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

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

Francesco Guerra, R. Marra

Published Oct 15, 1983• 35 citations
Publisher

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 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.

DOI
10.1103/physrevd.28.1916

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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}}$).

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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→(hi)\ensuremath∂\ensuremath∂xp\ensuremath{\rightarrow}(\frac{h}{i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}x}p\ensuremath→(ih​)\ensuremath∂x\ensuremath∂​, l\ensuremath→(hi)\ensuremath∂\ensuremath∂\ensuremathφl\ensuremath{\rightarrow}(\frac{h}{i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}\ensuremath{\varphi}}l\ensuremath→(ih​)\ensuremath∂\ensuremathφ\ensuremath∂​, 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.
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}}$).
Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function [Formula: see text] and phase function [Formula: see text] not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.
In the general framework of stochastic control theory we introduce a suitable form of stochastic action associated to the controlled process. A variational principle gives all the main features of Nelson's stochastic mechanics. In particular, we derive the expression for the current velocity field as the gradient of the phase action. Moreover, the stochastic corrections to the Hamilton-Jacobi equation are in agreement with the quantum-mechanical form of the Madelung fluid (equivalent to the Schr\"odinger equation). Therefore, stochastic control theory can provide a very simple model simulating quantum-mechanical behavior.
Abstract The quantization of the dynamics of point systems is closely connected with the Hamilton-Jacobi theory of the calculus of variations for simple integrals, the latter being a suitable mathematical formalism for describing the classical laws of point dynamics. This connexion is evident from the fact that, in order to quantize a dynamical system, one has first to know what the pairs of canonically conjugate variables are, and also from the fact that the quantum laws, if expressed in terms of commutation brackets, have exactly the same form as the classical laws, if expressed in terms of Poisson brackets. In dealing with the quantization of the dynamics of continuous media, e.g. the electromagnetic and other fields, which has been developed along different lines, following Heisenberg and Pauli (1929), it seems tempting to try to base the method of quantization on an extended Hamilton-Jacobi theory of the calculus of variations for multiple integrals, the latter being the appropriate formalism for describing most relevant systems of continuum physics. In a previous paper (Weiss 1936), referred to as I, such an attem pt has been made, by extending the notion of pairs of conjugate variables. That attem pt led to quantum relations on an arbitrary space-like section in space-time, provided that one postulated that, if the space-like section becomes especially a space section, the quantum relations should go over into those of Heisenberg and Pauli.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
Abstract We investigate operator algebraic origins of the classical Koopman–von Neumann wave function ψKvN\psi _{KvN}ψKvN​ ψ KvN as well as the quantum-mechanical one ψQM\psi _{QM}ψQM​ ψ QM . We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, ψKvN\psi _{KvN}ψKvN​ ψ KvN and ψQM\psi _{QM}ψQM​ ψ QM are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle E{{\mathcal {E}}}E E follows from this. ψKvN\psi _{KvN}ψKvN​ ψ KvN and ψQM\psi _{QM}ψQM​ ψ QM are shown to be sections in E{{\mathcal {E}}}E E . The difference between ψKvN\psi _{KvN}ψKvN​ ψ KvN and ψQM\psi _{QM}ψQM​ ψ QM originates from a quantization map interpreted as “twisting” of sections over E{{\mathcal {E}}}E E . We also show that the Schrödinger equation is obtained from the Koopman–von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.