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Stationary-Action Stochastic Control Representation of the Schrödinger Initial Value Problem

Ruobing Zhao

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

Hamilton-Jacobi partial differential equations (HJ PDEs) arise in many scientific fields and applications, especially in mechanics and optimal control. Solutions to second order Hamilton-Jacobi partial differential equations (HJ PDEs) have controlled diffusion process representations. Of particular interest is the diffusion representation of an action functional associated with the solution of Schrödinger initial value problems (IVPs). In existing work that connects stochastic control problems to Schrödinger IVPs, one searches for the minimum of an action functional which is the payoff of the control problem. Issues arise, however, as the action functional loses convexity over longer time durations. The time duration where such representation is valid is in fact infinitesimal when the system dimension is infinite. In this dissertation, we present an approach inspired by the Principle of Stationary Action in physics that removes this limitation, which leads to exceptional computational benefits. Instead of searching for local minima of the action functional as in traditional optimal control problems, we look for its stationary values instead. We introduce the ``staticization" operator and stationary-action control problems. A stationary-action stochastic control representation for the dequantized Schrödinger IVP in a non-inertial frame where the potential field is a polynomial is given. A solution approximation as a series expansion in a small parameter is obtained through the use of complex-valued diffusion-process representations. Following this, the staticization operator is studied in detail. A new approach to solving conservative dynamical systems (e.g. paths of objects in gravitational or Coulomb potential field) using stationary-action control-theoretic methods with promising computational benefits is introduced. Two examples of application, the N-body problem and the Schrödinger IVP, are given. Lastly, we demonstrate the existence of strong solutions of a class of degenerate stochastic differential equations (SDEs) that arises in the stationary-action stochastic control representation of Schrödinger IVPs with the Coulomb potential, which is non-smooth and has branch cuts in the complex range. The SDEs we consider have drift terms that have discontinuities and singularities along some manifolds, and diffusion coefficients that are degenerate, and there is no previously existing results regarding the existence of strong solutions for such SDEs.

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Hamilton-Jacobi partial differential equations (HJ PDEs) arise in many scientific fields and applications, especially in mechanics and optimal control. Solutions to second order Hamilton-Jacobi partial differential equations (HJ PDEs) have controlled diffusion process representations. Of particular interest is the diffusion representation of an action functional associated with the solution of Schrödinger initial value problems (IVPs). In existing work that connects stochastic control problems to Schrödinger IVPs, one searches for the minimum of an action functional which is the payoff of the control problem. Issues arise, however, as the action functional loses convexity over longer time durations. The time duration where such representation is valid is in fact infinitesimal when the system dimension is infinite. In this dissertation, we present an approach inspired by the Principle of Stationary Action in physics that removes this limitation, which leads to exceptional computational benefits. Instead of searching for local minima of the action functional as in traditional optimal control problems, we look for its stationary values instead. We introduce the ``staticization" operator and stationary-action control problems. A stationary-action stochastic control representation for the dequantized Schrödinger IVP in a non-inertial frame where the potential field is a polynomial is given. A solution approximation as a series expansion in a small parameter is obtained through the use of complex-valued diffusion-process representations. Following this, the staticization operator is studied in detail. A new approach to solving conservative dynamical systems (e.g. paths of objects in gravitational or Coulomb potential field) using stationary-action control-theoretic methods with promising computational benefits is introduced. Two examples of application, the N-body problem and the Schrödinger IVP, are given. Lastly, we demonstrate the existence of strong solutions of a class of degenerate stochastic differential equations (SDEs) that arises in the stationary-action stochastic control representation of Schrödinger IVPs with the Coulomb potential, which is non-smooth and has branch cuts in the complex range. The SDEs we consider have drift terms that have discontinuities and singularities along some manifolds, and diffusion coefficients that are degenerate, and there is no previously existing results regarding the existence of strong solutions for such SDEs.
A stochastic control representation for the Maslov dequantization of the solution of the Schrödinger equation is obtained. The representation employs complex-valued diffusion process and stationarity of an action functional. The use of stationarity removes the need for convexity of the action, and hence there is no restriction on the duration of the problem. Previous efforts along this line were restricted to potentials which had holomorphic extensions to the entire space. The use of a stationarity based representation for the Coulomb potential removes that restriction. The result is a representation in terms of iterated staticization, which is expected to yield exceptional computational benefits.
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.
A diffusion-based representation for the Maslov dequantization of the time-dependent Schrödinger equation is obtained, where the wave function corresponds to the statical value function of a complex-valued stochastic control problem. Whereas previous efforts along this line were restricted to holomorphic potentials, we here use an additional controller specifically constructed to generate the Coulomb potential via staticization. The solution to the Schrödinger equation is obtained from requantization of the representations. A demonstrative numerical example is included at the end.
Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. image
Abstract Recent studies have extended the use of the stochastic Hamilton-Jacobi-Bellman (HJB) equation to include complex variables for deriving quantum mechanical equations. However, these studies often assume that it is valid to apply the HJB equation directly to complex numbers, an approach that overlooks the fundamental problem of comparing complex numbers when finding optimal controls. This paper explores the application of the HJB equation in the context of complex variables. It provides an in-depth investigation of the stochastic movement of quantum particles within the framework of stochastic optimal control theory. We obtain the complex diffusion coefficient in the stochastic equation of motion using the Cauchy-Riemann theorem, considering that the particle’s stochastic movement is described by two perfectly correlated real and imaginary stochastic processes. During the development of the covariant form of the HJB equation, we demonstrate that if the temporal stochastic increments of the two processes are perfectly correlated, then the spatial stochastic increments must be perfectly anti-correlated, and vice versa. The diffusion coefficient we derive has a form that enables the linearization of the HJB equation. The method for linearizing the HJB equation, along with the subsequent derivation of the Dirac equation, was developed in our previous work [V. Yordanov, Scientific Reports 14, 6507 (2024)]. These insights deepen our understanding of quantum dynamics and enhance the application of stochastic optimal control theory to quantum mechanics.