Exploring the Rindler vacuum and the Euclidean plane

Abstract

In flat spacetime, two inequivalent vacuum states that arise rather naturally are the Rindler vacuum |R〉 and the Minkowski vacuum |M〉. We discuss 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 [J. Math. Phys. 17, 2101 (1976)], viz., that GR, the Feynman propagator corresponding to |R〉, can be expressed as a curious integral transform of GM, the Feynman propagator in |M〉. We show that this relation follows from the well-known result that GM can be written as a periodic sum of GR, 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 FM,FR, provided that FM can be represented as a periodic sum of FR with period 2πi. We provide an explicit procedure to retrieve FR from its periodic sum FM for a wide class of functions. An example of particular interest is the pair of Schwinger kernels KM,KR, corresponding to the Minkowski and Rindler vacua. We obtain an explicit expression for KR 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.