Scalar field theory in curved space and the definition of momentum


Simon Davis


Research Report 97-5
Date: 18 February 1997, revised 12 March 1998


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


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