Exploring the Rindler vacuum and the Euclidean plane

Abstract

In flat spacetime, two inequivalent vacuum states that arise rather naturally are the Rindler vacuum $|\mathcal{R}〉$ and the Minkowski vacuum $|\mathcal{M}〉$ . We discuss several aspects of the Rindler vacuum, concentrating on the propagator and Schwinger (heat) kernel defined using $|\mathcal{R}〉$ , both in the Lorentzian and Euclidean sectors. We start by exploring an intriguing result due to Candelas and Raine [J. Math. Phys. 17, 2101 (1976)], viz., that $G_{\mathcal{R}}$ , the Feynman propagator corresponding to $|\mathcal{R}〉$ , can be expressed as a curious integral transform of $G_{\mathcal{M}}$ , the Feynman propagator in $|\mathcal{M}〉$ . We show that this relation follows from the well-known result that $G_{\mathcal{M}}$ can be written as a periodic sum of $G_{\mathcal{R}}$ , in the Rindler time τ, with the period (in proper units) 2πi. We further show that the integral transform result holds for a wide class of pairs of bi-scalars $\left\{F_{\mathcal{M}},F_{\mathcal{R}}\right\}$ , provided that $F_{\mathcal{M}}$ can be represented as a periodic sum of $F_{\mathcal{R}}$ with period 2πi. We provide an explicit procedure to retrieve $F_{\mathcal{R}}$ from its periodic sum $F_{\mathcal{M}}$ for a wide class of functions. An example of particular interest is the pair of Schwinger kernels $\left\{K_{\mathcal{M}},K_{\mathcal{R}}\right\}$ , corresponding to the Minkowski and Rindler vacua. We obtain an explicit expression for $K_{\mathcal{R}}$ and clarify several conceptual and technical issues related to these biscalars both in the Euclidean and Lorentzian sectors. 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 requires four different types of analytic continuations based on one unifying principle. Our procedure allows the generalization of these results to any (bifurcate Killing) horizon in curved spacetime.