[Paper Review] The unitary representations of the Poincare group in any spacetime dimension

Xavier Bekaert, Nicolas Boulanger|arXiv (Cornell University)|Nov 24, 2006

Black Holes and Theoretical Physics4 citations

TL;DR

This paper provides a comprehensive group-theoretical classification of unitary irreducible representations (UIR) of the Poincaré group in arbitrary spacetime dimensions D > 2 using induced representations and Young diagrams. It systematically derives covariant field equations for massive, massless (helicity and infinite-spin), and tachyonic particles, establishing a one-to-one correspondence between relativistic field equations and UIRs of ISO(D−1,1). The key contribution is a unified, detailed framework for classifying relativistic particles via representation theory across all dimensions, with explicit field equations for each class.

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.

Motivation & Objective

• To provide a complete, group-theoretical classification of unitary irreducible representations (UIRs) of the Poincaré group ISO(D−1,1) in arbitrary spacetime dimensions D > 2.
• To establish a one-to-one correspondence between linear relativistic field equations and UIRs of the isometry group, generalizing the standard particle classification to any D.
• To systematically derive covariant field equations for all UIRs with non-negative mass-squared, including massive, helicity, infinite-spin, and tachyonic representations.
• To present a self-contained review of representation theory for general linear and orthogonal groups using Young diagrams, enabling explicit construction of field equations.

Proposed method

• Using the method of induced representations, the classification of Poincaré group UIRs is reduced to classifying representations of stability (little) groups for each orbit in momentum space.
• For massive particles, the little group is SO(D−1), and UIRs are classified via Young diagrams for the orthogonal group O(D−1).
• For massless particles, the little group is ISO(D−2), leading to two classes: helicity (finite-spin) and infinite-spin representations, with distinct field equations derived via Young diagram constraints.
• Covariant field equations are constructed using generalized Dirac and Klein-Gordon operators, with constraints enforcing tracelessness and symmetry via Young projectors.
• The framework incorporates auxiliary variables to ensure irreducibility and manifest Lorentz covariance of the field equations.
• The treatment includes dimensional reduction techniques and duality relations, particularly in D=3, to relate higher-spin fields to scalar or spinor duals.

Experimental results

Research questions

• RQ1 How can unitary irreducible representations of the Poincaré group ISO(D−1,1) be classified in arbitrary spacetime dimensions D > 2?
• RQ2 What is the precise correspondence between linear relativistic field equations and UIRs of the Poincaré group across all D?
• RQ3 How do helicity and infinite-spin massless representations differ in their field equation structure and transformation properties in higher dimensions?
• RQ4 What are the covariant field equations for tachyonic representations, and how do they relate to the Bargmann-Wigner programme?
• RQ5 How do Young diagrams and representation theory of O(n) classify the UIRs of the Poincaré group in arbitrary D?

Key findings

• The paper establishes a complete one-to-one correspondence between linear relativistic field equations and unitary irreducible representations of the Poincaré group ISO(D−1,1) in any spacetime dimension D > 2.
• For massive particles, the UIRs are classified by Young diagrams of the orthogonal group O(D−1), with field equations constructed via generalized Dirac and Klein-Gordon operators.
• Massless representations split into helicity (finite-spin) and infinite-spin types, with distinct field equations: helicity fields are described by symmetric traceless tensors, while infinite-spin fields require infinite-component gauge fields with continuous momentum labels.
• In D=3, the paper shows that all massless helicity fields (bosonic or fermionic) are dual to a single scalar or spinor field via Hodge duality, with the field strength dualizing to a symmetric tensor field.
• Tachyonic representations are classified via the Abelian little group SO(1,1), with UIRs labeled by a single real parameter s ∈ R, and their field equations are derived as solutions to the Bargmann-Wigner programme.
• The framework provides explicit, covariant field equations for all UIRs with non-negative mass-squared, including the derivation of the Siegel-Zwiebach operator as a dimensional reduction of higher-dimensional massless field equations.