Feynman Formula Research Articles

Feynman formulas for qp- and pq-quantization of some Vladimirov type time-dependent Hamiltonians on finite adeles

Let Q be the d-dimensional space of finite adeles over the algebraic number field K and let P=Q∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P=Q^*$$\\end{document} be its dual space. For a certain type of Vladimirov type time-dependent Hamiltonian HV(t):Q×P→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_V(t):Q\ imes P\\rightarrow {\\mathbb {C}}$$\\end{document} we construct the Feynman formulas for the solution of the Cauchy problem with the Schrödinger operator where the caret operator stands for the qp- or pq-quantization.

Price prediction and selling strategy optimization using the Feynman formula differential equations

Price prediction and selling strategy optimization using the Feynman formula differential equations

The Vladimirov operator with variable coefficients on finite adeles and the Feynman formulas for the Schrödinger equation

We construct the Hamiltonian Feynman, Lagrangian Feynman, and Feynman–Kac formulas for the solution of the Cauchy problem with the Schrödinger operator −MgDα − V, where Dα is the Vladimirov operator and Mg is the operator of multiplication by a real-valued function g defined on the d-dimensional space AKd of finite adeles over the algebraic number field K.

Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups

We study sequences of compositions of independent identically distributed random one-parameter semigroups of linear transformations of a Hilbert space and the asymptotic properties of the distributions of such compositions when the number of terms in the composition tends to infinity. To study the expectation of such compositions, we apply the Feynman-Chernoff iterations obtained via Chernoff’s theorem. By the Feynman-Chernoff iterations we mean prelimit expressions from the Feynman formulas; the latter are representations of one-parameter semigroups or related objects in terms of the limit of integrals over Cartesian powers of an appropriate space, or some generalizations of such representations. In particular, we study the deviation of the values of compositions of independent random semigroups from their expectation and examine the validity for such compositions of analogs of the limit theorems of probability theory such as the law of large numbers. We obtain sufficient conditions under which any neighborhood of the expectation of a composition of n random semigroups contains the (random) value of this composition with probability tending to one as n → ∞ (it is this property that is viewed as the law of large numbers for compositions). We also present examples of sequences of independent random semigroups for which the law of large numbers for compositions fails.

Solution-giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients

We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the C0-semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and then find the representation (based on the family of translation operators) of the solution in terms of coefficients of the equation and initial condition. It is proved that if the coefficients of the equation are bounded, infinitely smooth, and satisfy some other conditions, then there exists a solution-giving C0-semigroup of contraction operators. We also represent the solution as a Feynman formula (i.e., as a limit of a multiple integral with multiplicity tending to infinity) with generalized functions appearing in the integral kernel.

Randomizes hamiltonian mechanics

Randomized Hamiltonian mechanics is the Hamiltonian mechanics which is determined by a time-dependent random Hamiltonian function. Corresponding Hamiltonian system is called random Hamiltonian system. The Feynman formulas for the random Hamiltonian systems are obtained. This Feynman formulas describe the solutions of Hamilton equation whose Hamiltonian is the mean value of random Hamiltonian function. The analogs of the above results is obtained for a random quantum system (which is a random infinite dimensional Hamiltonian system). This random quantum Hamiltonians are the part of Hamiltonians of open quantum system.

Randomized Hamiltonian Mechanics

Hamiltonian mechanics determined by time-dependent random Hamiltonian functions is called randomized Hamiltonian mechanics. The corresponding Hamiltonian systems are called random. Feynman formulas for random Hamiltonian systems are obtained. It is shown that these formulas describe solutions of a Hamiltonian equation whose Hamiltonian is the mean value of the random Hamiltonian function. Analogous results are obtained for random quantum systems (which are known to be infinite-dimensional random Hamiltonian systems). The random quantum Hamiltonians can be used to describe the so-called open quantum systems (which are part of some lager quantum systems).

