptotically-flat space-like hypersurfaces $\Sigma_I$ and $\Sigma_F$; both are diffeomorphic to ${\Bbb R}^3$. Denote by $T$ the (real) Lorentzian proper-time interval between the surfaces, as measured at spatial infinity. Then rotate: $T\to{\mid}T{\mid}\exp(-i\theta),0<\theta\leq \pi/2$. The {\it classical} boundary-value problem is expected to be well-posed on a region of topology $I\times{\Bbb R}^3$, where $I$ is a closed interval. For a locally-supersymmetric theory, the quantum amplitude should be dominated by the semi-classical expression $\exp(iS_{\rm class})$, where $S_{\rm class}$ is the classical action. One finds the Lorentzian quantum amplitude from the limit $\theta\to 0_+$. In the usual approach, the only possible such final surfaces are in the strong-field region shortly before the curvature singularity. In our approach one can put arbitrary smooth gravitational data on $\Sigma_F$, provided that it has the correct mass $M$ -- the singularity is by-passed in the analytic continuation. Here, we consider Bogoliubov transformations and their possible relation to the probability distribution and density matrix in the traditional approach. We find that our probability distribution for configurations of the final scalar field cannot be expressed in terms of the diagonal elements of some non-trivial density-matrix distribution.