# Scalar field theory in curved space and the definition of momentum

## Author

**Simon Davis**

## Status

Research Report 97-5

Date: 18 February 1997, revised 12 March 1998

## Abstract

Some general remarks are made about the quantum field 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 the 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 the four-dimensional hyperboloid 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 then 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.

## Key phrases

dual space rules. generalized plane waves. propagator. Lehmann representation.
Goldstone's theorem. equivalence of vacua. second source. curvature.
momentum shifts

## AMS Subject Classification (1991)

Primary: 81T20

Secondary: 81T18, 81T30

## Content

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