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Transactions of the American Mathematical Society

The Stochastic Mechanics of the Pauli Equation

Timothy C. Wallstrom

Published Jan 1, 1990• 10 citations
Publisher

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.

DOI
10.1090/s0002-9947-1990-0986033-3

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In stochastic mechanics, the Bopp-Haag-Dankel diffusions on R3×SO⁡(3){\mathbb {R}^3} \times \operatorname {SO} (3)R3×SO(3) are used to represent particles with spin. Bopp and Haag showed that in the limit as the particle’s moment of inertia III 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 R3{\mathbb {R}^3}R3 converge to a Markovian limit process. In this paper, we prove this conjecture for spin spin⁡  12\operatorname {spin} \;\tfrac {1} {2}spin21​ and regular potentials, and identify the limit process as the diffusion naturally associated with the solution to the regular Pauli equation.
Recently the path integral quantization of the classical spin model was accomplished by Nielsen and Rohrlich. The configuration space of classical spin is the unit sphere with punctures at poles. Motivated by the fact that the Nelson's stochastic mechanics idea was to randomize the configuration space variable of the classical system, we give a review of the probabilistic approaches to the quantization of spin 1/2, with emphasis on the Dankel's problem of finding the probabilistic description of the Pauli equation in terms of stochastic diffusion processes. The original Dankel's analysis referred to the rigid top, and the problem of the point particle limit was left unsolved. Our observation is that if the spin stochastic process refers to the unit sphere with punctures, then Dankel's results provide a solution to the Pauli equation with no need of extra limiting procedures. The effects of the magnetic field can be successfully incorporated into the formalism.
In Nelson [10], it is shown that for each solution of a Schr\"odinger equation ϕ(x,t),\phi(x, t),ϕ(x,t), x∈Rd,x\in R^{d},x∈Rd, t≧0t\geqq 0t≧0 , with ∥ϕ∥=1\Vert\phi\Vert=1∥ϕ∥=1 , there exists a diffusion process which has ... probability density at time ttt given byMore precisely the following is shown in [10]: suppose that we are given a real valued function V(x),V(x),V(x), x∈Rdx\in R^{d}x∈Rd , and a complex valued function ϕ0(x),\phi_{0}(x),ϕ0​(x), x∈Rdx\in R^{d}x∈Rd , with ∥ϕ0∥=1\Vert\phi_{0}\Vert=1∥ϕ0​∥=1 , and consider the Schrodinger equation Then, under the assumption that ϕ\phiϕ and b~\tilde{b}b~ are sufficiently regular, there exists a diffusion process {Xt},\{X_{t}\},{Xt​}, t≧0t\geqq 0t≧0 , with initial density ∣ϕ0(x)∣2|\phi_{0}(x)|^{2}∣ϕ0​(x)∣2 and generator (1/2)Δ+b~(x,t)⋅∇(1/2)\Delta+\tilde{b}(x, t)\cdot\nabla(1/2)Δ+b~(x,t)⋅∇ such that the probability density of this diffusion at time t≧0t\geqq 0t≧0 is ∣ϕ(x,t)∣2|\phi(x, t)|^{2}∣ϕ(x,t)∣2 .In addition, in [10] it is also shown that the diffusion process {Xt},\{X_{t}\},{Xt​}, t≧0t\geqq 0t≧0 , solves the second order stochastic differential equation (1/2)(D∗D+DD∗)Xt=−∇V(Xt)(1/2)(D_{*}D+DD_{*})X_{t}=-\nabla V(X_{t})(1/2)(D∗​D+DD∗​)Xt​=−∇V(Xt​) , where DDD and D∗D_{*}D∗​ are Nelson's forward and backward stochastic derivatives.Generally sPeaking, the functions ϕ\phiϕ and b~\tilde{b}b~ may be singular, and the existence of the corresponding diffusion process must be studied carefully.Carlen [1,2][1, 2][1,2] , Meyer and Zheng [8] and Nagasawa [9] considered rigorously the construction of a diffusion process with a given initial density and a generator.lnlnln order to construct a Markovian propagator for the diffusion process, Carlen [1,2][1, 2][1,2] used partial differential equation methods.Meyer and Zheng [8] restricted themselves to the case when ϕ(x,t)=ϕ0(x),\phi(x, t)=\phi_{0}(x),ϕ(x,t)=ϕ0​(x), ∀t≧0\forall_{t\geqq 0}∀t≧0​ and considered the construction of the diffusion process through the relationships between Markov processes and Dirichlet forms.We may consult Nagasawa [9] about recent developments of this area.In Carlen [1], it is assumed that V(x),V(x),V(x), x∈Rdx\in R^{d}x∈Rd , is a Rellich class potential and ∥∇ϕ0∥2<∞\Vert\nabla\phi_{0}\Vert^{2}<\infty∥∇ϕ0​∥2<∞ , and shown that there exists a diffusion process {Xt},\{X_{t}\},{Xt​}, t≧0t\geqq 0t≧0 , which has the probability density ∣ϕ(x,t)∣2|\phi(x, t)|^{2}∣ϕ(x,t)∣2 at time t≧0t\geqq 0t≧0 and admits the representation
In this paper the Nelson’s stochastic mechanics is extended to general diffusion motions. A representation theorem is proved which gives a one-to-one correspondence between solutions of certain Schrödinger equations and diffusion processes satisfying appropriate regularity conditions. Exploiting results of stochastic mechanics on Riemannian manifolds it is shown that the real part of the Schrödinger equations corresponding to the considered diffusions can be interpreted as Newton’s second law where the force is produced by generalized electromagnetic potentials.