Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2305.03141

Dynamical Exclusion of Probability and Energy Conservation

Victor Atanasov

Published May 2, 2023• 0 citations
arXivHTMLPDF

Abstract

The interrelationship between energy and probability conservation is explored from the point of view of statistical physics and non-relativistic quantum mechanics. The simultaneous validity of the law of conservation of energy and the continuity equation (probability conservation) breaks for an interacting dynamical system. A separate and independent description of a physical system can be obtained by requiring that the law of conservation of probability is at the heart of the derivation of the ''equations of motion''. In effect, The Schrodinger equation can be viewed as an appropriate factorization of the continuity equation instead of an energy conservation relation per se.

DOI
10.48550/arxiv.2305.03141

Related papers

XCI.The conservation of probability and energy as criteria for the validity of wave equations

A. Lees

Summary There are two conditions which a wave equation should satisfy, one corresponding to the conservation of probability, the other expressing a relation which, in the case in which the Hamiltonian does not involve the time explicitly, corresponds to the conservation of energy. It is known that the first condition is satisfied by Schrödinger's non-relativistic equation and by Dirac's equation, but not by the second-order relativistic equation. We prove that the second condition also is satisfied in the same two cases. These facts seem to provide a stronger reason for rejecting the second-order relativistic equation than those usually given.

The London Edinburgh and Dublin Philosophical Magazine and Journal of Science•Jun 1, 1933•0 citations

Interference in Quantum Mechanics, A Violation of the Law of Conservation of Energy

Thomas Alexander Meyer

All measurable phenomena in physics are irreducibly particle like and the wave picture of physical systems only arises when explaining a statistically relevant set of measurements of individual systems as an average. The wave picture arises in the effects of interference which can always be explained in the context of the state description of the interfering system, interference necessitates that the measurement outcomes be calculated by squaring the amplitudes after adding and non-interference is calculated by squaring before adding. Even though the calculation for each is independently valid, the change in the statistics when one modulates between interference and noninterference, as in the quantum eraser, requires a violation of the law of conservation of energy. There is no “driving force ” in the change of statistics. It is always a change in preparation that leads to the change of interference statistics, whilst in the calculation there is only a change in the state description. The irreducibly particle like nature of the quanta requires that there be a causal “driving force ” in the change in statistics and a mere change in description is an insufficient explanation by the standards of energy conservation. We present the argument that indeed this has already all been proven in the historic 1991 study of the ZWM papers of quantum coherence. Wave Particle Duality by Statistical Correspondence The first conceptual argument to arise in the history of quantum mechanics is that which questions how the

•May 1, 2013•0 citations

Interpreting Quantum Mechanics in Terms of Random Discontinuous Motion of Particles

Shan Gao

This thesis is an attempt to reconstruct the conceptual foundations of quantum mechanics. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schrödinger equation due to the requirements of spacetime translation invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A discrete model of energy-conserved wavefunction collapse is proposed and shown to be consistent with existing experiments and our macroscopic experience. In addition, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and dynamical collapse theories, and briefly analyze the problem of the incompatibility between quantum mechanics and special relativity.

•Nov 3, 2011•29 citations

Quantum mechanics as a statistical theory

J. E. Moyal

An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics . In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics , deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.

Mathematical Proceedings of the Cambridge Philosophical Society•Jan 1, 1949•3,098 citations

The quantum laws of physics: a new description of dynamics and causality

Holger Friedrich Hofmann

It is possible to completely explain all aspects of quantum mechanics by expressing the relations between physical properties in terms of complex conditional probabilities (Phys. Rev. A 89, 042115(2014)). These fully deterministic probabilities replace the classical notions of phase space geometries and continuous trajectories with a more accurate description that takes into account the role of dynamics in the emergence of reality. We can then understand why so many previous attempts to find a detailed explanation of quantum phenomena have failed: the assumption of a static reality breaks down when the interaction needed to obtain a real effect exceeds Planck`s constant. Beyond that limit, complex conditional probabilities are the only valid expression of the fundamental laws of physics.

arXiv:1405.0053•Apr 30, 2014•0 citations

Wave Mechanics in Classical Phase Space, Brownian Motion, and Quantum Theory

Giacomo Della Riccia, Norbert Wiener

A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.

Journal of Mathematical Physics•Aug 1, 1966•40 citations
The interrelationship between energy and probability conservation is explored from the point of view of statistical physics and non-relativistic quantum mechanics. The simultaneous validity of the law of conservation of energy and the continuity equation (probability conservation) breaks for an interacting dynamical system. A separate and independent description of a physical system can be obtained by requiring that the law of conservation of probability is at the heart of the derivation of the ''equations of motion''. In effect, The Schrodinger equation can be viewed as an appropriate factorization of the continuity equation instead of an energy conservation relation per se.
Summary There are two conditions which a wave equation should satisfy, one corresponding to the conservation of probability, the other expressing a relation which, in the case in which the Hamiltonian does not involve the time explicitly, corresponds to the conservation of energy. It is known that the first condition is satisfied by Schrödinger's non-relativistic equation and by Dirac's equation, but not by the second-order relativistic equation. We prove that the second condition also is satisfied in the same two cases. These facts seem to provide a stronger reason for rejecting the second-order relativistic equation than those usually given.
All measurable phenomena in physics are irreducibly particle like and the wave picture of physical systems only arises when explaining a statistically relevant set of measurements of individual systems as an average. The wave picture arises in the effects of interference which can always be explained in the context of the state description of the interfering system, interference necessitates that the measurement outcomes be calculated by squaring the amplitudes after adding and non-interference is calculated by squaring before adding. Even though the calculation for each is independently valid, the change in the statistics when one modulates between interference and noninterference, as in the quantum eraser, requires a violation of the law of conservation of energy. There is no “driving force ” in the change of statistics. It is always a change in preparation that leads to the change of interference statistics, whilst in the calculation there is only a change in the state description. The irreducibly particle like nature of the quanta requires that there be a causal “driving force ” in the change in statistics and a mere change in description is an insufficient explanation by the standards of energy conservation. We present the argument that indeed this has already all been proven in the historic 1991 study of the ZWM papers of quantum coherence. Wave Particle Duality by Statistical Correspondence The first conceptual argument to arise in the history of quantum mechanics is that which questions how the
This thesis is an attempt to reconstruct the conceptual foundations of quantum mechanics. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schrödinger equation due to the requirements of spacetime translation invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A discrete model of energy-conserved wavefunction collapse is proposed and shown to be consistent with existing experiments and our macroscopic experience. In addition, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and dynamical collapse theories, and briefly analyze the problem of the incompatibility between quantum mechanics and special relativity.
An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics . In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics , deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.
It is possible to completely explain all aspects of quantum mechanics by expressing the relations between physical properties in terms of complex conditional probabilities (Phys. Rev. A 89, 042115(2014)). These fully deterministic probabilities replace the classical notions of phase space geometries and continuous trajectories with a more accurate description that takes into account the role of dynamics in the emergence of reality. We can then understand why so many previous attempts to find a detailed explanation of quantum phenomena have failed: the assumption of a static reality breaks down when the interaction needed to obtain a real effect exceeds Planck`s constant. Beyond that limit, complex conditional probabilities are the only valid expression of the fundamental laws of physics.
A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.