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Journal of Physics A Mathematical and General

Probabilistic solution of Pauli type equations

Gian Fabrizio De Angelis, Giovanni Jona-Lasinio, M. Sirugue

Published Aug 1, 1983• 45 citations
Publisher

Abstract

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).

DOI
10.1088/0305-4470/16/11/015

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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).
The ordinary Feynman-Kac-Itô formula gives a path integral representation of the Schrödinger semi-groups. We discuss here an analogous probabilistic expression for the (positive energy) solutions of the imaginary-time Klein-Gordon equation in a static external electromagnetic field. When the external field is not purely magnetic, our result is different from the path integral associated to the semi-group t ∊ [0, + ∞ ) → exp[− t ( H − mc 2 )/ℏ] of a relativistic Schrödinger operator H = { c 2 [− i ℏ∇−( e/c ) A ] 2 + m 2 c 4 } 1/2 + V .
A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.
In this paper we prove a Feynman-Kac-Itô formula for magnetic Schrödinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrödinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman-Kac-Itô formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman-Kac-Itô formula, we prove a Kato inequality, a Golden-Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials.