Feynman Formulae Research Articles

Representation of solutions of the Cauchy problem for a one-dimensional Schrödinger equation with a smooth bounded potential by quasi-Feynman formulae

We consider the Cauchy problem for a Schrödinger equation whose Hamiltonian is the difference of the operator of multiplication by the potential and the operator of taking the second derivative. Here the potential is a real differentiable function of a real variable such that this function and its derivative are bounded. This equation has been studied since the advent of quantum mechanics and is still a good model case for various methods of solving partial differential equations. We find solutions of the Cauchy problem in the form of quasi-Feynman formulae by using Remizov’s theorem. Quasi-Feynman formulae are relatives of Feynman formulae containing multiple integrals of infinite multiplicity. Their proof is easier than that of Feynman formulae but they give longer expressions for the solutions. We provide detailed proofs of all theorems and deliberately restrict the spectrum of our results to the domain of classical mathematical analysis and elements of real analysis trying to avoid general methods of functional analysis. As a result, the paper is long but accessible to readers who are not experts in the field of functional analysis.

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

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

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.

Representation of solutions of evolution equations on a ramified surface by Feynman formulae

We obtain solutions of parabolic second-order differential equations for functions in the class defined on a ramified surface . By using Chernoff's theorem, we prove that such solutions, whenever they exist, can be represented by Lagrangian Feynman formulae, that is, they can be written as limits of integrals over Cartesian powers of the configuration space as the number of factors tends to infinity.

Representation of solutions of evolution equations on a ramified surface by Feynman formulae

В работе получены решения параболических дифференциальных уравнений второго порядка относительно функций, определенных на разветвленной поверхности $K$, в классе $L_1(K)$. C помощью теоремы Чернова доказано, что такие решения, при условии их существования, представимы в виде лагранжевых формул Фейнмана, т. е. в виде пределов интегралов по декартовым степеням конфигурационного пространства при стремящейся к бесконечности степени. Библиография: 13 наименований.

Unbounded random operators and Feynman formulae

We introduce and study probabilistic interpolations of various quantization methods. To do this, we develop a method for finding the expectations of unbounded random operators on a Hilbert space by averaging (with the help of Feynman formulae) the random one-parameter semigroups generated by these operators (the usual method for finding the expectations of bounded random operators is generally inapplicable to unbounded ones). Although the averaging of families of semigroups generates a function that need not possess the semigroup property, the Chernoff iterates of this function approximate a certain semigroup, whose generator is taken for the expectation of the original random operator. In the case of bounded random operators, this expectation coincides with the ordinary one.

Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions

Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Lévy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure of quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computations and numerical modeling of the corresponding dynamics. Hamiltonian Feynman formulae allow to represent the considered semigroups by means of Feynman path integrals. In the article, a family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented as phase space Feynman path integrals with respect to these Feynman pseudomeasures, i.e., different quantizations correspond to Feynman path integrals with the same integrand but with respect to different pseudomeasures. This answers Berezin’s problem of distinguishing a procedure of quantization on the language of Feynman path integrals. Moreover, the obtained Lagrangian Feynman formulae allow also to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.

LAGRANGIAN AND HAMILTONIAN FEYNMAN FORMULAE FOR SOME FELLER SEMIGROUPS AND THEIR PERTURBATIONS

A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of n-fold iterated integrals of some elementary functions as n → ∞. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite-dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman–Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae give an effective tool to calculate functional integrals with respect to probability measures generated by these Feller processes and, in particular, to obtain simulations of Feller processes.

Формулы Фейнмана для решений уравнений типа Шредингера с полиномиальными потенциалами четвертого порядка

Формулы Фейнмана для решений уравнений типа Шредингера с полиномиальными потенциалами четвертого порядка

Feynman formulae for a parabolic equation with biharmonic differential operator on a configuration space

Feynman formulae for a parabolic equation with biharmonic differential operator on a configuration space

Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

A Feynman formula is a representation of the semigroup, generated by an initial- boundary value problem for some evolutionary equation, by a limit of integrals over Carte- sian powers of some space E, the integrands being some elementary functions. The multi- ple integrals in Feynman formulae approximate integrals with respect to some measures or pseudomeasures on sets of functions which take values in E and are defined on a real in- terval. Hence Feynman formulae can be used both to calculate explicitly solutions for such problems, to get some representations for these solutions by integrals over functions taking values in E (such representations are called Feynman-Kac formulae), to get approximations for transition probability of some diffusion processes and transition amplitudes for quantum dynamics and to get computer simulations for some stochastic and quantum dynamics. The Feynman formula is called a Hamiltonian Feynman formula if the space, Cartesian products of which are used, is the phase space of a classical Hamiltonian system; the corresponding Feynman-Kac formula is called a Hamiltonian Feynman-Kac formula. In the latter formula one integrates over functions taking values in the same phase space. In a similar way one can define Lagrangian Feynman formulae and Lagrangian Feynman-Kac formulae substituting the phase space by the configuration space. In the present paper we formulate and prove the Hamiltonian Feynman and Feynman- Kac formulae for heat semigroups describing the diffusion of particles with the position-

Two derivations of the Feynman formulae for the fields of a moving charge

Two derivations of the Feynman formulae for the fields of a moving charge

Two derivations of the Feynman formulae for the fields of a moving charge

The Feynman expressions for the electric and magnetic fields of a charge moving on an arbitrary trajectory are found by two simple methods. The first involves a direct integration of Maxwell's time-dependent equations. The second is based on the formalism of quantum optics which uses the first order equations of motion for creation and annihilation operators. As an illustration of the simplicity of these methods, the fields of an oscillating electric dipole are obtained.