Many Q-Particles from One: A New Approach to $*$-Hopf Algebras?

TL;DR

This paper introduces a novel method for constructing Hilbert space representations of inhomogeneous quantum groups, enabling the modeling of many q-particles from a single fundamental q-particle, with implications for quantum symmetries.

Contribution

It presents a new approach to reconcile coalgebra structures with complex conjugation in quantum groups, demonstrated through a q-deformation toy model.

Findings

All Hilbert space representations derived from tensor products of fundamental ones.

Spectrum of total momentum matches that of a single particle, of the form {μ q^n}.

Method allows modeling of many free q-particles from one basic q-particle.

Abstract

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 $\{\mu q^n\}_{n\in\zn}$.