  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.
