$Fun(SO_q(N))$-Isotropic Harmonic Oscillator on the Quantum Euclidean Space $R_q^N$
We briefly describe the construction of a consistent $q$-deformation of the quantum mechanical isotropic harmonic oscillator on ordinary $\rn^N$ space.
q-alg↗
arXiv subjects
Gaetano Fiore
Publications and source records attributed to Gaetano Fiore.
Many Q-Particles from One: A New Approach to $*$-Hopf Algebras?
We propose a nonstandard approach to solving the apparent incompatibility between the coalgebra structure of some inhomogeneous quantum groups and their natural complex conjugation. In this work we sketch the general idea and develop the method in detail on a toy-model; the latter is a q-deformation of the Hopf algebra of 1-dim translations + dilatations. We show how to get all Hilbert space representations of the latter from tensor products of the fundamental ones; physically, this corresponds to constructing composite systems of many free distinct q-particles in terms of the basic one-particle ones. The spectrum of the total momentum turns out to be the same as that of a one-particle momentum, i.e. of the form $\{μq^n\}_{n\in\zn}$.
hep-th↗
Realization of $U_q(so(N))$ within the differntial algebra on ${\bf R}_q^N$
We realize the Hopf algebra $U_{q^{-1}}(so(N))$ as an algebra of differential operators on the quantum Euclidean space ${\bf R}_q^N$. The generators are suitable q-deformed analogs of the angular momentum components on ordinary ${\bf R}^N$. The algebra $Fun({\bf R}_q^N)$ of functions on ${\bf R}_q^N$ splits into a direct sum of irreducible vector representations of $U_{q^{-1}}(so(N))$; the latter are explicitly constructed as highest weight representations.
hep-th↗
The $SO_q(N,{\bf R})$-Symmetric Harmonic Oscillator on the Quantum Euclidean Space ${\bf R}_q^N$ and its Hilbert Space Structure
We show that the isotropic harmonic oscillator in the ordinary euclidean space ${\bf R}^N$ ($N\ge 3$) admits a natural q-deformation into a new quantum mechanical model having a q-deformed symmetry (in the sense of quantum groups), $SO_q(N,{\bf R})$. The q-deformation is the consequence of replacing $ R^N$ by ${\bf R}^N_q$ (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over ${\bf R}^N_q$, which we use for the definition of the scalar product of states.
hep-th↗