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It is well-known that long ago, Wigner classified the unitary irreducible representations of the Poincare group in dimension 4.

I am looking for a convenient reference describing all unitary irreducible representations of the Poincare group in dimensions 2 and 3. (I know how it can be done in principle. But I am looking for a reference to cite.)

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  • $\begingroup$ The Poincaré group in dimension $n$ means $\mathbf{R}^n\rtimes\mathrm{SO}(n-1,1)$? $\endgroup$
    – YCor
    Jun 22, 2015 at 14:17
  • $\begingroup$ @YCor: Yes. It is also called the inhomogeneous Lorentz group, and sometimes denoted as $ISO(n-1,1)$ or $ISO(1,n-1)$. $\endgroup$ Jun 22, 2015 at 16:21
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    $\begingroup$ I think that for such a semidirect product you can apply the work of Mackey (Induced representations of locally compact groups. I. Ann. of Math. (2) 55, (1952), theorem 14.1) to obtain a classification of unitary representations. $\endgroup$
    – Pierre
    Jun 22, 2015 at 19:05
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    $\begingroup$ I continue my previous comment (which I posted by mistake before completing it). Basically the unitary representations will be parametrized by an SO(n-1,1) orbit on the dual group of R^n, and a unitary representation of the associated stabilizer $H < SO(n-1,1)$. (there is an hypothesis of "regularity" about the action of one group on the other in Mackey's theorem, but I think it is satisfied in the present situation) $\endgroup$
    – Pierre
    Jun 22, 2015 at 19:13

1 Answer 1

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I finally found a complete analysis for dimension 3 in

D.R. Grigore, The projective unitary irreducible representations of the Poincaré group in 1+2 dimensions, J. Math. Phys. 34 (1993), 4172-4189. (http://arxiv.org/abs/hep-th/9304142)

Now I also found a complete analysis for dimension 2 in

S.K. Bose, Projective representations of the 1+ 1‐dimensional Poincaré group, J. Math. Phys. 37 (1996), 2376-2387.

For completeness, Wigner's classification in 4 dimensions is in

E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40 (1939), 149–204.

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