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Journal of Mathematical Physics

Bernstein processes and spin-1/2 particles

Boualem Djehiche

Published Sep 1, 1992• 4 citations
Publisher

Abstract

A pure jump Bernstein process whose probability density is the product of the solution of the (imaginary time) Schrödinger equation and its adjoint equation, associated to a class of time dependent Hamiltonians describing spin-1/2 particles, is constructed. A path integral representation of these solutions, in terms of this process, is obtained as well as a calculus over path space for the associated Euclidean quantum observables.

DOI
10.1063/1.529525

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A pure jump Bernstein process whose probability density is the product of the solution of the (imaginary time) Schrödinger equation and its adjoint equation, associated to a class of time dependent Hamiltonians describing spin-1/2 particles, is constructed. A path integral representation of these solutions, in terms of this process, is obtained as well as a calculus over path space for the associated Euclidean quantum observables.
Bernstein diffusions belong to a new class of time symmetric (but not time homogeneous) stochastic processes associated with the quantum dynamics of nonrelativistic particles in potentials. It is shown in which sense Feynman’s formal path integral method can be interpreted in terms of those processes, specially for the subset of Gaussian Bernstein diffusions. The familiar Ornstein-Uhlenbeck process becomes, in this framework, a particular Gaussian process in a large class of Bernstein diffusions, all associated with the same dynamics. The method is also illustrated for some Hilbert space valued Gaussian Diffusions These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Extends the Feynman-Kac formula to the case of imaginary time Schrodinger equations (heat equations) for multicomponent wavefunctions. The approach covers in particular the usual Pauli equation for a spin- 1 / 2 particle in an arbitrary magnetic field. The formula contains, besides the expectation with respect to a Wiener process in ordinary space, an expectation with respect to a jump process over the discrete indices describing the internal degrees of freedom. To illustrate the method the authors evaluate the formula in some special cases and they derive also various inequalities. A detailed comparison is made with a recent work by Gaveau and Vauthier (1981).
A general description of Bernstein processes, a class of diffusion processes, relevant to the probabilistic counterpart of quantum theory known as Euclidean Quantum Mechanics, is given. It is compatible with finite or infinite dimensional state spaces and singular interactions. Although the relations with statistical physics concepts (Gibbs measure, entropy,…) is stressed here, recent developments requiring Feynman’s quantum mechanical tools (action functional, path integrals, Noether’s Theorem,…) are also mentioned and suggest new research directions, especially in the geometrical structure of our approach.
This paper is a review of path integral representations for semigroups {exp(-t H_r/ħ)}{t≥0} where H_r's are relativistic quantum Hamiltonians. We consider three different cases: in the first one Hr is a relativistic Schrödinger operator, in the second is the Hamiltonian associated to Klein-Gordon equation and in the third is that coming from the Dirac one. The paths are trajectories of diffusion processes and, for the Dirac case, they differ from previous constructions based on jump Markov processes. All formulas describe one relativistic particle in an external static electromagnetic field therefore they are relativistic versions of the Feymann-Kac representation for Schrödinger semigroups.
Abstract A class of stochastic particle models for the spatially discretized time-dependent Schrödinger equation is constructed. Each particle is characterized by a complex-valued weight and a position. The particle weights change according to some deterministic rules between the jumps. The jumps are determined by the creation of offspring. The main result is that certain functionals of the particle systems satisfy the Schrödinger equation. The proofs are based on the theory of piecewise deterministic Markov processes.