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Representations of the D=2 Euclidean and Poincaré groups

Giovanni Camilletti, María A. Lledó, Mariano A. del Olmo

TL;DR

We classify and realize the unitary irreducible representations of the 2D Euclidean and Poincaré groups $E(2)$ and $P(1,1)$ by applying Mackey’s orbit method for semidirect products. The work constructs explicit realizations as induced representations from abelian translations, deriving the corresponding equivariant function spaces, Lie-algebra actions, and bases of eigenvectors (notably $J$ for $E(2)$ and $K$ for $P(1,1)$), and presents the full matrix elements in those bases. For $E(2)$, the representations decompose into bosonic and fermionic sectors with Bessel-function matrix elements, while for $P(1,1)$ the matrix elements become distributional Fourier transforms, including a careful regularization. The results provide concrete, computable UIRs in 2D, clarify the role of Spin double covers, and supply explicit realizations relevant for 2D particle models and related symmetry analyses, with clear orbit classifications and explicit action formulas.

Abstract

In this paper we compute explicitly the Unitary Irreducible Representations of the Poincaré and Euclidean groups in dimension D=2, following, step by step, Mackey's theorem of induced representations.

Representations of the D=2 Euclidean and Poincaré groups

TL;DR

We classify and realize the unitary irreducible representations of the 2D Euclidean and Poincaré groups and by applying Mackey’s orbit method for semidirect products. The work constructs explicit realizations as induced representations from abelian translations, deriving the corresponding equivariant function spaces, Lie-algebra actions, and bases of eigenvectors (notably for and for ), and presents the full matrix elements in those bases. For , the representations decompose into bosonic and fermionic sectors with Bessel-function matrix elements, while for the matrix elements become distributional Fourier transforms, including a careful regularization. The results provide concrete, computable UIRs in 2D, clarify the role of Spin double covers, and supply explicit realizations relevant for 2D particle models and related symmetry analyses, with clear orbit classifications and explicit action formulas.

Abstract

In this paper we compute explicitly the Unitary Irreducible Representations of the Poincaré and Euclidean groups in dimension D=2, following, step by step, Mackey's theorem of induced representations.
Paper Structure (19 sections, 1 theorem, 109 equations, 1 figure)

This paper contains 19 sections, 1 theorem, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mackey's theorem for unitary representations of semidirect products
  3. Induced representations
  4. About the existence of invariant measures in $G/H$.
  5. Induced representations for semidirect products
  6. The Euclidean group $\mathrm{E}(2)$
  7. Unitary, irreducible representations of $\mathrm{E}(2)$
  8. Orbits and equivariant functions
  9. Lie algebra representation
  10. Basis of eigenvectors of $J$
  11. The Poincaré group $\mathrm{P}(1,1)$
  12. The UIR of the Poincaré group in $D=2$.
  13. Orbits and equivariant functions
  14. Lie algebra representation
  15. Basis of eigenvectors of $K$
  16. ...and 4 more sections

Key Result

Theorem 2.5

(Mackey)ma1ma2. Let $G=N\rtimes H$ be a locally compact, second countable group that is a semidirect product, with $N$ an abelian, normal subgroup, and with $H$ a closed subgroup. Let $\chi\in\hat{N}$ and let $H_\chi$ be the little group of $\chi$. Let $\sigma:H_\chi\rightarrow \mathrm{Aut} (\math Moreover, the representation $\mathrm{Ind}_{G_\chi}^G(\chi\sigma)$, (denoted usually as $\mathrm{In

Figures (1)

  • Figure 1: Action of $G$ on $\Gamma(E)$.

Theorems & Definitions (5)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5