Representations of the D=2 Euclidean and Poincaré groups

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.

💡 Research Summary

The paper presents a complete, step‑by‑step construction of all unitary irreducible representations (UIRs) of the two‑dimensional Euclidean group E(2) and the two‑dimensional Poincaré group ISO(1,1) by means of Mackey’s theory of induced representations. After a concise introduction that motivates the study—highlighting that in higher dimensions the explicit form of the Wigner rotation is generally unavailable—the authors devote Section 2 to a thorough exposition of Mackey’s theorem. They define a closed subgroup H of a locally compact second‑countable group G, the induced representation Ind_G^H(σ), and then specialize to semidirect products G = N ⋊ H with N abelian. The dual group ĤN consists of one‑dimensional characters χ_p(x)=e^{i p·x}, and the orbit method is introduced: for each character the orbit O_χ, the isotropy group G_χ, and the little group H_χ are identified.

Section 3 introduces the Euclidean group E(2) = T² ⋊ SO(2). The translation subgroup T² is identified with ℝ², and the rotation subgroup is replaced by its double cover Spin(2) to accommodate both bosonic and fermionic statistics. The authors give an explicit matrix realization of the double cover using complex numbers z=e^{iθ/2} and show how the map (a,z) ↦ (a,R_θ) is two‑to‑one.

In Section 4 the Mackey construction is applied to E(2). The characters of T² are χ_q(a)=e^{i q·a}. Two families of orbits appear: (i) the trivial orbit q=0, whose isotropy group is the whole group and whose little group is Spin(2). The induced representations are one‑dimensional and labeled by an integer m∈ℤ, with matrix element e^{i mθ/2}. Even m give genuine E(2) representations, odd m require the double cover. (ii) Non‑zero orbits |q|>0, which are circles in momentum space. Their isotropy group is T² ⋊ ℤ₂, and the little group is the discrete group ℤ₂ (±id). Equivariant functions have the form f(a,z)=e^{-i q·R_{θ}^{-1}a} · \tilde f(z) with the parity condition \tilde f(−z)=±\tilde f(z). The induced representation acts by a phase e^{i R_θ q·b} and a shift of the argument z→u^{-1}z. By passing to the Lie algebra, the generators become J=−i∂/∂θ and Q_i=i∂/∂a_i, and the matrix elements involve Bessel functions J_m(|q|r), reproducing the familiar plane‑wave expansion in two dimensions.

Section 5 briefly reviews the Poincaré group ISO(1,1) = ℝ^{1,1} ⋊ SO(1,1). The Lorentz part SO(1,1) has a non‑compact continuous component and a discrete reflection; its double cover Spin(1,1) is constructed in the appendix.

Section 6 repeats the Mackey analysis for ISO(1,1). The characters of the translation subgroup are χ_p(x)=e^{i p·x}. Three classes of orbits are distinguished: timelike (p²>0), spacelike (p²<0), and light‑like (p²=0). For timelike and spacelike orbits the little group reduces to ℤ₂, leading again to two inequivalent one‑dimensional representations distinguished by a sign. For the light‑like orbit the little group is the full ℝ of null‑rotations, whose double cover Spin(1,1) produces a two‑fold periodicity in the rapidity parameter. The induced representations are written explicitly; for non‑zero orbits the matrix elements involve modified Bessel functions K_ν and exponential phases, while for the light‑like case they reduce to simple phase factors e^{i mη/2} with η the rapidity.

Appendix A derives the Spin groups from Clifford algebras, giving the explicit isomorphisms Spin(2)≅U(1) and Spin(1,1)≅ℝ. Appendix B provides the concrete parametrization of the double covers and the homomorphisms ϕ:Spin→SO.

The authors conclude that in two dimensions the full machinery of Mackey induction can be carried out analytically, yielding concrete formulas for all UIRs of E(2) and ISO(1,1). This explicit knowledge is valuable for low‑dimensional quantum field theories, anyonic models, and for pedagogical purposes, as it avoids the need for abstract Casimir‑based arguments or numerical approximations that are unavoidable in higher dimensions.