Feynman formulas and path integrals for some evolution semigroups related to τ-quantization

Abstract

This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n-fold iterated integrals as n → ∞) and their connections with phase space Feynman path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamiltonian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of stochastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals.

Chernoff Approximation for Semigroups Generated by Killed Feller Processes and Feynman Formulae for Time-Fractional Fokker–Planck–Kolmogorov Equations

“…Note that different Feynman formulae for the same semigroup allow to establish relations between different path integrals. This leads, in particular, to some "change-of-variables" rules, connecting certain Feynman-Kac formulae and Feynman path integrals (see the papers [19,7]). Chernoff approximation can be understood in some particular cases also as a construction of Markov chains approximating a given Markov process [9] and as a numerical path integration method for solving corresponding PDEs/SDEs [23].…”

“…For example, this method has been used to investigate Schrödinger type evolution equations in [71,66,74,41,30,84,81,83]; stochastic Schrödinger type equations have been studied in [58,57,59,34]. Second order parabolic equations related to diffusions in different geometrical structures (e.g., in Eucliean spaces and their subdomains, Riemannian manifolds and their subdomains, metric graphs, Hilbert spaces) have been studied, e.g., in [19,15,69,14,67,82,70,7,20,90,18,89,17,13,12,86,11,10,85,56]. Evolution equations with non-local operators generating some Markov processes in R d have been considered in [16,19,21,22].…”

“…The Riemann-Liouville fractional derivative is more general since it does not require the first derivative of u to exist. Therefore, one may adopt the right hand side of the formula (7) to define the Caputo derivative. The further generalization is to consider the so-called distributed order fractional derivative D µ with the order µ determined by a finite Borel measure µ defined on the interval (0, 1) and such that µ(0, 1) > 0 (cf.…”

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

“…Since 2000, O. G. Smolyanov and members of his group succeeded in representing solutions of the Cauchy problem for many evolution equations in form of Feynman formulas (see [39,40,41,42,43,47,48,49,51,52,48,54,61,57,58,44] and refereces therein). The key idea in these representations lies in finding the Chernoff function G for operator L and then applying Chernoff's theorem to obtain the equality e tL u 0 = lim n→∞ G(t/n) n u 0 which apperas to be a Feynman formula, because in all known examples (until [50] was published in 2016, see also [44,63]) G(t) from the equation above was an integral operator, so G(t/n) n was an n-tuple integral operator, giving us a limit of multiple integral where miltiplicity tends to infinity.…”

Perturbation theory tomographic series

“…Поэтому с помощью формулы (5) можно искать решение уравнения Фоккера -Планка с учетом связи между томограммой и волновой функцией. Задачи о представлении решения для данного уравнения рассматривали ранее, поскольку су-ществует его тесная связь с формулировкой Фейнмана интегралов по траекториям и представлением решения уравнения Шредингера [14].…”