Advanced Studies in Theoretical Physics

A generalization of the Feynman-Kac formula through an operator algebra

Oswaldo Gonzalez-Gaxiola, Francisco Bulnes

Published Jan 1, 2013• 0 citations

Abstract

In this work is introduced the harmonic oscillator process that in probability is known as Ornstein-Uhlenbeck process or velocities process of Ornstein-Uhlenbeck and is established that the mathematical expectation of certain functionals of this process coincides with the expected value of certain observables in the fundamental state or basis state to the oscillator and with it is found a relation between the spaces of classical and quantum probability. Furthermore is established a certain algebra of operators that gives rise to the dynamics of quantum harmonic oscillator and to consider certain class of perturbations of the Hamiltonian corresponding to their harmonic oscillator is obtained a version of the Feynman-Kac formula with a dilatation generated by the unitary group of translations for a co-cycle. Finally are given some generalizations of the Feynman-Kac formula.

DOI: 10.12988/astp.2013.310121

Related papers

Quantum white noise Feynman–Kac formula

Aymen Ettaieb, Narjes Turki Khalifa, Habib Ouerdiane

Abstract In this paper, we give a probabilistic representation of the heat equation associated with the quantum K-Gross Laplacian using infinite-dimensional stochastic calculus in two variables. Applying the heat semigroup to the particular case where the operator is the multiplication one, we establish a relation between the classical and the quantum heat semigroup. Finally, using a combination between convolution calculus and the generalized stochastic calculus, we give a generalization of the Feynman–Kac formula.

Random Operators and Stochastic EquationsMay 15, 20182 citations

Feynman-Kac formula for Levy processes and semiclassical (Euclidean) momentum representation

Nicolas Privault, Xiangfeng Yang, Jean-Claude Zambrini

We prove a version of the Feynman-Kac formula for Levy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve Lévy-type potentials. Large deviation techniques are used to obtain the limiting behavior of the systems as the Planck constant approaches zero. It turns out that the limiting behavior coincides with fresh aspects of the semiclassical limit of (Euclidean) quantum mechanics. Non-trivial examples of Levy processes are considered as illustrations and precise asymptotics are given for the terms in both configuration and momentum representations.

arXiv:1308.2363Markov Processes And Related FieldsAug 11, 20131 citation

G-Gaussian Processes under Sublinear Expectations and q-Brownian Motion in Quantum Mechanics

Shige Peng A

We provide a general approach to construct a stochastic process with a given consistent family of finite dimensional distributions under a nonlinear expectation space. We use this approach to construct a generalized Gaussian process under a sublinear expectation and a q-Brownian motion. The later one is under a complex-valued linear expectation, with which a new type of Feynman-Kac formula can be derived to represent the solution of a Schrödinger equation.

arXiv:1105.1055May 5, 20115 citations

Quantum dynamics of the classical harmonic oscillator

Dimitrios Giannakis

A correspondence is established between measure-preserving, ergodic dynamics of a classical harmonic oscillator and a quantum mechanical gauge theory on two-dimensional Minkowski space. This correspondence is realized through an isometric embedding of the L2(μ) space on the circle associated with the oscillator’s invariant measure, μ, into a Hilbert space H of sections of a C-line bundle over Minkowski space. This bundle is equipped with a covariant derivative induced from an SO+(1, 1) gauge field on the corresponding inertial frame bundle, satisfying the Yang–Mills equations. Under this embedding, the Hamiltonian operator of a Lorentz-invariant quantum system, constructed as a geometrical Laplace-type operator on bundle sections, pulls back to the generator of the unitary group of Koopman operators governing the evolution of classical observables of the harmonic oscillator, with Koopman eigenfunctions of zero, positive, and negative eigenfrequencies corresponding to quantum eigenstates of zero (“vacuum”), positive (“matter”), and negative (“antimatter”) energies. The embedding also induces a pair of operators acting on classical observables of the harmonic oscillator, exhibiting canonical position–momentum commutation relationships. These operators have the structure of order-1/2 fractional derivatives and therefore display a form of non-locality. In a second part of this work, we study a quantum mechanical representation of the classical harmonic oscillator using a one-parameter family of reproducing kernel Hilbert spaces, K^τ, associated with the time-τ transition kernel of a fractional diffusion on the circle. As shown in recent work, these spaces are unital Banach *-algebras of functions. It is found that the evolution of classical observables in these spaces takes place via a strongly continuous, unitary Koopman evolution group, which exhibits a stronger form of classical–quantum consistency than the L2(μ) case. Specifically, for every real-valued classical observable in K^τ, there exists a quantum mechanical observable, whose expectation value is consistent with classical function evaluation. This allows for a description of classical state space dynamics, classical statistics, and quantum statistics of the harmonic oscillator within a unified framework.

arXiv:1912.12334Journal of Mathematical PhysicsApr 1, 20217 citations

A derivation of quantum Jarzynski equality using quantum Feynman-Kac formula

Fei Liu

We present a quantum mechanical analogue of the famous Feynman-Kac formula that was originally established for classical stochastic processes. By constructing a backward-time invariable, we find that this formula can be used to derive the quantum Jarzynski equality. Different from the previous approaches, the time-ordered correlated function in the quantum mechanical work definition is naturally emerged in terms of mathematical rigor.

Jan 7, 20121 citation

Fundamental phase space formula for the similitude group

Laarni B. Natividad, Job A. Nable

In this work, the statement and proof of a fundamental formula in the phase space representation of quantum systems will be carried out for the similitude group, 𝑆𝑖𝑚(2).This formula takes the form∫ 𝒂(𝒀)𝑷(𝒀)𝒅(𝒀) = 〈𝑨 ̂〉, where 𝑌 is the phase space variable and 〈𝐴 ̂〉 is a linear operator on Hilbert space representing a quantum dynamical observable.〈𝐴 ̂〉 is the quantum expected value of the observable in a state of the system.The focus on the similitude group is due to current interest in signal analysis, localization operators and pseudo-differential operators.The fundamental formula states that this may be computed in a classical manner, as an integral against a probability distribution.The formula is intimately related to the quantization-dequantization problem 𝑎(𝑌) ⟷ 𝐴 ̂ which assigns a quantum operator to the classical phase space function 𝑎(𝑌).

AIP conference proceedingsJan 1, 20240 citations