Masslessness in $n$-dimensions

TL;DR

The paper extends the group-theoretic analysis of massless representations from four to n dimensions by classifying unitary irreducible representations of the n-dimensional Poincaré group via the orbit method and then determining which extend irreducibly to the conformal group ${\overline{G}}_n=SO_0(2,n)$. Masslessness is characterized by the condition $P_\mu P^\mu=0$, with massless representations induced from degenerate little groups ${\rm Spin}(n-2)\,T_{n-2}$; finite-dimensional and infinite-dimensional massless ${\mathfrak{so}}(N)$-types are parameterized by extremal weights $s_a$, with explicit Casimir values. The authors show that extension to the conformal group exists only for massless inductions and is unique, and they analyze the behavior under De Sitter restrictions ${\mathcal{S}}^\pm_n$ and Wigner–Inönü contraction back to ${\overline{\mathcal{P}}}_n$, revealing that higher dimensions impose stronger constraints on the inducing representation $S$, especially distinguishing even and odd $n$. The results affirm the special status of 4D spacetime and provide a comprehensive framework for massless representations in higher dimensions, including their decomposition, contraction limits, and potential physical implications for higher-dimensional gauge theories.

Abstract

We determine the representations of the ``conformal'' group ${\bar{SO}}_0(2, n)$, the restriction of which on the ``Poincaré'' subgroup ${\bar{SO}}_0(1, n-1).T_n$ are unitary irreducible. We study their restrictions to the ``De Sitter'' subgroups ${\bar{SO}}_0(1, n)$ and ${\bar{SO}}_0(2, n-1)$ (they remain irreducible or decompose into a sum of two) and the contraction of the latter to ``Poincaré''. Then we discuss the notion of masslessness in $n$ dimensions and compare the situation for general $n$ with the well-known case of 4-dimensional space-time, showing the specificity of the latter.