Riemann surface and quantization

Abstract

This paper proposes an approach of the unified consideration of classical and quantum mechanics from the standpoint of the complex analysis effects. It turns out that quantization can be interpreted in terms of the Riemann surface corresponding to the multivalent Ln  function. A visual interpretation of «trajectories» of the quantum system and of the Feynman's path integral is presented. A magnetic dipole having a magnetic charge that satisfies the Dirac quantization rule was obtained.

Ψ-model of micro- and macrosystems

“…coincides with the known expression for the magnetic charge and satisfies the quantization condition [28,32].…”

“…Thus, path  may be shrunk as much as desired around the OZ axis, at that, integral value (1.1.10) will not change. 7 One can see from (1.1.9) that integral over d is different from zero, and integral over d gives zero, therefore the calculation of rot v is analogue to the procedure described in our paper [32]. As a result, we obtain…”

“…The system obtained (1. . If 0 Q  , for example, at 0 n  , then the original problem will be reduced to the problem considered in article [32].…”

“…As the rotation about the OZ axis determines the preferred direction in space, it seems natural to consider not the spherically symmetric density distribution 6), which leads also to the principle of Bohr-Sommerfeld quantization at considering multivalent Riemannian surface [32]. At that there has been no need to modify g -factor, as it is done in the Dirac and Pauli equations.…”

“…where A corresponds to the solenoidal component of the velocity field. Paper [32] showed that field  can be solenoidal as well for non-smooth scalar potential  , for example 12 Value Q corresponds to the density of the source of field v from the physical standpoint, and as Q is different from zero in general case (1.1.7), there are such sources.…”