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.
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.
In this paper we establish three variational principles that provide new foundations for Nelson’s stochastic mechanics in the case of nonrelativistic particles without spin. The resulting variational picture is much richer and of a different nature with respect to the one previously considered in the literature. We first develop two stochastic variational principles whose Hamilton–Jacobi-like equations are precisely the two coupled partial differential equations that are obtained from the Schrödinger equation (Madelung equations). The two problems are zero-sum, noncooperative, stochastic differential games that are familiar in the control theory literature. They are solved here by means of a new, absolutely elementary method based on Lagrange functionals. For both games the saddle-point equilibrium solution is given by the Nelson’s process and the optimal controls for the two competing players are precisely Nelson’s current velocity v and osmotic velocity u, respectively. The first variational principle includes as special cases both the Guerra–Morato variational principle [Phys. Rev. D 27, 1774 (1983)] and Schrödinger original variational derivation of the time-independent equation. It also reduces to the classical least action principle when the intensity of the underlying noise tends to zero. It appears as a saddle-point action principle. In the second variational principle the action is simply the difference between the initial and final configurational entropy. It is therefore a saddle-point entropy production principle. From the variational principles it follows, in particular, that both v(x,t) and u(x,t) are gradients of appropriate principal functions. In the variational principles, the role of the background noise has the intuitive meaning of attempting to contrast the more classical mechanical features of the system by trying to maximize the action in the first principle and by trying to increase the entropy in the second. Combining the two variational principles, we get the quantum Hamilton principle, i.e., a variational characterization of the logarithm of the wave function ψ. The Lagrangian is the Lagrangian of classical mechanics with the complex-valued velocity v−iu replacing the classical velocity. The dynamics is given by a stochastic differential equation for real-valued diffusions with complex-valued drift and driving noise processes. From the variational principle we derive a Newton-type law. We finally define the momentum process and show that its mean and variance coincide with those of the quantum momentum operator.
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 consider stochastic variational principles for random processes taking values on a discrete configuration space. For a suitable time-reversal-invariant choice of the stochastic action, the resulting programming equations can be related to the Schr\"odinger equation for a discrete system. If the discrete system is considered as an approximation of a continuous system, then the limit reproduces Nelson's stochastic mechanics and allows one to derive the assumptions on the random noise acting on the system. In fact, the variational principle also gives information about the osmotic behavior of the process. Finally, we show that there are also critical processes whose behavior can be interpreted as a model for quantum measurement, because they relax in time to mixtures of processes. Therefore, stochastic variational principles can provide a very simple conceptual model simulating quantum behavior, both from the point of view of unperturbed time evolution and of the measurement phenomena leading to wave-function collapse.
In the framework of Nelson’s stochastic mechanics [E. Nelson, Dynamical Theories of Brownian Motion (Princeton University, Princeton, 1967); F. Guerra, Phys. Rep. 77, 263 (1981); E. Nelson, Quantum Fluctuations (Princeton University, Princeton, 1985)] we seek to develop the particle counterpart of the hydrodynamic results of M. Pavon [J. Math. Phys. 36, 6774 (1995); Phys. Lett. A 209, 143 (1995)]. In particular, a first form of Hamilton’s principle is established. We show that this variational principle leads to the correct equations of motion for the classical particle, the Brownian particle in thermodynamical equilibrium, and the quantum particle. In the latter case, the critical process q satisfies a stochastic Newton law. We then introduce the momentum process p, and show that the pair (q,p) satisfies canonical-like equations.
The Lagrangian variational principle with the classical action leads, in stochastic mechanics, to Madelung’s fluid equations, if only irrotational velocity fields are allowed, while new dynamical equations arise if rotational velocity fields are also taken into account. The new equations are shown to be equivalent to the (gauge invariant) system of a Schrödinger equation involving a four-vector potential (A,Φ) and the coupled evolution equation (of magnetohydrodynamical type) for the vector field A. A general energy theorem can be proved and the stability properties of irrotational and rotational solutions investigated.