Scalar field theory and the definition of momentum in curvedspace

Abstract

String propagation in ten-dimensionalMinkowski space or the direct product ofMinkowski space and a six-dimensionalKähler manifold or orbifold might beregarded as an approximation to a theorywhich allows for the local curvature ofspacetime by the energy-momenta of thecomponent fields. String scattering in theinteraction region might then be based onquantum field theory in a local region witha curved geometry. Special emphasis isgiven to field theory in anti-de Sitterspace, as it represents a maximallysymmetric spacetime of constant curvaturewhich could arise in the description ofmatter interactions in local regions ofspacetime. Curvature shifts in themomentum and squared mass are evaluatedfor scalar fields in anti-de Sitter space,and it is shown that the shift in p2 + m2 compensates the ground-statecontribution to the bosonic stringHamiltonian, implying the consistency ofcomputing the scattering entirely in flatspace. Dual space rules for evaluatingFeynman diagrams in Euclideananti-de Sitter space are initially definedusing eigenfunctions based on generalizedplane waves. Loop integrals can beevaluated even more easily using momentumspace rules in conformally flatcoordinates for anti-de Sitter space,which admits flat three-dimensionalsections that are analytic continuationsof horospheres in hyperbolic space H4.An additional argument in favour of themodel of string propagation described inthis paper is based on the removal ofreflective boundary conditions on quantumfields interacting in a locallyanti-de Sitter region without spatialinfinity, implying the existence of aone-parameter family of O(3,2)-invariantvacua in this region consistent with thedegree of freedom in defining the stringtheory vacuum.