Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constrained Least Action and Quantum Mechanics

Winfried Lohmiller, Jean-Jacques Slotine

TL;DR

The paper extends Euler–Lagrange dynamics to a constrained configuration space $\mathbb{G}^n \subset \mathbb{R}^n$ with Dirac impulse forces at the boundary, showing that non-Lipschitz constraint activation yields multiple deterministic extremals of the action $S = \int L\,dt$. For standard quantum contexts such as the double-slit, quantum amplitudes can be obtained by a minimization over these constrained extremals rather than the full $\int \mathcal{D}{\bf q}\, e^{\tfrac{i}{\hbar}S}$, effectively producing polygonal (diffraction-like) trajectories in place of continuous classical straight lines. In the quadratic-Lagrangian case, the quantum state becomes a superposition over extremals with $S(\bar{\mathbf q})$, linking constrained least-action paths to the Feynman framework. The authors discuss potential computational simplifications and the need to establish consistency with the full Feynman path integral for nonlinear cases, as well as implications for classical simulations of some quantum systems.

Abstract

The wave-particle duality and its probabilistic interpretation are at the heart of quantum mechanics. Here we show that, in some standard contexts like the double slit experiment, a deterministic interpretation can be provided. This interpretation arises from the fact that if one mininizes a deterministic action {\it subject to spatial inequality contraints}, in general the solution is not unique. Thus, the probabilistic setting is replaced in part by the non-uniqueness of solutions of certain constrained optimization problems, itself often a result of the non-Lipschitzness of the constrained dynamics. While the approach leaves the results of associated Feynman integrals unchanged, it may considerably simplify their computation as the problems grow more complex, since only specific deterministic paths computed from the constrained minimization need to be included in the integral.

Constrained Least Action and Quantum Mechanics

TL;DR

The paper extends Euler–Lagrange dynamics to a constrained configuration space with Dirac impulse forces at the boundary, showing that non-Lipschitz constraint activation yields multiple deterministic extremals of the action . For standard quantum contexts such as the double-slit, quantum amplitudes can be obtained by a minimization over these constrained extremals rather than the full , effectively producing polygonal (diffraction-like) trajectories in place of continuous classical straight lines. In the quadratic-Lagrangian case, the quantum state becomes a superposition over extremals with , linking constrained least-action paths to the Feynman framework. The authors discuss potential computational simplifications and the need to establish consistency with the full Feynman path integral for nonlinear cases, as well as implications for classical simulations of some quantum systems.

Abstract

The wave-particle duality and its probabilistic interpretation are at the heart of quantum mechanics. Here we show that, in some standard contexts like the double slit experiment, a deterministic interpretation can be provided. This interpretation arises from the fact that if one mininizes a deterministic action {\it subject to spatial inequality contraints}, in general the solution is not unique. Thus, the probabilistic setting is replaced in part by the non-uniqueness of solutions of certain constrained optimization problems, itself often a result of the non-Lipschitzness of the constrained dynamics. While the approach leaves the results of associated Feynman integrals unchanged, it may considerably simplify their computation as the problems grow more complex, since only specific deterministic paths computed from the constrained minimization need to be included in the integral.
Paper Structure (3 sections, 1 theorem, 21 equations, 2 figures)

This paper contains 3 sections, 1 theorem, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Constrained Classical Least Action and Quantum Mechanics
  3. Summary

Key Result

Theorem 1

The action of a Lagrangian $L = \frac{1}{2}\dot{\bf q}^T {\bf M}({\bf q}, t) \dot{\bf q} - V({\bf q}, t)$ constrained by ${\bf q} \in \mathbb{G}^n \subset \mathbb{R}^n$, where $\mathbb{G}^n$ is defined by the $j=1, ..., J$ inequality constraints is extremized between two points ${\bf q}_o(t_o)$ and ${\bf q}(t)$ by a solution $\bar{\bf q}(t) = {\bf q}(t)$ of where the Lagrange multipliers $\lamb

Figures (2)

  • Figure 1: Multiple Deterministic Least Action solutions of the double slit experiment
  • Figure 2: Multiple deterministic Least Action solutions of the single slit experiment

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Theorem 1