Symplectic areas, quantization, and dynamics in electromagnetic fields

A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: a choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the quantum evolution problem for a charged particle in an electromagnetic field is represented in an exact continual form and in the semiclassical approximation via the area of dynamical membranes.Comment: 39 pages, 17 figure

The genus and the category of configuration spaces

In this paper configuration spaces of smooth manifolds are considered. The accent is made on actions of certain groups (mostly $p$-tori) on this spaces by permuting their points. For such spaces the cohomological index, the genus in the sense of Krasnosel'skii-Schwarz, and the equivariant Lyusternik-Schnirelmann category are estimated from below, and some corollaries for functions on configuration spaces are deduced