Lebesgue-Feynman Measures on Infinite Dimensional Spaces

The Lebesgue-Feynman measure on a linear space E is a generalized measure on E which is defined as a linear functional μ on some linear space F(E,ℂ) of functions E → ℂ which is invariant with respect to the shift on any vector. There exists two different approach to definition of Lebesgue-Feynman measure. The first definition is given by Feynman formula and described in the works (Smolyanov and Shavgulidze 2015; Sadovnichiy et al., Doklady Math. 86(2), 644–647, 2012, Doklady Math. 93(1), 46–48, 2016). In this paper we consider the second approach which is based on the definition of translation-invariant function of a set on some ring of elementary subsets. As it is well-known there exist Sobolev-Schwartz distributions which do not correspond to any function of a set. In contrast, the Lebesgue-Feynman generalized measure as we show below is closely connected to the shift-invariant function of a set which is determined on the ring of elementary subsets.

Explicit formula for evolution semigroup for diffusion in Hilbert space

A parabolic partial differential equation [Formula: see text] is considered, where [Formula: see text] is a linear second-order differential operator with time-independent (but dependent on [Formula: see text]) coefficients. We assume that the spatial coordinate [Formula: see text] belongs to a finite- or infinite-dimensional real separable Hilbert space [Formula: see text]. The aim of the paper is to prove a formula that expresses the solution of the Cauchy problem for this equation in terms of initial condition and coefficients of the operator [Formula: see text]. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup using a Feynman formula (i.e. we write it in the form of a limit of a multiple integral over [Formula: see text] as the multiplicity of the integral tends to infinity), which gives us a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on [Formula: see text]. This solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in [Formula: see text] is zero, we prove that the strongly continuous resolving semigroup indeed exists (which implies the existence of a unique solution to the Cauchy problem in the class mentioned above), and that the solution to the Cauchy problem depends continuously on the coefficients of the equation.

Field by uniformly moving charge: a new derivation via Feynman’s formula

One hundred and thirty years ago Oliver Heaviside (1850–1925) first published the exact expression of the electromagnetic field generated by a uniformly moving electric charge. A didactical route to establish such an important result is here proposed via a new treatment of the Feynman formula for the electric field intensity of a moving charge, when the charge moves uniformly. We believe the present approach could be a useful starting point for introducing as gently as possible a complex analysis of electrodynamics even to undergraduates not yet familiar with Maxwell’s equations and Lorentz transformation.

Feynman Formulas for Solutions to Evolution Equations in Domains of Multidimensional Ramified Surfaces

Solutions of second-order parabolic differential equations for functions defined in domains of a K ramified surface in the class L2(K) are obtained. With the help of Chernoff’s theorem, such solutions (if they exist) can be represented in the form of Lagrangian Feynman formulas, i.e., in the form of limits of integrals over Cartesian powers of the configuration space as the number of factors tends to infinity.

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

Abstract We consider operator semigroups generated by Feller processes killed upon leaving a given domain. These semigroups correspond to Cauchy–Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation leads to a representation of the solution of the corresponding Cauchy–Dirichlet type problem by a Feynman formula, i.e. by a limit of n-fold iterated integrals of certain functions as n → ∞. Feynman formulae can be used for direct calculations, modelling of underlying dynamics, simulation of underlying stochastic processes. Further, a method to approximate solutions of time-fractional evolution equations is suggested. The method is based on connections between time-fractional and time-non-fractional evolution equations as well as on Chernoff approximations for the latter ones. This method leads to Feynman formulae for solutions of time-fractional evolution equations. A class of distributed order time-fractional equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy–Dirichlet type problems are obtained.

Feynman formulas for nonlinear evolution equations

Transformations of measures, generalized measures, and functions generated by evolution differential equations on a Hilbert space E are studied. In particular, by using Feynman formulas, a procedure for averaging nonlinear random flows is described and an analogue of the law of large number for such flows is established (see [1, 2]).

