Path integrals for a particle in curved space

Abstract

We consider a particle obeying the Schr\"odinger 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, $〈x,s|{x}^{\ensuremath{'}},0〉$, for the particle to go from ${x}^{\ensuremath{'}}$ 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, $〈{x}_{l+1},{s}_{l}+\ensuremath{\epsilon}|{x}_{l},{s}_{l}〉$, appearing in the definition of the path integral, differ even to first order in $\ensuremath{\epsilon}$ (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.