# Quantum Field Theory on Curved Backgrounds. II. Spacetime Symmetries

## TL;DR

This paper investigates spacetime symmetries in scalar quantum field theories on static backgrounds, demonstrating how Euclidean symmetries extend to Lorentzian manifolds and illustrating the method with Anti-de Sitter space.

## Contribution

It develops a method to construct unitary representations of spacetime isometry groups from Euclidean quantum field theories on static manifolds.

## Key findings

- Isometry groups generated by self-adjoint or unitary operators.
- Analytic continuation from Euclidean to Lorentzian spacetime.
- Application to hyperbolic space and Anti-de Sitter space.

## Abstract

We study space-time symmetries in scalar quantum field theory (including interacting theories) on static space-times. We first consider Euclidean quantum field theory on a static Riemannian manifold, and show that the isometry group is generated by one-parameter subgroups which have either self-adjoint or unitary quantizations. We analytically continue the self-adjoint semigroups to one-parameter unitary groups, and thus construct a unitary representation of the isometry group of the associated Lorentzian manifold. The method is illustrated for the example of hyperbolic space, whose Lorentzian continuation is Anti-de Sitter space.