Scalar Field Theory in Curved Space and the Definition of Momentum

TL;DR

This paper explores scalar quantum field theory in curved spaces, especially anti-de Sitter space, analyzing how curvature affects momentum, propagators, and symmetry theorems, with implications for understanding matter interactions in curved spacetime.

Contribution

It provides a detailed analysis of quantum scalar fields in anti-de Sitter space, including Feynman rules, vacuum structure, and a proof of Goldstone's theorem in curved spacetime.

Findings

Curvature shifts momentum and mass scales in anti-de Sitter space.

Transform space rules can be simplified using conformally flat coordinates.

Goldstone's theorem is valid in anti-de Sitter space with modifications.

Abstract

Some general remarks are made about the quantum theory of scalar fields and the definition of momentum in curved space. Special emphasis is given to field theory in anti-de Sitter space, as it represents a maximally symmetric space-time of constant curvature which could arise in the local description of matter interactions in small regions of space-time. Transform space rules for evaluating Feynman diagrams in Euclidean anti-de Sitter space are initially defined using eigenfunctions based on generalized plane waves. It is shown that, for a general curved space, the rules associated with the vertex are dependent on the type of interaction being considered. A condition for eliminating this dependence is given. It is demonstrated that the vacuum and propagator in conformally flat coordinates in anti-de Sitter space are equivalent to those analytically continued from $H^4$ and that transform space rules based on these coordinates can be used more readily. A proof of the analogue of Goldstone's theorem in anti-de Sitter space is given, using a generalized plane wave representation of the commutator of the current and the scalar field. It is shown that the introduction of curvature in the space-time shifts the momentum by an amount which is determined by the Riemann tensor to first order, and it follows that there is a shift in both the momentum and mass scale in anti-de Sitter space.