The unitary representations of the Poincare group in any spacetime dimension

TL;DR

The work provides a comprehensive, dimension-agnostic group-theoretical classification of unitary irreducible representations of the Poincaré group in $D>2$ via induced representations from little groups, tying each UIR to a momentum-orbit and its stabilizer. It develops covariant field equations through the Bargmann–Wigner and Siegel–Zwiebach formalisms, covering massive, massless (helicity and infinite-spin), tachyonic, and zero-momentum sectors, and uses Young diagrams and IW contractions to organize tensor representations across $GL(D)$ and $O(D)$. Key results include explicit little-group structures for each orbit, systematic DOF counting, and the link between representation theory and covariant relativistic field equations. The framework underpins the one-particle content of relativistic quantum fields in arbitrary dimensions and clarifies how dimensional reduction and gauge formulations arise from the underlying group-theoretical structure.

Abstract

An extensive group-theoretical treatment of linear relativistic field equations on Minkowski spacetime of arbitrary dimension D>2 is presented in these lecture notes. To start with, the one-to-one correspondence between linear relativistic field equations and unitary representations of the isometry group is reviewed. In turn, the method of induced representations reduces the problem of classifying the representations of the Poincare group ISO(D-1,1) to the classication of the representations of the stability subgroups only. Therefore, an exhaustive treatment of the two most important classes of unitary irreducible representations, corresponding to massive and massless particles (the latter class decomposing in turn into the ``helicity'' and the "infinite-spin" representations) may be performed via the well-known representation theory of the orthogonal groups O(n) (with D-4<n<D). Finally, covariant field equations are given for each unitary irreducible representation of the Poincare group with non-negative mass-squared. Tachyonic representations are also examined. All these steps are covered in many details and with examples. The present notes also include a self-contained review of the representation theory of the general linear and (in)homogeneous orthogonal groups in terms of Young diagrams.