Source: Journal of Generalized Lie Theory and Applications, Volume 9, Number 1, 10 pages.
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.