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Annals of Mathematics and Physics

Strong Mathematical Completion of Quantum Mechanics

Zoran Majkić

Published Jun 5, 2026• 0 citations
Publisher

Abstract

SQM (Statistical QM) is a rigorously no-hidden-variables theory, based on the Schrödinger equation for a non-relativistic case where the eigenvalues of the Hamiltonian operator are possible outcomes of total energies, and wavefunctions are corresponding eigenvectors representing probability density (epistemic functions) that the outcome of measurement (in quantum ensemble) will have a particular eigenvalue. So, Schrödinger’s wavefunction can not represent a real individual particle and hence shows that SQM is an incomplete theory as noted by Einstein and de Broglie. Thus, it was necessary to provide the QM theory with missed individual particles theory IQM. The individual massive non-point-like particles in IQM are represented by the 3D rest-mass energy-density topology (like the stars), generating an autogravitational force, and with hidden variables represented by the energy-density flow inside the particle’s finite volume. So, a particle in QM is represented by an ontological complex wavefunction, based on its energy density and the principle of least action.In this paper, based on Noether’s continuity equation of energy-density flow, we take into consideration the ontological version of the Schrödinger equation, where the Hamiltonian operator is substituted by a fundamental classical ontological operator derived from a particle’s hidden variables. Then, we demonstrate how, by quantization of this ontological PDE of a massive particle, we derive the statistical Schrödinger equation with a physical explanation why, in this process, the hidden variables disappear. Finally, we define the classical ensemble of repeated measurements of a real particle with identically prepared measurement systems and derive the Born rule for it.In this way, not only are IQM and SQM two complementary parts of QM, giving the same results as demonstrated previously, but we show that the statistical QM can be derived from the more sophisticated IQM classical theory of individual particles. That is, we obtained the strong completion of Quantum Mechanics.

