Noncommutative Geometry and Dynamical Models on $U(u(2))$ Background

Abstract

In our previous publications we have introduced a differential calculus on the algebra $U(gl(m))$ based on a new form of the Leibniz rule which differs from that usually employed in Noncommutative Geometry. This differential calculus includes partial derivatives in generators of the algebra $U(gl(m))$ and their differentials. The corresponding differential algebra $Ω(U(gl(m)))$ is a deformation of the commutative algebra $Ω(\operatorname{Sym}(gl(m)))$. A similar claim is valid for the Weyl algebra $W(U(gl(m)))$ generated by the algebra $U(gl(m))$ and the mentioned partial derivatives. In the particular case m=2 we treat the compact form $U(u(2))$ of this algebra as a quantization of the Minkowski space algebra. Below, we consider non-commutative versions of the Klein-Gordon equation and the Schrodinger equation for the hydrogen atom. To this end we de ne an extension of the algebra $U(u(2))$ by adding to it meromorphic functions in the so-called quantum radius and quantum time. For the quantum Klein-Gordon model we get (under an assumption on momenta) an analog of the plane wave, for the quantum hydrogen atom model we find the first order corrections to the ground state energy and the wave function.