$q$-Heisenberg Algebra in $\otimes^{2}-$Tensor Space
Julio César Jaramillo Quiceno
TL;DR
This work constructs a $q$-deformed Heisenberg algebra on the $\otimes^{2}$-tensor space by introducing generators $\hat{x}$ and $\hat{p}$ linked through $q$-commutation relations and a coordinate-dependent function $f$, recovering the classical case when $f=1$. It derives explicit tensor-space commutators such as $\hat{x} \otimes \hat{p}_{k} - q_{jk} \hat{p}_{k} \otimes \hat{x} = -i\hbar \delta_{jk} f$ and a momentum-commutator involving $Df$ and $\partial f/\partial x_k$. It extends the formalism to non-monogenic Clifford-valued $f$, obtaining generalized relations and showing the monogenic limit $Df=0$ simplifies the tensor momentum relations. The results provide a bridge between $q$-deformed algebras, Clifford analysis, and noncommutative functional analysis, with potential impact on quantum geometry and high-energy physics.
Abstract
In this paper, we introduce the $q$-Heisenberg algebra in the tensor product space $\otimes^2$. We establish its algebraic properties and provide applications to the theory of non-monogenic functions. Our results extend known constructions in $q$-deformed algebras and offer new insights into functional analysis in non-commutative settings.