A Feynman-Kac formula for magnetic monopoles
J. Dimock
TL;DR
This paper develops a Feynman-Kac formula for quantum particles influenced by magnetic monopoles, connecting geometric bundle theory with stochastic methods to analyze the Schrödinger equation.
Contribution
It introduces a Feynman-Kac representation for the Schrödinger equation with magnetic monopoles, using bundle connections and eigenfunction expansions.
Findings
Established essential self-adjointness of the Hamiltonian.
Derived a stochastic integral representation of solutions.
Connected geometric bundle theory with quantum stochastic analysis.
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 a Feynman-Kac formula expressing solutions of the imaginary time Schrodinger equation as stochastic integrals.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Scientific Research and Discoveries · advanced mathematical theories