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arXiv:2005.09438 · Infinite Dimensional Analysis Quantum Probability and Related Topics

A Feynman–Kac formula for magnetic monopoles

Jonathan Dimock

Published Jun 1, 2021• 0 citations
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Abstract

We consider the quantum mechanics of a charged particle in the presence of Dirac’s magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We use a continuum eigenfunction expansion to find an invariant domain of essential self-adjointness for the Hamiltonian. This leads to a proof of the Feynman–Kac formula expressing solutions of the imaginary time Schrödinger equation as stochastic integrals.

DOI
10.1142/s0219025721500156

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We consider the quantum mechanics of a charged particle in the presence of Dirac’s magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We use a continuum eigenfunction expansion to find an invariant domain of essential self-adjointness for the Hamiltonian. This leads to a proof of the Feynman–Kac formula expressing solutions of the imaginary time Schrödinger equation as stochastic integrals.
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In this paper, we consider Dirac equations on relativistic phase spaces [Formula: see text], where [Formula: see text] is Minkowski space with [Formula: see text]. We use the geometric quantization approach in which the wave functions are polarized sections of a complex line bundle [Formula: see text] over [Formula: see text]. The covariant derivatives with connection [Formula: see text] in this bundle define canonical commutation relations. Fermions are charged with respect to the field [Formula: see text], so lifting the Dirac equations from space-time [Formula: see text] to phase space [Formula: see text] results in their solutions being localized in the space [Formula: see text] or in space-time [Formula: see text]. We describe the explicit form of these solutions.
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).