# Exploring the Rindler vacuum and the Euclidean Plane

## TL;DR

This paper investigates the properties of the Rindler vacuum in flat spacetime, exploring propagators, heat kernels, and their relations to Minkowski vacuum, with implications for understanding quantum fields near horizons.

## Contribution

It provides explicit relations and procedures to derive Rindler vacuum quantities from Minkowski ones, extending the analysis to Euclidean and Lorentzian sectors and general horizons.

## Key findings

- Expressed Rindler propagator as an integral transform of Minkowski propagator.
- Derived explicit Rindler Schwinger kernel from Minkowski kernel.
- Clarified analytic continuation procedures for Rindler wedges.

## Abstract

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.