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Force as a Quantum Observable?

Raheem Adom

Published Dec 29, 2025• 0 citations
Publisher

Abstract

This work prominently proposes an internally consistent fundamental observable for force on par with energy, momentum, position and angular momentum. It demonstrate self-adjointness on natural domains. Importantly the new observable is presented in a new quantum wave equation. It is open access and published in the International Journal of Quantum Foundations.

DOI
10.5281/zenodo.18087843

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In a recent study (Adom R., Int. J. Quantum Found., 12 (2025) 379), I developed a mathematical formulation of force as an essential quantum observable, connecting its quantum wave equation to the violation of translational symmetries that govern the behavior of open quantum systems in a particularly coherent manner. In this letter, I demonstrate that, following the same methodological principles that underpin the well-established quantum wave equations, the proposed force wave equation constitutes a theoretically sound framework that emerges naturally when energy-momentum exchange, instigated by force, is viewed as the fundamental organizing principle for non-relativistic quantum dynamics. This validates the conceptual and mathematical foundations of the framework.

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A conceptually rigorous validation of the discovery of force as a fundamental quantum observable

Raheem Adom

•Jan 1, 2026•0 citations
This work prominently proposes an internally consistent fundamental observable for force on par with energy, momentum, position and angular momentum. It demonstrate self-adjointness on natural domains. Importantly the new observable is presented in a new quantum wave equation. It is open access and published in the International Journal of Quantum Foundations.
In a recent study (Adom R., Int. J. Quantum Found., 12 (2025) 379), I developed a mathematical formulation of force as an essential quantum observable, connecting its quantum wave equation to the violation of translational symmetries that govern the behavior of open quantum systems in a particularly coherent manner. In this letter, I demonstrate that, following the same methodological principles that underpin the well-established quantum wave equations, the proposed force wave equation constitutes a theoretically sound framework that emerges naturally when energy-momentum exchange, instigated by force, is viewed as the fundamental organizing principle for non-relativistic quantum dynamics. This validates the conceptual and mathematical foundations of the framework.
Abstract In a recent study ( Adom R. , Int. J. Quantum Found. , 12 (2025) 379), I developed a mathematical formulation of force as an essential quantum observable, connecting its quantum wave equation to the violation of translational symmetries that govern the behavior of open quantum systems in a particularly coherent manner. In this letter, I demonstrate that, following the same methodological principles that underpin the well-established quantum wave equations, the proposed force wave equation constitutes a theoretically sound framework that emerges naturally when energy-momentum exchange, instigated by force, is viewed as the fundamental organizing principle for non-relativistic quantum dynamics. This validates the conceptual and mathematical foundations of the framework.
A quantum mechanics representation based on position (r⃗\vec{r}r), linear momentum(p⃗\vec{p}p​) and energy(EEE) eigenvalues is presented here. A set of equations, explicitly independent on wave function, was derived relating these observables. In this view, a particle has a known trajectory and at any point on space there is a linear momentum associated. Trajectory here, can be viewed as if a measurement were taken continuously. This picture does not change current quantum mechanics interpretation, rather it comes as a new route of calculation. Also, wave function can be retrieved performing an integration in all space for linear momentum. Equations derived in this work, present a potential dependent on linear momentum originating what we call quantum force. A particle experiences a total force whose resultant is composed by classic and quantum force, evolving on time according to it. In terms of evaluation, a single particle could be described by a set of auxiliary particles named as wave particles(WP). They will start with a range of initial conditions and will collectively describe probabilistic aspects of quantum mechanics. We expect this route could be applied to many problems, such as multi-body systems.
A quantum mechanics representation based on position (r⃗\vec{r}r), linear momentum(p⃗\vec{p}p​) and energy(EEE) eigenvalues is presented here. A set of equations, explicitly independent on wave function, was derived relating these observables. In this view, a particle has a known trajectory and at any point on space there is a linear momentum associated. Trajectory here, can be viewed as if a measurement were taken continuously. This picture does not change current quantum mechanics interpretation, rather it comes as a new route of calculation. Also, wave function can be retrieved performing an integration in all space for linear momentum. Equations derived in this work, present a potential dependent on linear momentum originating what we call quantum force. A particle experiences a total force whose resultant is composed by classic and quantum force, evolving on time according to it. In terms of evaluation, a single particle could be described by a set of auxiliary particles named as wave particles(WP). They will start with a range of initial conditions and will collectively describe probabilistic aspects of quantum mechanics. We expect this route could be applied to many problems, such as multi-body systems.
Recent developments are reviewed and some new results are presented in the study of time in quantum mechanics and quantum electrodynamics as an observable, canonically conjugate to energy. This paper deals with the maximal Hermitian (but nonself‐adjoint) operator for time which appears in nonrelativistic quantum mechanics and in quantum electrodynamics for systems with continuous energy spectra and also, briefly, with the four‐momentum and four‐position operators, for relativistic spin‐zero particles. Two measures of averaging over time and connection between them are analyzed. The results of the study of time as a quantum observable in the cases of the discrete energy spectra are also presented, and in this case the quasi‐self‐adjoint time operator appears. Then, the general foundations of time analysis of quantum processes (collisions and decays) are developed on the base of time operator with the proper measures of averaging over time. Finally, some applications of time analysis of quantum processes (concretely, tunneling phenomena and nuclear processes) are reviewed.