arXiv 1002.39792010-02-21gr-qc

黑洞蒸发中的量子振幅：复近似和自旋0振幅

Quantum Amplitudes in Black-Hole Evaporation: Complex Approach and Spin-0 Amplitude

A. N. St. J. Farley, P. D. D'Eath

备注：: 27 pages
DOI：: 10.1142/S021827181101872X

摘要：

研究了类Schwarzschild黑洞在引力坍缩过程中的量子力学衰变。我们评估了从初始状态到最终状态的跃迁的量子振幅（而不仅仅是概率），并表明在坍缩到黑洞的过程中没有信息丢失。引力场和标量场的边界数据被放置在初始类空超曲面$Σ_I$和最终曲面$Σ_F$上。这些渐近平坦的3-曲面由洛伦兹固有时间间隔$T$分开，在空间无穷远处测量。通过将$T$旋转到下半复平面$T\to {\mid}T{\mid}\exp(-iθ), 0<θ\leqπ/2$中，使边值问题在经典和量子力学上都是适定的。这对应于Feynman的$+iε$处方。对于经典边值问题，我们计算了二阶变分经典洛伦兹作用量$S^{(2)}_{class}$作为边界数据的泛函。根据费曼的理论，从定义良好的复$T$振幅恢复到极限$θ\to 0_{+}$中的洛伦兹量子振幅.狄拉克对约束系统量子化的正则方法表明，对于局部超对称引力理论，振幅正好是半经典的：除了超对称约束的δ泛函之外，弱扰动的振幅是$\exp(i S^{(2)}_{class})$。我们把这样的量子振幅弱标量场配置$Σ_F$，采取弱最终引力场是球对称的。治疗涉及绝热解决方案的标量波动方程。这扩展了我们以前的工作，给出了这种量子振幅的实部和虚部的明确表达式。

英文摘要：

We consider the quantum-mechanical decay of a Schwarzschild-like black hole formed by gravitational collapse into almost-flat space-time and weak radiation at a late time. We evaluate quantum amplitudes (not just probabilities) for transitions from initial to final states, and show that no information is lost in collapse to a black hole. Boundary data for the gravitational field and a scalar field are posed on an initial space-like hypersurface $Σ_I$ and a final surface $Σ_F$. These asymptotically-flat 3-surfaces are separated by a Lorentzian proper-time interval $T$, measured at spatial infinity. The boundary-value problem is made well-posed, classically and quantum-mechanically, by rotating $T$ into the lower-half complex plane: $T\to {\mid}T{\mid}\exp(-iθ), 0<θ\leqπ/2$. This corresponds to Feynman's $+iε$ prescription. For the classical boundary-value problem, we calculate the second-variation classical Lorentzian action $S^{(2)}_{class}$ as a functional of the boundary data. Following Feynman, the Lorentzian quantum amplitude is recovered in the limit $θ\to 0_{+}$ from the well-defined complex-$T$ amplitude. Dirac's canonical approach to the quantisation of constrained systems shows that for locally-supersymmetric theories of gravity the amplitude is exactly semi-classical: $\exp(i S^{(2)}_{class})$ for weak perturbations, apart from delta-functionals of the supersymmetry constraints. We treat such quantum amplitudes for weak scalar-field configurations on $Σ_F$, taking the weak final gravitational field to be spherically symmetric. The treatment involves adiabatic solutions of the scalar wave equation. This extends our previous work, giving explicit expressions for the real and imaginary parts of such quantum amplitudes.

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