角分裂法中$\left\langle ϕ^{2} \right\rangle$的模和正则化
Mode-sum regularization of $\left\langle ϕ^{2} \right\rangle$ in the angular-splitting method
摘要：
弯曲时空中的重整化应力-能量张量（$\left\langleϕ^{2}\right\rangle_{ren}$）的计算在概念和技术层面上都是一项具有挑战性的任务。最近，我们开发了一种新的方法来计算这样的重整化量在渐近平坦的弯曲时空，基于点分裂过程。我们的方法要求时空具有某种对称性。我们已经实现了这种方法来计算$\left\langle ϕ^{2}\right\rangle _{ren}$在一个固定的时空使用t分裂，即分裂的时间平移方向。在这里，我们提出了这种方法的角度分裂版本，旨在计算重整化量在一般（可能是动态的）球对称时空。为了说明角分裂方法是如何工作的，我们在这里用它来计算史瓦西背景中的量子无质量标量场在各种量子态（Boulware、Unruh和Hartle-Hawking态）下的$\left\langle ϕ^{2}\right\rangle _{ren}$。我们发现很好的协议从t-分裂的变体，并与其他方法得到的结果。我们追求这种新的模式和方法的主要目标是使重整化的应力能量张量的计算在一个动态球对称的背景下，例如，蒸发黑洞。这里提出的角度分裂变体最适合于此目的。
英文摘要：
The computation of the renormalized stress-energy tensor or $\left\langleϕ^{2}\right\rangle_{ren}$ in curved spacetime is a challenging task, at both the conceptual and technical levels. Recently we developed a new approach to compute such renormalized quantities in asymptotically-flat curved spacetimes, based on the point-splitting procedure. Our approach requires the spacetime to admit some symmetry. We already implemented this approach to compute $\left\langle ϕ^{2}\right\rangle _{ren}$ in a stationary spacetime using t-splitting, namely splitting in the time-translation direction. Here we present the angular-splitting version of this approach, aimed for computing renormalized quantities in a general (possibly dynamical) spherically-symmetric spacetime. To illustrate how the angular-splitting method works, we use it here to compute $\left\langle ϕ^{2}\right\rangle _{ren}$ for a quantum massless scalar field in Schwarzschild background, in various quantum states (Boulware, Unruh, and Hartle-Hawking states). We find excellent agreement with the results obtained from the t-splitting variant, and also with other methods. Our main goal in pursuing this new mode-sum approach was to enable the computation of the renormalized stress-energy tensor in a dynamical spherically symmetric background, e.g. an evaporating black hole. The angular-splitting variant presented here is most suitable to this purpose.
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