Momentum-space representation in curved space

A non-Hermitian momentum-space representation of functions of a manifold (curved space) isomorphic to $R^n$ is constructed out of pure plane waves. We prove the covariance of the coordinate-and momentum-space representations. We give the momentum-space representation of the trace of operators. In interacting field theory this representation enables us to calculate the part of the effective action that encodes local properties of a general curved space. Thus the trace anomaly and the beta function can be evaluated. An explicit calculation is displayed at one loop for a scalar field coupled to gravity. In flat space the formalism applies to Cartesian and curvilinear coordinates. Application to the study of the stability of the vacuum of a field theory in curved space is possible.