Off-Diagonal Elements of the DeWitt Expansion from the Quantum Mechanical Path Integral

Abstract

The DeWitt expansion of the matrix element $M_{xy} = \left\langle x \right| \exp -[\case{1}{2} (p-A)^2 + V]t \left| y \right\rangle$, $(p=-i\partial )$ in powers of $t$ can be made in a number of ways. For $x=y$ (the case of interest when doing one-loop calculations) numerous approaches have been employed to determine this expansion to very high order; when $x\ne y$ (relevant for doing calculations beyond one-loop) there appear to be but two examples of performing the DeWitt expansion. In this paper we compute the off-diagonal elements of the DeWitt expansion coefficients using the Fock-Schwinger gauge. Our technique is based on representing $M_{xy}$ by a quantum mechanical path integral. We also generalize our method to the case of curved space, allowing us to determine the DeWitt expansion of $\tilde M_{xy} = \langle x| \exp \case{1}{2} [\case{1}{\sqrt {g}} (\partial_μ- i A_μ)g^{μν}{\sqrt{g}}(\partial_ν- i A_ν) ] t| y \rangle$ by use of normal coordinates. By comparison with results for the DeWitt expansion of this matrix element obtained by the iterative solution of the diffusion equation, the relative merit of different approaches to the representation of ${\tilde{M}}_{xy}$ as a quantum mechanical path integral can be assessed. Furthermore, the exact dependence of ${\tilde{M}}_{xy}$ on some geometric scalars can be determined. In two appendices, we discuss boundary effects in the one-dimensional quantum mechanical path integral, and the curved space generalization of the Fock-Schwinger gauge.

16pp, REVTeX. One additional appendix concerning end-point effects for finite proper-time intervals; inclusion of these effects seem to make our results consistent with those from explicit heat-kernel methods

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Cited by in corpus (8)

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• Gravitational corrections to the Euler-Heisenberg Lagrangian
• Scalar heat kernel with boundary in the worldline formalism
• Quantum effective actions from nonperturbative worldline dynamics
• Heat-kernel approach for scattering
• The Universal One-Loop Effective Action with Gravity
• Photon-graviton mixing in an electromagnetic field
• Covariant perturbation expansion of off-diagonal heat kernel