Nelson and Wiener

Charles R. Johnson, Carlos M. Saiago

Frederik Jozef Belinfante

Maurice A. de Gosson

Folkert Kuipers

We study a (relativistic) Wiener process on a complexified (pseudo-)Riemannian manifold. Using Nelson’s stochastic quantization procedure, we derive three equivalent descriptions for this problem. If the process has a purely real quadratic variation, we obtain the one-sided Wiener process that is encountered in the theory of Brownian motion. In this case, the result coincides with the Feyman–Kac formula. On the other hand, for a purely imaginary quadratic variation, we obtain the two-sided Wiener process that is encountered in stochastic mechanics, which provides a stochastic description of a quantum particle on a curved spacetime.

Donald G. Babbitt

Introduction.It has been known for a long time that the Wiener integral can be effectively used in studying various properties of the Schrödinger operator A-V(x) where A = d 2 /dxl + ' ' ' +d 2 /dx 2 n , the Laplacian in m-dimensional Euclidean space E m and V(x) is a realvalued function on E m satisfying certain regularity conditions.For example, see works by Kac [6], Getoor [2; 3], Ray [9] and Nelson [8].Getoor [2] was the first to point out the connection between a Wiener type integral and the perturbation theory of certain operators of the form Q -V(x) where 0 is the infinitesimal generator of a homogeneous Markov process on a locally compact metric space X and V(x) is a real-valued Borel measurable function on X, bounded on bounded sets and bounded below everywhere.When X -E m and the Markov process is a suitably normed Wiener process, 2 12= A and his results apply to the Schrödinger operator A-V(x).In this paper we will only consider the case X = E m and Q -A but V(x) will be allowed to have certain singularities precluded by Getoor's conditions on V.For example, we can consider the perturbation of A+e/r, where X = E Z and r= (x 2 -\-y 2 +z 2 ) 112 , i.e., the Schrödinger operator with attractive Coulomb potential.In particular the essential condition on V(x) is that A -V is semi-bounded above on a certain dense domain D in L 2 (£ w ), the Hubert space of complexvalued square summable function on E m .z Main results.In this section we will introduce some notation, state our theorems and corollaries, sketch one proof and remark on the remaining proofs.Co(E m ) will denote the space of infinitely differentiable, complexvalued functions on E m with compact support.Let L l ™(E m ) denote 1 The preparation of this paper was sponsored in part by the National Science Foundation.2 In particular we assume a 2 = 2 where a 2 is the variance per unit of time of the Wiener process.See [6]. DEFINITION. A symmetric operator A with domain DÇ-L2(E m) is said to be semibounded from above if there exists a real number k< oo such that (A^, ^)^k{\p, \j/) s&||^|| 2 for all ^££> and where (</>, xl/)=f<f>^dx.

Fusao Ishii

Quantum mechanics is defined here as the theoretical framework crystallised at the fifth Solvay Conference of 1927, comprising five postulates: the wavefunction, the canonical commutation relation [x, p] = iℏ, the Born rule, the Schrödinger equation, and the transition probability rule. We present a complete derivation of all five Solvay postulates from purely classical foundations, starting from the directly observable Coulomb field of the electron. The single empirical input is ℏ—measured from pre-Solvay experiments (Planck 1900–1911, Einstein 1905, Casimir 1948/Lamoreaux 1997) without quantum mechanical interpretation, in the same way G was measured by Cavendish (1798) without general relativity. By Einstein’s photoelectric theory (1905) generalized to all electromagnetic fields, the Coulomb field energy is carried in electromagnetic quanta, each carrying both electric and magnetic fields with energy density ϵ0E^2. Lorentz invariance uniquely determines the ZPF spectral energy density u(ω) ∝ ω^3 (Boyer 1969, 1975). The Casimir force measurement gives the total ZPF energy per mode B = ℏ; classical E/B equipartition from Maxwell’s equations gives the electric energy per mode ℏω/2. The central original contribution is the physical justification of the Compton frequency cutoff ωc = mc^2/ℏ: an electron cannot emit an electromagnetic quantum with energy exceeding mc^2, because the residual electron would require negative mass, never observed in nature. By equating two independent expressions for the momentum variance of the ZPF-driven harmonic oscillator—one from the Abraham–Lorentz equation, one from Nelson’s osmotic velocity formula—D = ℏ/2m is derived without any quantum assumption. The ZPF-driven motion is then proved to be Brownian motion in the precise sense of Wiener (1923): all four Wiener conditions are verified from the properties of the ZPF established in Sections 3–4. This answers the question Nelson (1966) explicitly left open: “We do not attempt to give a physical explanation of why the electron should undergo Brownian motion.” The physical explanation is that the electron cannot escape its own Coulomb field. By Itô’s stochastic calculus and Nelson’s theorem, this gives the Schrödinger equation as an effective description valid for timescales Δt ≫ ℏ/mc^2. The transformation ψ = √ρe^(iS/ℏ) linearises the nonlinear Fokker–Planck equation, explaining why superposition holds. All five Solvay postulates emerge as consequences of classical physics plus the single measured constant ℏ. No Solvay postulate is assumed anywhere.