Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

Abstract

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 Minkowskian 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 operator families such that the corresponding path integrals are welldefined in some half space of 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 ϕ 4 theory.

Zeta-regularized vacuum expectation values

“…Nevertheless it was shown [9] that both traces, tr(U A) and tr U , can be constructed in a non-perturbative way using Fourier Integral Operator ζ-functions. In this formulation, we assume that the Hamiltonian H and the operator A are pseudo-differential operators on a compact Riemannian C ∞ -manifold X without boundary (a Cauchy surface of the "universe"; the infinite volume limit X → "non-compact manifold" is taken after ζregularization, cf.…”

“…Finally, we are interested in computing A G (0) which is "almost always" (cf. [9,11]) independent of the choice of G and, in general, very difficult to compute. At this stage we therefore have a fully regularized expectation value, but the physical meaning of A G (0) is still unclear.…”

Zeta-regularized vacuum expectation values fromquantum computing simulations

“…In a more general setup, as shown in ref. [6], a family of operators G(z) with the property G(0) = 1 is introduced, gauging the time evolution operator to be of the form U(T, 0)G(z). This too leads to the gauged Fourier integral kernel of eq.…”

“…Maybe, this appears to be a most complicated way to obtain the ground state energy of the free Dirac operator, but this simple computation illustrates the steps involved in obtaining ζ -regulated vacuum expectation values through solving -possibly very high dimensional -spherical integrals. More examples can be found in [6,7,8].…”

“…In this proceeding, following refs. [6,7,8], we present a proposal to provide exactly such a setup. In particular, there are two main messages to convey:…”