Geometric Origin of Quantization

For almost a century, quantum mechanics has remained one of the most successful physical theories. Nevertheless, its mathematical framework is built upon a system of postulates: the existence of the wave function, Planck’s constant, quantization rules, and other fundamental assumptions are accepted as primary facts.

The work presented below attempts to view this problem from a different perspective. Instead of taking the axiomatic foundations of quantum mechanics as the starting point, it proposes to replace them with the principle of existence of self-consistent solutions of a complete nonlinear dynamical system.

Instead of asking

“Which axioms should be chosen as the foundation of quantum mechanics?”

we ask a different question:

“Can quantization arise not from a system of postulates, but as a natural consequence of spacetime geometry and the laws of electrodynamics?”

The results obtained in this work provide an affirmative answer to this question.

The proposed approach is based on the complete self-consistent system

charged particles + transverse electromagnetic field + adiabatically evolving spacetime geometry.

Within this framework, the discreteness of physical states does not originate from an independent postulate. Instead, it follows from the fact that only a discrete set of periodic self-consistent solutions of the complete nonlinear system is mathematically admissible.

The logical structure of the proposed approach may be summarized as follows:

Non-stationary spacetime geometry 
(adiabatic evolution of the Universe)
                    ↓
Geometric adiabatic invariant of the transverse 
electromagnetic field 
(numerically equal to Planck's constant ) 
                    ↓
Self-consistency of the coupled system
"particles + field + geometry"
                    ↓
Phase resonance
                    ↓
Quantization of motion
                    ↓
Discrete energy spectrum

Within this framework several results are obtained.

• It is shown that the transverse electromagnetic field on an adiabatically evolving manifold possesses a geometric adiabatic invariant having the dimension of action.
• Its numerical value, calculated entirely from the observed cosmological parameters, agrees with the experimentally measured Planck constant within the observational uncertainties of those parameters.
• The equations of Complete Electrodynamics are derived, naturally incorporating the non-stationarity of spacetime geometry.
• It is demonstrated that quantization emerges as a spectral condition for the existence of periodic self-consistent solutions of the complete nonlinear system of equations and therefore does not require an independent axiomatic introduction.

The proposed work is not intended as a modification of conventional quantum mechanics. Rather, it explores a deeper level of physical description in which the familiar quantum rules appear as consequences of a single geometric mechanism.

The resulting equations of Complete Electrodynamics provide a unified geometric framework from which, in the appropriate limits, both relativistic electrodynamics and quantum mechanics naturally emerge.

The full preprint is available on Zenodo:

DOI: 10.5281/zenodo.21255165

Preprint: https://zenodo.org/records/21349419