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I am looking for any reference on Wigner's classification of irreducible representations of the Poincaré group. I know the classification, but is there any reference where the representations are constructed and explained. This classification gives the different spin particles in Quantum mechanics. Thanks.

Edit (Qiaochu Yuan, 7/12/11): I am also interested in the answer to this question and unsatisfied with the current answer, so I have offered a bounty. I don't currently have institutional access to Wigner's original paper and in any case find it a little difficult to read, and would appreciate a modern, thorough, mathematical account.

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@Qiaochu: I'm far from a library at the moment, but I'd start looking here: Barut-Rączka, Theory of group representations and applications, Sternberg, Group theory and physics, and the books by Mackey. – t.b. Jul 12 '11 at 21:43
@Theo: thanks for the reference to Sternberg. Looking through it now. – Qiaochu Yuan Jul 13 '11 at 1:01
@Theo: it seems that Sternberg describes how to construct the representations but doesn't prove that they exhaust the physically meaningful possibilities. Barut-Rączka is extremely thorough but it would take me quite awhile to digest the necessary background... – Qiaochu Yuan Jul 13 '11 at 2:43
Is it question about irreducible unitary representations? – Alex 'qubeat' Jul 17 '11 at 21:30
@Alex: yes, I assume so, since that is what Wigner studied. – Qiaochu Yuan Jul 18 '11 at 0:26

You could try to look at:

  • Geometry of Quantum Theory - V. S. Varadarajan - Second Edition, on Chapter 9 (Relativistic Free Particles), in particular to the Theorem 9.4 (p.347), that is the classification theorem obtained by Wigner.
  • A course in abstract harmonic analysis - G. B. Folland, on Chapter 6 (Induced Representations), in particular in the section 6.7.3 (The Poincaré Group, p.190).
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Why was this answer downvoted? Varadarajan seems to prove the classification (it is Theorem 9.4 on p.347 in the second edition). – t.b. Jul 15 '11 at 12:11
Oh yes, it's theorem 9.4, I'm sorry for the typing error, I edit the post. – Robert Jul 15 '11 at 12:16
It is good style to make references as easy as possible on the reader. In particular, mention the edition you have in mind and the page number on which the theorem appears, especially if you're referring to a book (or a longer article). The more redundancy you add, the easier it will be for the reader to find what she's looking for (you can never exclude typos in what you write). – t.b. Jul 15 '11 at 12:21
I wrote the edition, but forgot to write the page. I add it right now! Thanks for the hints, and sorry again. – Robert Jul 15 '11 at 12:24
@RobertPi: thanks. Varadarajan looks pretty good. – Qiaochu Yuan Jul 18 '11 at 0:28

Here is an article that explains Wigner's classification (but not in the exact same way as Wigner himself). To see Wigner's classification explicitly, you probably should check out the following[E.P. Wigner, Ann. Math., 40, 149, (1939)].

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The first .pdf you linked to only describes the classification; it neither constructs the representations nor proves that they are a complete list. – Qiaochu Yuan Jul 12 '11 at 18:51

A review treating the construction of the unitary representations of the Poincare group for any space dimension is given in the following arxiv article by Xavier Bekaerta and Nicolas Boulanger.

This article is written for readers with quantum mechanics background. It explains the method of induced representations for the Poincare group representations construction and the complete classification of all unitary irreducible representations. In particular the description includes the tachyonic and infinite spin representations, which do not have extensive applications in physics.

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Thanks for this answer, but I would really prefer an exposition by mathematicians, not physicists (so more from the representation-theoretic perspective than the quantum-mechanical one). I'm not particularly comfortable with Einstein notation yet. – Qiaochu Yuan Jul 15 '11 at 23:23
  1. Original Wigner paper was reprinted also in Nucl. Phys. B (Proc. Suppl.) 6, pp 9 - 64 (1989) – it is more accessible and the whole issue devoted to the theme and there are other useful topics like Weinberg comments together with his own article on nonlinear representations (pp 67 – 75).

  2. N.N. Bogolubov, A. A. Logunov, A.I. Oksak, I. Todorov, General principles of quantum field theory, Springer, 1989. (chapter 7.2 and maybe also Appendix I)


Rolf Berndt, Representations of Linear Groups - An Introduction Based on Examples from Physics and Number Theory, Vieweg Wiesbaden, 2007 (Ch. 7.5) there is also recommended (together with already mentioned Barut et al, with that I am absolutely agree):

J. F. Cornwell. Group Theory in Physics. Academic Press, London 1984.

Yet I may only see an abridgment of the book issued 1997, there the Poincare group only briefly mentioned.

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