In the present work, a transition is made from the axiomatic formulation of quantummechanics to the principle of existence of stable self-consistent solutions of the non-linearsystem ”charged particles - transverse electromagnetic field - spacetime geometry”. Withinthe framework of this approach, quantization arises not as a consequence of a set of initialpostulates, but as a mathematical condition for the existence of a non-trivial periodic solutionfor 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 bephysically admissible, which leads first to the quantization of motion and then to discreteenergy spectra. It is shown that a transverse electromagnetic field on a non-stationary manifoldpossesses a geometric adiabatic invariant having the dimension of action. Its calculated valuecoincides with the Planck constant within the observational uncertainties of the cosmologicalparameters used. It can be argued that the Planck constant is a measure of the deviation ofthe local geometry of the Universe from the stationary Minkowski space. The same geometricstructure leads to the equations of complete electrodynamics, from which, in the appropriatelimits, quantum mechanics, Maxwell’s electrodynamics, and a number of observable physicaleffects follow. In the proposed approach, the Planck constant, quantization, the cosmologicalredshift, and other phenomena turn out to be different manifestations of a single geometricmechanism associated with the adiabatic evolution of spacetime.