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Journal of Mathematical Physics

The dequantization programme for stochastic quantum mechanics

Franklin E. Schroeck

Published Feb 1, 1985• 13 citations
Publisher

Abstract

The classical limit ℏ→0 of stochastic quantum mechanics is investigated. An algebra of stochastic classical observables is canonically associated to the algebra of stochastic quantum observables with a product inherited from sequential instruments and also with a Lie commutator. It is shown that the sharp-point limit (limit of no stochasticity), which implies ℏ→0, yields the usual algebra of classical observables.

DOI
10.1063/1.526659

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The classical limit ℏ→0 of stochastic quantum mechanics is investigated. An algebra of stochastic classical observables is canonically associated to the algebra of stochastic quantum observables with a product inherited from sequential instruments and also with a Lie commutator. It is shown that the sharp-point limit (limit of no stochasticity), which implies ℏ→0, yields the usual algebra of classical observables.
In previous work the approach to stochastic quantization, originally proposed by Nelson, has been formulated in the frame of the stochastic variational principles of control theory. Then the Hamilton-Jacobi-Madelung equation is interpreted as the programming equation of the controlled problem, to be associated with the hydrodynamical continuity equation. Here we point out explicitly the canonical Hamiltonian structure of these equations, by introducing a suitable symplectic structure on the underlying phase space in various representations. One possible representation leads to the Schr\"odinger equation, which, together with its complex conjugate, can be recognized as a particular form of the Hamilton canonical equations in this frame. Then a suitably selected time-invariant subalgebra of the classical hydrodynamical algebra, closed under Poisson bracket pairing, is shown to be connected to the standard quantum observable operator algebra. In this correspondence Poisson brackets for hydrodynamical observables become averages of quantum observables in the given state. From this point of view stochastic quantization can be interpreted as giving an explanation for the standard quantization procedure of replacing the classical particle (or field) observables with operators, according to the scheme $p\ensuremath{\rightarrow}(\frac{h}{i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}x}$, $l\ensuremath{\rightarrow}(\frac{h}{i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}\ensuremath{\varphi}}$, etc. This discussion shows also the relevance of the canonical symplectic structure of the quantum state space, a feature which seems to have been overlooked in the axiomatic approaches to quantum mechanics.
We start by rewriting classical mechanics in a quantum mechanical fashion and point out that the only difference with quantum theory re- sides at one point. There is a classical analogon of the collapse of the wavefunction and an extension of the usual Born rule is proposed which might solve this problem. We work only with algebra’s over the real (com- plex) numbers, general number fields of finite characteristic allowing for finite dimensional representations of the commutation relations are not considered given that such fields are not well ordered and do not give rise to a well defined probability interpretation. Our theory generalizes however to discrete spacetimes and finite dimensional algebra’s. Looking at physics this way, spacetime itself distinguishes itself algebraically by means of well chosen commutation relations and there is further nothing special about it meaning it has also a particle interpretation just like any other dynamical variable. Likewise, there is no reason for the dynamics to be Hamiltonian and therefore we have a nonconservative formulation of quantum physics at hand. The harmonic oscillator (amongst few oth- ers) distinguishes itself because the algebra forms a finite dimensional Lie algebra; the classical and quantum (discrete) harmonic oscillator are stud- ied in a some more generality and examples are given which are neither classical, nor quantum.
The deduction by Guerra and Marra of the usual quantum operator algebra from a canonical variable Hamiltonian treatment of Nelson's hydrodynamical stochastic description of real nonrelativistic Schr\"odinger waves is extended to the causal stochastic interpretation given by Guerra and Ruggiero and by Vigier of relativistic Klein-Gordon waves. A specific representation shows that the Poisson brackets for canonical hydrodynamical observables become ``averages'' of quantum observables in the given state. Stochastic quantization thus justifies the standard procedure of replacing the classical particle (or field) observables with operators according to the scheme ${p}_{\ensuremath{\mu}}$\ensuremath{\rightarrow}-i\ensuremath{\Elzxh}${\ensuremath{\partial}}_{\ensuremath{\mu}}$ and ${L}_{\ensuremath{\mu}\ensuremath{\nu}}$\ensuremath{\rightarrow}-i\ensuremath{\Elzxh}(${x}_{\ensuremath{\mu}}$${\ensuremath{\partial}}_{\ensuremath{\nu}}$-${x}_{\ensuremath{\nu}}$${\ensuremath{\partial}}_{\ensuremath{\mu}}$).
A theory of stochastic flows over the algebra of observables of a dynamical system is presented in which the main objective is to ensure that the overall canonical/symplectic structure on the algebra is preserved. We study both classical and quantum systems and the importance of physical interpretation in the Stratonovich interpretation is stressed. We find the natural formulation of quantum dissipative systems to be given in terms of quantum stochastic calculus. This treatment allows for a physically meaningful treatment of both constant and nonlinear dissipation. As an application, we quantize a mechanical system with the same nonlinear damping mechanism as the van der Pol oscillator.
A quantum effect is an operator A on a complex Hilbert space H that satisfies 0⩽A⩽I. We denote the set of quantum effects by E(H). The set of self-adjoint projection operators on H corresponds to sharp effects and is denoted by P(H). We define the sequential product of A,B∈E(H) by A∘B=A1/2BA1/2. The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on A∘B imply that A and B commute for the usual operator product. For example, if A∘B satisfies certain distributive or associative laws, then AB=BA. Moreover, if A∘B∈P(H), then AB=BA and A∘B=B or B∘A=B if and only if AB=BA=B. A natural definition of stochastic independence is introduced and briefly studied.