Feynman formulas for semigroups generated by an iterated Laplace operator

In the present paper, we find representations of a one-parameter semigroup generated by a finite sum of iterated Laplace operators and an additive perturbation (the potential). Such semigroups and the evolution equations corresponding to them find applications in the field of physics, chemistry, biology, and pattern recognition. The representations mentioned above are obtained in the form of Feynman formulas, i.e., in the form of a limit of multiple integrals as the multiplicity tends to infinity. The term “Feynman formula” was proposed by Smolyanov. Smolyanov’s approach uses Chernoff’s theorems. A simple form of representations thus obtained enables one to use them for numerical modeling the dynamics of the evolution system as a method for the approximation of solutions of equations. The problems considered in this note can be treated using the approach suggested by Remizov (see also the monograph of Smolyanov and Shavgulidze on path integrals). The representations (of semigroups) obtained in this way are more complicated than those given by the Feynman formulas; however, it is possible to bypass some analytical difficulties.

Analogues of Feynman formulas for ill-posed problems associated with the Schrödinger equation

Representations of Schrodinger semigroups and groups by Feynman iterations are studied. The compactness, rather than convergence, of the sequence of Feynman iterations is considered. Approximations of solutions of the Cauchy problem for the Schrodinger equation by Feynman iterations are investigated. The Cauchy problem for the Schrodinger equation under consideration is ill-posed. From the point of view of the approach of the paper, this means that the problem has no solution in the sense of integral identity for some initial data. The well-posedness of the Cauchy problem can be recovered by extending the operator to a selfadjoint one; however, there exists continuum many such extensions. Feynman iterations whose partial limits are the solutions of all Cauchy problems obtained for various self-adjoint extensions are studied.

A note on “Measuring propagation speed of Coulomb fields” by R. de Sangro, G. Finocchiaro, P. Patteri, M. Piccolo, G. Pizzella

In connection with the discussion and the measurements fulfilled in [R. de Sangro et al., Eur. Phys. J. C 75 (2015) 137], the full identity is demonstrated between the Feynman formula for the field of a moving charge and the Lienard-Wiechert potentials.

Representations of solutions of Lindblad equations by randomized Feynman formulas

Representations of solutions of Lindblad equations by randomized Feynman integrals over trajectories are obtained by averaging similar representations for solutions of stochastic Schrodinger equations (Schrodinger–Belavkin equations). An approach based on the application of Chernoff’s theorem is applied. First, (randomized) Feynman formulas approximating Feynman path integrals are obtained; these formulas contain integrals over finite Cartesian powers of the space of values of the functions over which the Feynman integrals are taken.

Dynamics of particles with anisotropic mass depending on time and position

Representations of solutions of equations describing the diffusion and quantum dynamics of particles in a Riemannian manifold are discussed under the assumption that the mass of particles is anisotropic and depends on both time and position. These equations are evolution differential equations with secondorder elliptic operators, in which the coefficients depend on time and position. The Riemannian manifold is assumed to be isometrically embedded into Euclidean space, and the solutions are represented by Feynman formulas; the representation of a solution depends on the embedding.

Feynman approximations of the dynamics of the Wigner function

A Feynman formula for the solving semigroup describing the evolution of the Wigner function (defined on the phase space of a finite-dimensional Hamiltonian system) is obtained. For an information concerning Feynman formulas, see V. A. Dubravina, Feynman formulas for solutions of evolution equations on ramified surfaces, Russ. J. Math. Phys. 21 (2), 285–288 (2014).

SOLUTION TO A PARABOLIC DIFFERENTIAL EQUATION IN HILBERT SPACE VIA FEYNMAN FORMULA I

A parabolic partial differential equation u′ t (t, x) = Lu(t, x) is considered, where L is a linear second-order differential operator with time-independent coefficients, which may depend on x . We assume that the spatial coordinate x belongs to a finiteor infinite-dimensional real separable Hilbert space H . Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup by a Feynman formula, i.e. we write it in the form of the limit of a multiple integral over H as the multiplicity of the integral tends to infinity. This representation gives a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on H . Moreover, this solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in L vanishes we prove that the strongly continuous resolving semigroup exists (this implies the existence of the unique solution to the Cauchy problem in the class mentioned above) and that the solution to the Cauchy problem depends continuously on the coefficients of the equation. The article is published in the author’s wording.