Origin of the quantum observable operator algebra in the frame of stochastic mechanics

Francesco Guerra and Rossana Marra
Phys. Rev. D 28, 1916 – Published 15 October 1983
PDFExport Citation

Abstract

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 Schrödinger 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(hi)x, l(hi)φ, 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.

  • Received 23 May 1983

DOI:https://doi.org/10.1103/PhysRevD.28.1916

©1983 American Physical Society

Authors & Affiliations

Francesco Guerra

  • Departement of Mathematics, Istituto "G. Castelnuovo," Università di Roma, I-00185, Roma, Italy

Rossana Marra

  • Istituto di Fisica dell'Università, I-84100, Salerno, Italy

References (Subscription Required)

Click to Expand
Issue

Vol. 28, Iss. 8 — 15 October 1983

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review D

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×