Unitary Representations of Quantum Lorentz Group and Quantum Relativistic Toda Chain

TL;DR

This paper links quantum Lorentz groups and quantum Lobachevsky spaces to the relativistic Toda chain, providing a group-theoretical framework for Bessel-Jackson functions and their role as wave functions in a quantum integrable system.

Contribution

It constructs the principal series of unitary irreducible representations of quantum Lorentz groups and relates special matrix elements to Bessel-Macdonald-Jackson functions as Toda lattice wave functions.

Findings

Eigen-functions are modified Bessel-Jackson functions.

Matrix elements are Bessel-Macdonald-Jackson functions.

Integral representations of these functions are derived.

Abstract

The aim of this paper is to give a group theoretical interpretation of the three types of Bessel-Jackson functions. We consider a family of quantum Lorentz groups and a family of quantum Lobachevsky spaces. For three members of quantum Lobachevsky spaces the Casimir operators give rise to the two-body relativistic open Toda lattice Hamiltonians. Their eigen-functions are the modified Bessel-Jackson functions of three types. We construct the principal series of unitary irreducible representations of the quantum Lorentz groups. Special matrix elements in the irreducible spaces are the Bessel-Macdonald-Jackson functions. They are the wave functions of the two-body relativistic open Toda lattice. We obtain integral representations for these functions.