OFF-DIAGONAL ELEMENTS OF THE DEWITT EXPANSION FROM THE QUANTUM-MECHANICAL PATH-INTEGRAL

The DeWitt expansion of the matrix element M(xy)=[x\exp{-[1/2(p-A)(2)V]t}\y] (p=-i partial derivative), in powers of t can be made in a num ber of ways. For x=y (the case of interest when doing one-loop calcula tions), numerous approaches have been employed to determine this expan sion to very high order; when x not equal y (relevant for doing calcul ations beyond one loop), there appear to be but two examples of perfor ming the DeWitt expansion. In this paper we compute the off-diagonal e lements of the DeWitt expansion coefficients using the Fock-Schwinger gauge. Our technique is based on representing M(xy) by a quantum-mecha nical path integral. We also generalize our method to the case of curv ed space, allowing us to determine the DeWitt expansion of (M) over ti lde(xy)=[x\exp{1/2[g(-1/2)(partial derivative(mu)-iA(mu))g(mu nu)root g(partial derivative(nu)-iA(nu))]t}\y] by use of normal coordinates. B y comparison with results for the DeWitt expansion of this matrix elem ent obtained by the iterative solution of the diffusion equation, the relative merits of the different approaches to the representation of ( M) over tilde(xy) as a quantum-mechanical path integral can be assesse d. Furthermore, the exact dependence of (M) over tilde(xy) on some geo metric scalars can be determined. In two appendices, we discuss bounda ry effects in the one-dimensional quantum-mechanical path integral, an d the curved space generalization of the Fock-Schwinger gauge.