Continuous spin representations from group contraction

Abstract

We consider how the continuous spin representation (CSR) of the Poincaré 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 Poincaré 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→∞, 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 Inönü–Wigner contraction, with the inverse Kaluza–Klein 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 Kaluza–Klein radius fixed.