The stochastic mechanics of the Pauli equation

Abstract:

In stochastic mechanics, the Bopp-Haag-Dankel diffusions on ${\mathbb {R}^3} \times \operatorname {SO} (3)$ are used to represent particles with spin. Bopp and Haag showed that in the limit as the particle’s moment of inertia $I$ goes to zero, the solutions of the Bopp-Haag equations converge to that of the regular Pauli equation. Nelson has conjectured that in the same limit, the projections of the Bopp-Haag-Dankel diffusions onto ${\mathbb {R}^3}$ converge to a Markovian limit process. In this paper, we prove this conjecture for spin $\operatorname {spin} \;\tfrac {1} {2}$ and regular potentials, and identify the limit process as the diffusion naturally associated with the solution to the regular Pauli equation.