In flat spacetime, two inequivalent vacuum states which arise rather naturally are the Rindler vacuum (R) and the Minkowski vacuum (M). We disuss several aspects of the Rindler vacuum, concentrating on the propagator and Schwinger (heat) kernel defined using R, both in the Lorentzian and Euclidean sectors. We start by exploring an intriguing result due to Candelas and Raine , viz., that $G_{R}$, the Feynman propagator corresponding to R, can be expressed as a curious integral transform of $G_{M}$, the Feynman propagator in M. We show that, this relation actually follows from the well known result that, $G_{M}$ can be written as a periodic sum of $G_{R}$, in the Rindler time $\tau$, with the period $2\pi i$. We further show that, the integral transform result holds for a wide class of pairs of bi-scalars $(F_{M},F_{R})$, provided $F_{M}$ can be represented as a periodic sum of $F_{R}$ with period $2\pi i$. We provide an explicit procedure to retrieve $F_{R}$ from its periodic sum $F_{M}$, for a wide class of functions. An example of particular interest is the pair of Schwinger kernels $(K_{M},K_{R})$, corresponding to the Minkowski and the Rindler vacua. We obtain explicit expression for $K_{R}$ and clarify several conceptual and technical issues related to these biscalars both in the Euclidean and Lorentzian sector. In particular we address the issue of retrieving the information contained in all the four wedges of the Rindler frame in the Lorentzian sector, starting from the Euclidean Rindler (polar) coordinates. This is possible but require four different types of analytic continuations, based on one unifying principle. Our procedure allows generalisation of these results to any (bifurcate Killing) horizon in curved spacetime.

中文翻译：

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探索林德勒真空和欧几里得平面

在平坦的时空中，自然产生的两种不等价真空状态是林德勒真空 (R) 和闵可夫斯基真空 (M)。我们讨论了林德勒真空的几个方面，集中在洛伦兹扇区和欧几里得扇区中使用 R 定义的传播子和施温格（热）核。我们首先探索由 Candelas 和 Raine 引起的一个有趣的结果，即 $G_{R}$，对应于 R 的费曼传播子，可以表示为费曼传播子 $G_{M}$ 的一个奇怪的积分变换我们证明，这种关系实际上是从众所周知的结果得出的，$G_{M}$ 可以写成 $G_{R}$ 的周期和，在 Rindler 时间 $\tau$，具有期间 $2\pi i$。我们进一步表明，积分变换结果适用于广泛的双标量对 $(F_{M},F_{R})$，提供的 $F_{M}$ 可以表示为 $F_{R}$ 的周期和，周期为 $2\pi i$。我们提供了一个显式的过程来从它的周期总和 $F_{M}$ 中检索 $F_{R}$，用于广泛的函数类。一个特别有趣的例子是一对 Schwinger 核 $(K_{M},K_{R})$，对应于 Minkowski 和 Rindler vacua。我们获得了 $K_{R}$ 的明确表达式，并在欧几里得和洛伦兹部门中阐明了与这些双标量相关的几个概念和技术问题。特别是，我们解决了从欧几里得林德勒（极坐标）坐标开始，检索洛伦兹扇区中林德勒框架的所有四个楔形中包含的信息的问题。这是可能的，但需要基于一个统一原则的四种不同类型的分析延续。