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arXiv:math-ph/0610042

Energy and Momentum Conservation for Diffusion - a Stochastic Mechanics Approximation - Part I

Johan G. B. Beumee

Published Jan 1, 2006• 0 citations
arXivHTMLPDF
DOI
10.2139/ssrn.934570

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The dynamics and kinematics of reciprocal diffusions were examined in a previous paper [J. Math. Phys. 34, 1846 (1993)], where it was shown that reciprocal diffusions admit a chain of conservation laws, which close after the first two laws for two disjoint subclasses of reciprocal diffusions, the Markov and quantum diffusions. For the case of quantum diffusions, the conservation laws are equivalent to Schrödinger’s equation. The Markov diffusions were employed by Schrödinger [Sitzungsber. Preuss. Akad. Wiss. Phys. Math Kl. 144 (1931); Ann. Inst. H. Poincaré 2, 269 (1932)], Nelson [Dynamical Theories of Brownian Motion (Princeton University, Princeton, NJ, 1967); Quantum Fluctuations (Princeton University, Princeton, NJ, 1985)], and other researchers to develop stochastic formulations of quantum mechanics, called stochastic mechanics. We propose here an alternative version of stochastic mechanics based on quantum diffusions. A procedure is presented for constructing the quantum diffusion associated to a given wave function. It is shown that quantum diffusions satisfy the uncertainty principle, and have a locality property, whereby given two dynamically uncoupled but statistically correlated particles, the marginal statistics of each particle depend only on the local fields to which the particle is subjected. However, like Wigner’s joint probability distribution for the position and momentum of a particle, the finite joint probability densities of quantum diffusions may take negative values.
Based on the assumption that the standard Schrödinger equation becomes gravitationally modified for massive macroscopic objects, two independent proposals have survived from the 1980s. The Schrödinger–Newton equation (1984) provides well-localized solitons for free macro-objects but lacks the mechanism of how extended wave functions collapse on solitons. The gravity-related stochastic Schrödinger equation (1989) provides the spontaneous collapse, but the resulting solitons undergo a tiny diffusion, leading to an inconvenient steady increase in the kinetic energy. We propose the stochastic Schrödinger–Newton equation, which contains the above two gravity-related modifications together. Then, the wave functions of free macroscopic bodies will gradually and stochastically collapse to solitons, which perform inertial motion without momentum diffusion: conservation of momentum and energy is restored.
The present paper shows that Edward Nelson's stochastic mechanics approach for quantum mechanics can be derived from the two classical elastically colliding particles with masses M and m satisfying a collision momentum preserving equation. The properties of the classical elastic momentum collision expression determine the full Edward Nelson energy collision energy for both particles. This classical total energy expression does not require a statistical expectation since no process was defined for the energy and it models the main and incident particle velocities perfectly. Quantum mechanics can be obtained by modelling the incident particle as a non-random potential using stochastic processes modelling the forward, post-collision and backward pre-collision velocities of the main particle. This presents the Schroedinger equation exactly the way that Nelson proposed in 1966 except for the diffusion constant. In this case the average energy is conserved in time and the forward, post-collision and backward pre-collision velocities of the system are related using statistical methods. If the incident particle does not have a potential the additional constraints for the movement of the incident particle leads to another Schroedinger equation. Finally, under suitable conditions it will be shown that the colliding particles satisfy Minkowski metric in special relativity. This last example shows how gravity can be quantized using details of this energy expression.