Continuous Spin Representations from Group Contraction

TL;DR

This paper explores how continuous spin representations of the Poincare group in four dimensions can be derived through dimensional reduction and group contraction techniques from five-dimensional theories.

Contribution

It demonstrates a method to generate four-dimensional CSR from five-dimensional models using specific singular limits and group contraction procedures.

Findings

CSR can be obtained via Inonu-Wigner contraction of 5D models.

Two different limits produce CSR from single spin and Majorana theories.

The approach links higher-dimensional theories to four-dimensional continuous spin representations.

Abstract

We consider how the continuous spin representation (CSR) of the Poincare group in four dimensions can be generated by dimensional reduction. The analysis uses the front-form little group in five dimensions, which must yield the Euclidean group E(2), the little group of the CSR. We consider two cases, one is the single spin massless representation of the Poincare group in five dimensions, the other is the infinite component Majorana equation, which describes an infinite tower of massive states in five dimensions. In the first case, the double singular limit j,R go to infinity, with j/R fixed, where R is the Kaluza-Klein radius of the fifth dimension, and j is the spin of the particle in five dimensions, yields the CSR in four dimensions. It amounts to the Inonu-Wigner contraction, with the inverse K-K radius as contraction parameter. In the second case, the CSR appears only by taking a triple singular limit, where an internal coordinate of the Majorana theory goes to infinity, while leaving its ratio to the KK radius fixed.