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

A generalization of stochastic mechanics and its relation to quantum mechanics

Reinhard F. Werner

Published Jul 15, 1986• 49 citations
Publisher

Abstract

Nelson's assignment of a Markov process in configuration space to each Schr\"odinger Hamiltonian and each pure state is generalized to mixed states, arbitrary configuration spaces given by a maximal set of commuting observables, and general Hamiltonians given by integral kernels. Conditions for the quantum-mechanical description to be reconstructable from this stochastic description are discussed. It is shown that those stochastic expectation values which are stable with respect to the operation of coupling a dynamically independent but statistically correlated system to the given one are precisely those already expressible as quantum-mechanical expectation values.

DOI
10.1103/physrevd.34.463

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Nelson's assignment of a Markov process in configuration space to each Schr"odinger Hamiltonian and each pure state is generalized to mixed states, arbitrary configuration spaces given by a maximal set of commuting observables, and general Hamiltonians given by integral kernels. Conditions for the quantum-mechanical description to be reconstructable from this stochastic description are discussed. It is shown that those stochastic expectation values which are stable with respect to the operation of coupling a dynamically independent but statistically correlated system to the given one are precisely those already expressible as quantum-mechanical expectation values.
Nelson's stochastic mechanics may be understood as a stochastic underpinning, or reconstruction, of nonrelativistic quantum mechanics, once the diffusion scale is fixed by ℏ\hbarℏ and the admissible states are restricted by the usual single-valuedness condition on the wavefunction. In this note I briefly indicate what this route achieves and why it remains conceptually attractive. Four advantages are emphasized. First, it supplies a clear configuration-space stochastic picture of the underlying processes. Second, the Born rule is built in from the outset, with ∣ψ∣2|ψ|^2∣ψ∣2 arising as the probability density ρρρ of the underlying diffusion process rather than as an independent postulate. Third, it offers a markedly different perspective on measurement and nonlocality: in particular, collapse need not be treated as an extra axiom, and the nonlocality associated with entangled states is softened relative to the deterministic Bohmian guidance picture. Fourth, by tying quantumness to a diffusion scale, it naturally suggests a continuum of physical descriptions ranging from the strictly classical to the strictly quantum-mechanical regime. I conclude by proposing a natural distance scale in stochastic mechanics and examining its implications for testing possible limits of Bell correlations.
Statistically interpretable axioms are formulated that define a quantum stochastic process (QSP) as a causally ordered operator field in an arbitrary space-time region T of an open quantum system under a sequential observation at a discrete space-time localization. It is shown that to every QSP described in the weak sense by a self-consistent system of causally ordered correlation kernels there corresponds a unique, up to unitary equivalence, minimal QSP in the strong sense. It is shown that the proposed QSP construction, which reduces in the case of the linearly ordered discrete T=Z to the construction of the inductive limit of Lindblad's canonical representations, corresponds to Kolmogorov's classical reconstruction if the order on T is ignored and leads to Lewis construction if one uses the system of all (not only causal) correlation kernels, regarding this system as lexicographically preordered on T. The approach presented encompasses both nonrelativistic and relativistic irreversible dynamics of open quantum systems and fields satisfying the conditions of local commutativity and semigroup covariance. Also given are necessary and sufficient conditions of dynamicity (or conditional Markovianity) and regularity, these leading to the properties of complete mixing (relaxation) and ergodicity of the QSP.
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.
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.
Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.
Nelson's stochastic interpretation of quantum mechanics is extended from the case of pure states to that of mixed states. It is shown that a pure probabilistic formalism, which applies the Newton-Nelson law to the initial position and velocity distributions, does not reproduce the time evolution predicted by quantum mechanics. In order to recover the latter, a new notion must be introduced, that of pure quantum states, over which the mixture has to be decomposed, and which then satisfy the Newton-Nelson law independently.