The Quantum Double as Quantum Mechanics

TL;DR

This paper explores the structure of quantum doubles as quantum algebras of observables, providing new examples of quantum systems with Hopf algebraic observable algebras in the context of q-deformed spaces.

Contribution

It introduces $*$-structures on braided groups and shows that quantum doubles can be interpreted as quantum observable algebras for particles on q-deformed spaces, extending the understanding of quantum symmetries.

Findings

Quantum double $D(U_q(su_2))$ models a quantum particle on a hyperboloid in q-Minkowski space.

The dual Hopf algebra also serves as a quantum observable algebra with q-deformed position and momentum spaces.

Results generalize to quantum doubles of other quantum groups.

Abstract

We introduce $*$-structures on braided groups and braided matrices. Using this, we show that the quantum double $D(U_q(su_2))$ can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in q-Minkowski space (a three-sphere in the Lorentz metric), and with the role of angular momentum played by $U_q(su_2)$. This provides a new example of a quantum system whose algebra of observables is a Hopf algebra. Furthermore, its dual Hopf algebra can also be viewed as a quantum algebra of observables, of another quantum system. This time the position space is a q-deformation of $SL(2,\R)$ and the momentum group is $U_q(su_2^*)$ where $su_2^*$ is the Drinfeld dual Lie algebra of $su_2$. Similar results hold for the quantum double and its dual of a general quantum group.