The irreducible unitary representations of the extended Poincare group in (1+1) dimensions

TL;DR

This paper classifies all irreducible unitary representations of the extended Poincare group in (1+1) dimensions, revealing their structure and physical relevance, especially for anomaly-free relativistic particles.

Contribution

It provides a complete classification of irreducible unitary representations of the extended Poincare group in (1+1) dimensions using the orbit method, including physical interpretations.

Findings

The extended Poincare group in (1+1) dimensions is non-nilpotent solvable exponential and of type I.

All irreducible unitary representations are constructed via the orbit method.

A covariant maximal polynomial quantization is established for the physically relevant representations.

Abstract

We prove that the extended Poincare group in (1+1) dimensions is non-nilpotent solvable exponential, and therefore that it belongs to type I. We determine its first and second cohomology groups in order to work out a classification of the two-dimensional relativistic elementary systems. Moreover, all irreducible unitary representations of the extended Poincare group are constructed by the orbit method. The most physically interesting class of irreducible representations corresponds to the anomaly-free relativistic particle in (1+1) dimensions, which cannot be fully quantized. However, we show that the corresponding coadjoint orbit of the extended Poincare group determines a covariant maximal polynomial quantization by unbounded operators, which is enough to ensure that the associated quantum dynamical problem can be consistently solved, thus providing a physical interpretation for this particular class of representations.