Strong Mathematical Completion of Quantum Mechanics

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.