PATH INTEGRALS FOR A PARTICLE IN CURVED SPACE

We consider a particle obeying the Schrödinger equation in a general curved n-dimensional space, with arbitrary linear coupling to the scalar curvature of the space. We give the Feynman path-integral expressions for the probability amplitude, x,s|x′,0, for the particle to go from x′ to x in time s. This generalizes results of DeWitt, Cheng, and Hartle and Hawking. We show in particular, that there is a one-parameter family of covariant representations of the path integral corresponding to a given amplitude. These representations are different in that the covariant expressions for the incremental amplitudes, xl+1,sl+ε|xl,sl, appearing in the definition of the path integral, differ even to first order in ε (after dropping common factors). Finally, using the proper-time representation, we give the corresponding generally covariant expressions for the propagator of a scalar field with arbitrary linear coupling to the scalar curvature of the spacetime. © 1979 The American Physical Society.