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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Stochastic space-time and the concept of potential in classical and quantum mechanics

R. K. Roy Choudhury, S. Roy

Published Sep 15, 1982• 6 citations
Publisher

Abstract

The generation of a potential from the fluctuation of the metric of micro-space-time due to an averaging procedure at the macrolevel as proposed in a previous paper in one dimension has been generalized to three dimensions. Also, an attempt is made to study the behavior of quantum processes in this framework. It raises a new possibility of rethinking the role of the potential in quantum mechanics.

DOI
10.1103/physrevd.26.1335

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The generation of a potential from the fluctuation of the metric of micro-space-time due to an averaging procedure at the macrolevel as proposed in a previous paper in one dimension has been generalized to three dimensions. Also, an attempt is made to study the behavior of quantum processes in this framework. It raises a new possibility of rethinking the role of the potential in quantum mechanics.
A new idea for the quantization of dynamic systems, as well as space time itself, using a stochastic metric is proposed. The quantum mechanics of a mass point is constructed on a space time manifold using a stochastic metric. A stochastic metric space is, in brief, a metric space whose metric tensor is given stochastically according to some appropriate distribution function. A mathematically consistent model of a space time manifold equipping a stochastic metric is proposed in this work. The quantum theory in the local Minkowski space can be recognized as a classical theory on the stochastic Lorentz-metric-space. A stochastic calculus on the space time manifold is performed using white noise functional analysis. A path-integral quantization is introduced as a stochastic integration of a function of the action integral, and it is shown that path-integrals on the stochastic metric space are mathematically well-defined for large variety of potential functions. The Newton–Nelson equation of motion can also be obtained from the Newtonian equation of motion on the stochastic metric space. It is also shown that the commutation relation required under the canonical quantization is consistent with the stochastic quantization introduced in this work. The quantum effects of general relativity are also analyzed through natural use of the stochastic metrics. Some example of quantum effects on the Universe is discussed.
Preface Space-Time Geometry & Quantum Events The Transaction as a Quantum Concept Stochastic Foundation of Quantum Mechanics & the Origin of Particle Spin Fractal Space-Time as an Underlying Structure of the Standard Model Relativity of Scales, Fractal Space & Quantum Potentials Three Possible Implications of Space-Time Discreteness Quantum Computing Space-Time Metric Gauge Fields in Deformed Special Relativity Non-Local Granular Space-Time Foam as an Ultimate Arena at the Planck Scale Structural Approach to the Elementary Particle Theory Forming Physical Fields & Pseudometric & Metric Manifolds. Noncommutativity & Discrete Structures in Classical & Quantum Physics Index.
In this paper we formulate a new stochastic description of quantum mechanics in phase space. The theory of phase-space representations of quantum mechanics, initiated by Wigner, Groenewold, and Moyal and systematized recently by Agarwal and Wolf is essentially a single-time theory, in that it deals only with the quantum-mechanical joint distribution functions for position and momentum at a single instant of time. We develop a natural multitime extension of such a single-time theory. We consider a class of multitime phase-space distribution functions such that an arbitrary quantum multitime correlation function can be expressed as a phase-space average of the form encountered in classical stochastic theories. We study the nonclassical features of these multitime distribution functions and show that they may be considered as characterizing a generalized stochastic process in phase space. We demonstrate that the multitime distribution functions that correspond to Hamiltonian evolution of isolated quantum systems satisfy a certain condition that may be regarded as characterizing a generalized Markov process. We also investigate certain special features of the generalized stochastic processes that characterize the evolution of open systems.