Quantum geometry and quantization on U(u(2)) background. Noncommutative Dirac monopole

TL;DR

This paper develops a differential calculus on noncommutative algebras related to U(u(2)) and applies it to quantize the Dirac monopole, providing a new approach to noncommutative geometry in gauge theories.

Contribution

It introduces a method to extend quantum partial derivatives on noncommutative algebras and applies this to quantize the Dirac monopole within a noncommutative geometric framework.

Findings

Extended differential calculus on U(u(2)) algebra.

Constructed a de Rham complex for the noncommutative algebra.

Quantized the Dirac monopole using the developed framework.

Abstract

In our previous publications we introduced differential calculus on the enveloping algebras U(gl(m)) similar to the usual calculus on the commutative algebra Sym(gl(m)). The main ingredient of our calculus are quantum partial derivatives which turn into the usual partial derivatives in the classical limit. In the particular case m=2 we prolonged this calculus on a central extension A of the algebra U(gl(2)). In the present paper we consider the problem of a further extension of the quantum partial derivatives on the skew-field of the algebra A and define the corresponding de Rham complex. As an application of the differential calculus we suggest a method of transferring dynamical models defined on Sym(u(2)) to the algebra U(u(2)) (we call this procedure the quantization with noncommutative configuration space). In this sense we quantize the Dirac monopole and find a solution of this model.