Symplectic areas, quantization, and dynamics in electromagnetic fields

Abstract

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.

Cotangent bundle quantization: entangling of metric and magnetic field

“…The integration in (3.12) is taken with respect to the measure dm r(n ′ ) = dm l (n ′′ ) over the manifold M. Formula (3.12) belongs to the class of Connes' type tangential groupoid quantization formulas [27,31,4]. Note that in the convolution integrand (3.12) we have an additional groupoid cocycle…”

“…In the Euclidean case, membrane formulas of a similar type were discovered by M. Berry [72] for the asymptotics of the Wigner function (see also [73] for solutions of the Cauchy problem). The magnetic version of these formulas was first obtained and investigated in detail in [4,5], the case of symmetric spaces was studied in [74], and for general manifolds, see in [75,76].…”

“…In the Euclidean case M = R n the 1-form (3.22) coincides with the Valatin potential [66], and condition (3.21) is the Dirac gauge condition (see details in [4]). …”

“…[37,38,4]. In the Euclidean case, conditions which uniquely identify the Groenewold-Moyal product and Weyl ordering are known [39][40][41]; in the context of formal deformation theory see the discussion in [42][43][44].…”

“…There is also a magnetic analog [4,5] of this quantum product, wherep plays the role of the kinetic momenta and satisfies the commutation relations…”