Transformations of Feynman path integrals and generalized densities of Feynman pseudomeasures

Applications of transformations of Feynman path integrals and Feynman pseudomeasures to explain arising quantum anomalies are considered. A contradiction in the literature is also explained.     

Feynman path integrals for polynomially growing potentials

A general class of infinite dimensional oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proved, as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. These results are applied to provide a rigorous Feynman path integral representation for the solution of the time-dependent Schrödinger equation with a quartic anharmonic potential. The Borel summability of the asymptotic expansion of the solution in power series of the coupling constant is also proved.     

Feynman path integrals for the time dependent quartic oscillator

The Schrödinger equation with a time dependence in both a quadratic and a quartic potential is considered. Existence of solutions is shown and a rigorous Feynman path integral representation for the solution is given in terms of well-defined infinite-dimensional oscillatory integrals. To cite this article: S. Albeverio, S. Mazzucchi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations

We give a fairly general class of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets of the phase space. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the Riemann integrals, the interchange of the order with a limit and the perturbation expansion formula hold in the phase space path integrals. The use of piecewise bicharacteristic paths naturally leads us to the semiclassical approximation on the phase space.     

Feynman path integrals as analysis on path space by time slicing approximation

Using the time slicing approximation, we give a mathematically rigorous definition of Feynman path integrals for a general class of functionals on the path space. As an application, we prove the interchange with Riemann-Stieltjes integrals, the interchange with a limit, the perturbation expansion formula, the semiclassical approximation, and the fundamental theorem of calculus in Feynman path integral.     

Smooth functional derivatives in Feynman path integrals by time slicing approximation

We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, functional differentiation, translation and real linear transformation. The integration by parts and Taylor's expansion formula with respect to functional differentiation holds in Feynman path integral. Feynman path integral is invariant under translation and orthogonal transformation. The interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the semiclassical approximation and the fundamental theorem of calculus in Feynman path integral stay valid as well as N. Kumano-go [Bull. Sci. Math. 128 (3) (2004) 197-251].     

A method of calculating massive Feynman integrals

Scientific-Research Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 89, No. 1, pp. 56–72, October, 1991.     

Phase space Feynman path integrals with smooth functional derivatives by time slicing approximation

We give two general classes of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to each class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the starting point of momentum paths and the endpoint of position paths. Each class is closed under addition, multiplication, translation, real linear transformation and functional differentiation. Therefore, we can produce many functionals which are phase space path integrable. Furthermore, though we need to pay attention for use, the interchange of the order with the integrals with respect to time, the interchange of the order with some limits, the semiclassical approximation of Hamiltonian type, the natural property under translation, the integration by parts with respect to functional differentiation, and the natural property under orthogonal transformation are valid in the phase space path integrals.     

Quadpack computation of Feynman loop integrals

The paper addresses a numerical computation of Feynman loop integrals, which are computed by an extrapolation to the limit as a parameter in the integrand tends to zero. An important objective is to achieve an automatic computation which is effective for a wide range of instances. Singular or near singular integrand behavior is handled via an adaptive partitioning of the domain, implemented in an iterated/repeated multivariate integration method. Integrand singularities possibly introduced via infrared (IR) divergence at the boundaries of the integration domain are addressed using a version of the Dqags algorithm from the integration package Quadpack, which uses an adaptive strategy combined with extrapolation. The latter is justified for a large class of problems by the underlying asymptotic expansions of the integration error. For IR divergent problems, an extrapolation scheme is presented based on dimensional regularization.     

Representation of solutions to evolution equations with Vladimirov operator in terms of Feynman path integrals

Doklady Mathematics -