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Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace. (English) Zbl 1377.81088

Summary: A fully regulated definition of Feynman’s path integral is presented here. The proposed re-formulation of the path integral coincides with the familiar formulation whenever the path integral is well defined. In particular, it is consistent with respect to lattice formulations and Wick rotations, i.e., it can be used in Euclidean and Minkowski space-time. The path integral regularization is introduced through the generalized Kontsevich-Vishik trace, that is, the extension of the classical trace to Fourier integral operators. Physically, we are replacing the time-evolution semi-group by a holomorphic family of operators such that the corresponding path integrals are well defined in some half space of \(\mathbb{C}\). The regularized path integral is, thus, defined through analytic continuation. This regularization can be performed by means of stationary phase approximation or computed analytically depending only on the Hamiltonian and the observable (i.e., known a priori). In either case, the computational effort to evaluate path integrals or expectations of observables reduces to the evaluation of integrals over spheres. Furthermore, computations can be performed directly in the continuum and applications (analytic computations and their implementations) to a number of models including the non-trivial cases of the massive Schwinger model and a \(\phi^{4}\) theory.{
©2017 American Institute of Physics}

MSC:

81S40 Path integrals in quantum mechanics
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
35S30 Fourier integral operators applied to PDEs
30B40 Analytic continuation of functions of one complex variable
81T10 Model quantum field theories
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