The unitary representations of the Poincare group in any spacetime dimension

TL;DR

This paper provides a comprehensive group-theoretical analysis of linear relativistic field equations in Minkowski spacetime of any dimension, classifying unitary representations of the Poincare group and deriving covariant field equations.

Contribution

It offers an exhaustive classification of unitary irreducible representations of the Poincare group in arbitrary dimensions, including massive, massless, and tachyonic cases, with detailed examples and covariant equations.

Findings

Classified all unitary irreducible representations for D>2.

Derived covariant field equations for each representation.

Included analysis of tachyonic and infinite-spin representations.

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.