Irreducible representations of Poincaré group

Question

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.

Answer 1

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)].

Answer 2

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).

Answer 3

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)

[EDIT]

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.

Answer 4

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.