DOI
10.17352/amp.000190

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SQM (Statistical QM) is a rigorously no-hidden-variables theory, based on the Schrödinger equation for a non-relativistic case where the eigenvalues of the Hamiltonian operator are possible outcomes of total energies, and wavefunctions are corresponding eigenvectors representing probability density (epistemic functions) that the outcome of measurement (in quantum ensemble) will have a particular eigenvalue. So, Schrödinger’s wavefunction can not represent a real individual particle and hence shows that SQM is an incomplete theory as noted by Einstein and de Broglie. Thus, it was necessary to provide the QM theory with missed individual particles theory IQM. The individual massive non-point-like particles in IQM are represented by the 3D rest-mass energy-density topology (like the stars), generating an autogravitational force, and with hidden variables represented by the energy-density flow inside the particle’s finite volume. So, a particle in QM is represented by an ontological complex wavefunction, based on its energy density and the principle of least action.In this paper, based on Noether’s continuity equation of energy-density flow, we take into consideration the ontological version of the Schrödinger equation, where the Hamiltonian operator is substituted by a fundamental classical ontological operator derived from a particle’s hidden variables. Then, we demonstrate how, by quantization of this ontological PDE of a massive particle, we derive the statistical Schrödinger equation with a physical explanation why, in this process, the hidden variables disappear. Finally, we define the classical ensemble of repeated measurements of a real particle with identically prepared measurement systems and derive the Born rule for it.In this way, not only are IQM and SQM two complementary parts of QM, giving the same results as demonstrated previously, but we show that the statistical QM can be derived from the more sophisticated IQM classical theory of individual particles. That is, we obtained the strong completion of Quantum Mechanics.
I present a reconstruction of general Hamiltonian action mechanics that eliminates all foundational problems of quantum mechanics. The key advance is the completion of Hamiltonian mechanics to the universal mechanics of particles based on action-waves, consistent with the inclusive validity of the principle of stationary action. It is found that irreducible indeterminism is intrinsic and universal at all scales of dynamics. The new action-wave equation is the complete description of single dynamical histories, dissolving the classical-quantum divide. The statistical theory of quantum mechanics emerges as the ensemble average of modified action dynamics. The ensemble average of the new action mechanics leads to a hybrid function consisting of the action-waves and the probability density of the ensemble. This hybrid wavefunction obeys the Schrödinger equation, which is not a single particle dynamical equation. The reconstructed mechanics without matter waves is free of the cardinal problem known as the collapse of the wavefunction and with that, the vexing issue of quantum measurement is resolved. Another significant advance is the correct decoding of quantum entanglement and purging of nonseparability and nonlocality in quantum correlations. The action-waves do not carry the burden of divergent zero-point energy. The reconstructed mechanics is in complete agreement with all empirical requirements and in harmony with credible physical ontology.
This year we celebrate 100 years of quantum mechanics (QM). Incorrect interpretations of QM and incorrect mental models of the invisible details of quantum phenomena lead to paradoxes. To explain these, we advocate the statistical contextual interpretation (SCI) of quantum mechanics. State vectors (wave functions) and various operators are purely mathematical entities that permit quantitative probabilistic predictions. “State vector” describes an ensemble of identically prepared physical systems, and a specific “operator” represents a class of equivalent measurements of a physical observable. A collapse of wavefunction is not a mysterious and instantaneous physical process; a collapsed quantum state describes a new ensemble of physical systems prepared in a particular way. A value of a physical observable, such as a spin projection, associated with a pure quantum ensemble is a characteristic of this ensemble created by its interaction with measuring instruments. Probabilities are objective properties of random experiments in which empirical frequencies stabilize. Following Einstein, SCI rejects the claim that QM provides a complete description of individual physical systems, but it remains agnostic about whether a more detailed subquantum description can be found or is necessary. In conformity with Bohr contextuality, SCI rejects Bell-local and Bell-causal hidden variable models. Nevertheless, by incorporating into a probabilistic model contextual hidden variable measuring instruments, long distance quantum correlations studied in Bell tests can be explained without evoking quantum nonlocality or retro-causality. SCI allows the explanation of several quantum phenomena without evoking quantum magic. SCI does not claim to provide a complete description of quantum phenomena; in fact, it is unknown whether quantum probabilities even provide a complete description of existing experimental data. Time series of experimental data may contain much more information than is obtained using empirical frequencies and histograms. Therefore, predictable completeness of QM must be tested and not taken for granted.
The Schrödinger equation has been the central mathematical instrument of quantum mechanics for a century. This paper argues that it is not a quantum mechanical equation at all. It is classical energy conservation applied inside a sphere, with a classical wave form assumed rather than derived. The wavefunction ψ is not a physical object but a lossy geometric encoding. The Born rule—the requirement to square ψ to obtain probabilities—is not a law of nature but a decompression artifact produced by discarding the imaginary phase. The Heisenberg uncertainty principle is a measurement limit, not an ontological statement about particles, and its use to derive the Bohr radius is circular. All distances obtained from this framework are ratios pretending to be absolutes in a universe where only ratios persist. Relativistic Orthogonal Selection Theory (ROST) dissolves each of these confusions by returning to the geometric substrate the equation was always implicitly describing.
In this paper we shall develop a hidden-beable theory which not only reproduce all the predictions of quantum mechanics but also gives an objective-ontological derivations to all postulates that give the foundation of the quantum mechanics. This is done by assuming that the state of a single system can be described completely by two complementary irreducible primitive realities: First is system's position and orientation which serve as local and explicit beables, and second is a new, hidden and global beable named "system's imaginability". System's imaginability is defined as degree of imagination attached to the system that it "could have" other states than it actually has. It is thus of ontological nature which refers to a single system, replacing the conventional epistemological concept of probability which tells the relative frequency of occurrence of various events in an ensemble of systems. We then derive a dynamical equation from which neglecting subquantum fluctuations of the order of \\hbar^2 will lead to the Schrodinger equation in configuration space. An objective measurement theory without observer is then developed, by introducing a new notion of "a controllable apparatus". The superposition principle, Dirac-von Neumann projection postulate and Born's probability rule will be given an objective derivation. Finally we shall prove that the new dynamics fits Einstein's locality postulate in a very simple manner.
We show that quantum theory (QT) is a substructure of classical probabilistic physics. The central quantity of the classical theory is Hamilton’s function, which determines canonical equations, a corresponding flow, and a Liouville equation for a probability density. We extend this theory in two respects: (1) The same structure is defined for arbitrary observables. Thus, we have all of the above entities generated not only by Hamilton’s function but also by every observable. (2) We introduce for each observable a phase space function representing the classical action. This is a redundant quantity in a classical context but indispensable for the transition to QT. The basic equations of the resulting theory take a “quantum-like” form, which allows for a simple derivation of QT by means of a projection to configuration space reported previously [Quantum Stud Math Found 5:219–227, 2018]. We obtain the most important relations of QT, namely the form of operators, Schrödinger’s equation, eigenvalue equations, commutation relations, expectation values, and Born’s rule. Implications for the interpretation of QT are discussed, as well as an alternative projection method allowing for a derivation of spin.
The foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as ccc-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, the wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as the Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity that persists throughout an experiment and therefore does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions, like a Bose-Einstein condensate system, for which a single measurement yields a real ensemble. We comment on the relation between the reality of the wave function and the derivation of non-relativistic quantum mechanics from relativistic quantum field theory.