Scalar field theory and the definition of momentum in curved space

String propagation in ten-dimensional Minkowski space or the direct product of Minkowski space and a six-dimensional Kähler manifold or orbifold might be regarded as an approximation to a theory which allows for the local curvature of spacetime by the energy-momenta of the component fields. String scattering in the interaction region might then be based on quantum field theory in a local region with a curved geometry. Special emphasis is given to field theory in anti-de Sitter space, as it represents a maximally symmetric spacetime of constant curvature which could arise in the description of matter interactions in local regions of spacetime. Curvature shifts in the momentum and squared mass are evaluated for scalar fields in anti-de Sitter space, and it is shown that the shift in p² + m² compensates the ground-state contribution to the bosonic string Hamiltonian, implying the consistency of computing the scattering entirely in flat space. Dual space rules for evaluating Feynman diagrams in Euclidean anti-de Sitter space are initially defined using eigenfunctions based on generalized plane waves. Loop integrals can be evaluated even more easily using momentum space rules in conformally flat coordinates for anti-de Sitter space, which admits flat three-dimensional sections that are analytic continuations of horospheres in hyperbolic space H⁴. An additional argument in favour of the model of string propagation described in this paper is based on the removal of reflective boundary conditions on quantum fields interacting in a locally anti-de Sitter region without spatial infinity, implying the existence of a one-parameter family of O(3,2)-invariant vacua in this region consistent with the degree of freedom in defining the string theory vacuum.