[Physics] Unitary representations of the Poincare group in $(2+1)$ dimensions

poincare-symmetryrepresentation-theoryspecial-relativity

I am interested in classifying the unitary irreducible representations (UIR) of the Poincare group in (2+1) dimensions. I have done this in the (3+1) dimensional case but with the guidance of the abundant material available. For obvious reasons, the (2+1)-D case has not received as much attention and full reviews are few and far between.

I have proved that $\text{SO}(2,1)_{_0}\cong\text{SL}(2,\mathbb{R})/\mathbb{Z}_2$ and determined all irreducible finite dimensional representations of $\text{SL}(2,\mathbb{R})$ and of the Lorentz algebra $\mathfrak{so}(2,1)$. The irreps are determined by an integer $n=0,1,2,…$ and are realised on totally symmetric spin-tensors $\psi_{\alpha_1\alpha_2\cdots\alpha_n}=\psi_{(\alpha_1\alpha_2\cdots\alpha_n)}$. I have found the Casimir operator of the Lorentz group and found that it has eigenvalues $-\frac{1}{2}n(n+2)$ in the irrep labelled by $n$.

What are the next steps I should take in order to classify the UIR's of the (2+1) Poincare group?

I think the general procedure should go like

  1. Define a unitary operator acting on the Hilbert space of one particle states which implements an infinitesimal Poincare transformation $(\Lambda,b)\approx(1+\omega,\epsilon)$ by $$U(\Lambda,b)=1+\frac{i}{2}\omega_{ab}\mathcal{J}^{ab}-i\epsilon_a\mathcal{P}^a$$
    where $\mathcal{J}^{ab}$ are the Lorentz generators and $\mathcal{P}^{a}$ are the translation generators.

  2. Determine the commutation algebra obeyed by the above generators.

  3. Determine the Casimir operators of the Poincare group. One will be the four momentum squared and one will be related to the Pauli-Lubanski vector (Although I am not sure what $W^a$ would look like in this dimension… perhaps $W_a=\epsilon_{abc}\mathcal{J}^b\mathcal{P}^c$ where $\mathcal{J^a}$ is the vector which is hodge dual to the antisymmetric $\mathcal{J}^{ab}$).
  4. Use the first Casimir to separate into the massive and massless cases.
  5. Determine little groups of a standard momentum vector for each case. In the massive case I found it to be $\text{SO}(2)$ but for the massless case I am unsure, it seems to be a combination of a rotation and boost, but I don't know how to classify the irreps of such a group.
  6. Use the Little groups to fully classify the UIR's of the Poincare group.

I have read the first part of Binegar's paper (here) and the way in which he classifies the UIR's is unfamiliar to me. He classifies the UIR's via Mackey's method, which seems more complicated than the standard method used in the (3+1)-d case. Is it absolutely neccesary to use this method or can I do it in an analogous way to (say) Weinbergs method in volume 1 of his QFT series?

Best Answer

The $2+1$ dimensional case has been reviewed e.g. in Section 3 https://inspirehep.net/record/1286880 and with even more details in Section 4.3 in https://inspirehep.net/record/1494790 .

Since the Poincare group is a semidirect product one can construct unitary representations via induced representations. For details see e.g., Section 2.1 of the first and Section 3 of the second link above or Barut, Raczka -Theory of group representations and applications.

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