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IEEE Transactions on Automatic Control

Doing quantum mechanics with control theory

H.H. Rosenbrock

Published Jan 1, 2000• 28 citations
Publisher

Abstract

Bellman and Dreyfus (1964) showed how classical mechanics can be obtained from Hamilton's principle by dynamic programming. If we add noise in a particular way, we obtain Schrodinger's equation and many other results from the elementary theory of quantum mechanics. Some new results can also be obtained, and there are important consequences for the philosophy of science.

DOI
10.1109/9.827357

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Bellman and Dreyfus (1964) showed how classical mechanics can be obtained from Hamilton's principle by dynamic programming. If we add noise in a particular way, we obtain Schrodinger's equation and many other results from the elementary theory of quantum mechanics. Some new results can also be obtained, and there are important consequences for the philosophy of science.
This brief paper continues a development in earlier publications by Rosenbrock, in which results in quantum mechanics were obtained by adding a random disturbance to Hamilton's principle. The disturbance is complex, and the resulting variational problem is solved by dynamic programming, which computes the causal relations needed to accomplish a purpose. It is shown here that this procedure leads to many results that agree with those obtained in the standard theory of quantum mechanics, but also to some that are not readily obtained in that theory. For example, certain changes at a time t m to the end condition x f , which will be reached at t f >t m , can instantaneously change some aspects of the particle's behaviour at t m while it is still in transit. This accounts for Einstein's `spooky action at a distance'. An approach is also given to interference and entanglement, which is simpler than the standard account and may therefore be useful in quantum control and quantum computation.
Abstract An earlier development of some results in quantum mechanics from a stochastic variational principle is extended in several directions. An outline is first given of the methods of control theory upon which the development is based, and earlier results are briefly described. Extensions are then given to relativistic systems, to Dirac’s equation, and to elementary quantum field theory. The aim thoughout is to show that results in the standard theory can be obtained in a uniform way from an extended form of Hamilton’s principle, which has the advantage of conciseness and a relatively close relationship to the classical theory. The wave function appears as a modified form of the optimal cost function, and the photon can be identified with a singularity in the electromagnetic field. Interference is explained by optimization of an expected value, the ensemble over which the expectation is taken being dependent upon the information available.
In this paper, the relations between theories of the quantum mechanics and the stochastic control are briefly surveyed based on the recent works of the author (Ohsumi, $1989a(Ohsumi, $ ,$ $1989b(Ohsumi, $ $ , 1992)).Concretely, two types of (nonlinear) Schr\"odinger eqations from the stochsatic control theory by introducing wavefunctions which connect two solutions of nonlinear filtering equation and stochastic optimal control problem.Furthermore, an inverse prob- lem is investigated to derive the optimal control problem from a given Schr\"odinger equation.
We introduce a transformation of the quantum phase S′=S+ℏ2log⁡ρS'=S+\frac{\hbar}{2}\logρS′=S+2ℏ​logρ, which converts the deterministic equations of quantum mechanics into the Lagrangian reference frame of stochastic particles. We show that the quantum potential can be removed from the transformed quantum Hamilton-Jacobi equations if they are solved as stochastic Hamilton-Jacobi-Bellman equations. The system of equations provide a local description of quantum mechanics, which is enabled by the inherently retrocausal nature of stochastic Hamilton-Jacobi-Bellman equations. We also investigate the stochastic transformation of the classical system, where is it shown that quantum mechanics with the quantum potential reduced by a factor of 12\frac{1}{2}21​ has a classical representation, which may have interesting implications. Finally, we discuss the notion of a subsystem correspondence principle, which constrains the ontology of the total quantum system.