Symmetry

The Heisenberg Uncertainty Principle as an Endogenous Equilibrium Property of Stochastic Optimal Control Systems in Quantum Mechanics

Jussi Lindgren, Jukka Liukkonen

Published Sep 17, 2020• 17 citations

Abstract

We provide a natural derivation and interpretation for the uncertainty principle in quantum mechanics from the stochastic optimal control approach. We show that, in particular, the stochastic approach to quantum mechanics allows one to understand the uncertainty principle through the “thermodynamic equilibrium”. A stochastic process with a gradient structure is key in terms of understanding the uncertainty principle and such a framework comes naturally from the stochastic optimal control approach to quantum mechanics. The symmetry of the system is manifested in certain non-vanishing and invariant covariances between the four-position and the four-momentum. In terms of interpretation, the results allow one to understand the uncertainty principle through the lens of scientific realism, in accordance with empirical evidence, contesting the original interpretation given by Heisenberg.

DOI: 10.3390/sym12091533

Related papers

Entropy Constrained Knowledge Principle

Sanctus, J, Varda, Shubhang

We often marvel at the beautiful randomness in quantum mechanics, but what if it is simply entropy fulfilling its role as dictated by causality? This paper explores the idea that Heisenberg’s uncertainty principle may not represent an intrinsic limitation, but rather a thermodynamic outcome, a cost we pay for achieving precision. We examine Gaussian wavefunctions, investigate entropic uncertainty, and demonstrate that increased knowledge of one variable (such as position) leads to a corresponding increase in entropy in its complementary variable (such as momentum). Nature does not conceal the truth; rather, the process of acquiring knowledge alters the equilibrium. By associating entropy with quantum uncertainty, we are not simply reinterpreting mathematics; We are redefining the implications of quantum indeterminacy. Perhaps the universe is not inherently uncertain; it is simply economical with certainty. We propose that the Heisenberg Uncertainty Principle can be interpreted through the concept of entropy, framing it as an emergent characteristic of the system’s overall information content rather than merely a constraint imposed by quantum mechanics. Investigating this relationship may yield profound insights into the essence of quantum systems and their relationship with the macroscopic realm governed by thermodynamics. This paper offers both a philosophical and mathematical framework that unifies the ideas of entropy and uncertainty, providing a novel viewpoint on one of the most essential principles of quantum mechanics. By analyzing the influence of entropy on quantum uncertainty, this study contests traditional interpretations and encourages a renewed dialogue regarding the foundational principles of physics. It posits that uncertainty transcends mere measurement concerns and may be embedded in the very structure of the universe, prompting a reevaluation of how quantum mechanics integrates into the wider context of physical laws.

May 27, 20250 citations

Quantum Mechanics can be understood through stochastic optimization on spacetimes

Jussi Lindgren, Jukka Liukkonen

The main contribution of this paper is to explain where the imaginary structure comes from in quantum mechanics. It is shown how the demand of relativistic invariance is key and how the geometric structure of the spacetime together with the demand of linearity are fundamental in understanding the foundations of quantum mechanics. We derive the Stueckelberg covariant wave equation from first principles via a stochastic control scheme. From the Stueckelberg wave equation a Telegrapher's equation is deduced, from which the classical relativistic and nonrelativistic equations of quantum mechanics can be derived in a straightforward manner. We therefore provide meaningful insight into quantum mechanics by deriving the concepts from a coordinate invariant stochastic optimization problem, instead of just stating postulates.

Scientific ReportsDec 27, 201924 citations

Schrödinger equation for various quantum systems based on Heisenberg's uncertainty principle

Kexin An

This article establishes the proof of the Schrödinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrödinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrödinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.

Theoretical and Natural ScienceSep 27, 20240 citations

Derivation of Dirac equation from the stochastic optimal control principles of quantum mechanics

Vasil G. Yordanov

In this paper, we present a stochastic approach to relativistic quantum mechanics. We formulate the three fundamental principles of this theory and derive the Dirac equations based on them. This approach enables us to bring more insight into the nature of Dirac's spinors. Furthermore, we provide a physical interpretation of the stochastic optimal control theory of quantum mechanics.

Scientific ReportsMar 18, 20243 citations

Anti Heisenberg—Refutation Of Heisenberg’s Uncertainty Relation

Ilija Barukčić

The quantum mechanical uncertainty principle for position and momentum plays an important role in many treatments on the (philosophical, physical and other) implications of quantum mechanics. Roughly speaking, the more precisely the momentum (position) of a (quantum mechanical) object is given, the less precisely can one say what its position (momentum) is. This quantum mechanical measurement problem is not just an interpretational difficulty, it raises broader issues as well. The measurement (of a property) of a (quantum mechanical) object determines the existence of the measured. In brief, the quantum mechanical uncertainty principle challenges some fundamental principles of Science and especially the principle of causality. In particular, an independently existing (external) objective reality is denied. As we shall see, that the quantum mechanical uncertainty principle for position and momentum is based on the assumption that 1⩽0, which is a logical contradiction.

AIP conference proceedingsJan 1, 201115 citations

The Uncertainty Principle as an Ontological Boundary of Determinism Articulations with Chaos Theory and Experimental Confirmation

Juan Carlos Ruíz Castillo

The Heisenberg uncertainty principle constitutes one of the foundational pillars of quantum mechanics and represents a definitive rupture with the deterministic ideal of classical physics. This work develops a comprehensive analysis of the uncertainty principle from a mathematical, physical, and philosophical perspective, demonstrating that it should not be interpreted as an instrumental or epistemological limitation, but rather as an ontological property inherent to the formalstructure of quantum theory.Within the framework of Hilbert spaces, self-adjoint operators, and non-commutative algebras, it is shown that uncertainty emerges necessarily as a consequence of the structural incompatibility between conjugate observables. The study articulates the uncertainty principle with classical chaos theory and the sciences of complexity, establishing a fundamental distinction between epistemological unpredictability and ontological indeterminacy. While classical chaos preserves well-defined trajectories that are practically inaccessible, quantum mechanics eliminates the very notion of trajectory as a fundamental entity.This rupture manifests itself in the granularity of phase space, the impossibility of defining exact initial conditions, and the consequent reformulation of determinism. Contemporary experimental developments designed to exclude instrumental disturbances are analyzed, reinforcing the ontological interpretation of the uncertainty principle.Finally, the implications for simulation, predictability, and quantumOverall, this work argues that the uncertainty principle does not undermine scientific rationality, but rather redefines itsscope and deepens our understanding of physical reality. Keywords: uncertainty principle; quantum mechanics; mathematical structure; non-commutative operators; chaos theory; complexity; determinism; physical ontology.

Dec 17, 20250 citations