Geometric Origin of Quantization on an Adiabatically Evolving Manifold

In the present work, a transition is made from the axiomatic formulation of quantum
mechanics to the principle of existence of stable self-consistent solutions of the non-linear
system ”charged particles - transverse electromagnetic field - spacetime geometry”. Within
the framework of this approach, quantization arises not as a consequence of a set of initial
postulates, but as a mathematical condition for the existence of a non-trivial periodic solution
for the complete system of field and matter equations (a spectral condition on the monodromy)
on an adiabatically evolving manifold. Only a discrete set of such solutions turns out to be
physically admissible, which leads first to the quantization of motion and then to discrete
energy spectra. It is shown that a transverse electromagnetic field on a non-stationary manifold
possesses a geometric adiabatic invariant having the dimension of action. Its calculated value
coincides with the Planck constant within the observational uncertainties of the cosmological
parameters used. It can be argued that the Planck constant is a measure of the deviation of
the local geometry of the Universe from the stationary Minkowski space. The same geometric
structure leads to the equations of complete electrodynamics, from which, in the appropriate
limits, quantum mechanics, Maxwell’s electrodynamics, and a number of observable physical
effects follow. In the proposed approach, the Planck constant, quantization, the cosmological
redshift, and other phenomena turn out to be different manifestations of a single geometric
mechanism associated with the adiabatic evolution of spacetime.