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When I studied representation theory for the first time it was only focused on finite groups. It was the second half of a one semester course in group theory, and the book employed was "Representation Theory: A First Course" by Harris and Fulton.

Currently I'm studying Quantum Field Theory, and regarding spinors and quantum fields with spin, I'm interested in the representation theory of the Lorentz group.

I know the description of the fields with various spins is tightly connected to the irreducible representations of the Lorentz group.

Unfortunately, though, most traditional textbooks in QFT take a stand to not do a rigorous treatment of the matter, nor present a mathematicaly pleasant notation or definitions. Most of them speak quite loosely of the subject, that it doesn't even look connected to the representation theory I saw in the course I took.

Considering this, what I'm looking for is a book or lecture notes which presents a mathematicaly rigorous approach to the representation theory of the Lorentz group and also establish in a mathematicaly correct way the connection to quantum fields with spin and the representations of the Lorentz group. Is there such a resource out there?

Gold
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  • Math. sources that I know would give a rigorous treatment of the representation theory but will not discuss QFT connection. Also, you should specify if you are interested in finite dimensional representation theory or infinite-dimensional one (they are quite different); also, is your Lorentz group $SO(3,1)$? – Moishe Kohan May 24 '17 at 03:04
  • I have the same question, so I've started a bounty. Yes the Lorentz group (of physics) is $SO(1, 3)$. I'd also like if the resource discusses the Poincaré group, which is the inhomogeneous extension of $SO(1, 3)$ (includes translations in all four directions). – WillG Nov 22 '21 at 18:36
  • I've seen physics books set up some kind of relationship between the Poincaré group and $SL(2, \mathbb C)$ and then (non-rigorously) describe representations of that (see e.g. Wightman's "PCT, Spin & Statistics, and All That"). – WillG Nov 22 '21 at 18:37
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    Weinberg's QFT is a good example of the physicist's way of exploring such representations: He already assumes the groups are represented as $n\times n$ matrices, then perturbs them with infinitesimals, and finds constraints on the infinitesimals. I (and the OP, I think) would like to see a more rigorous, geometric approach that remains coordinate-free where possible. – WillG Nov 22 '21 at 18:41
  • Hi @WillG. It's long since I have asked this question, so perhaps I could share some thoughts on what worked for me. There are essentially two at first disjoint (but later on connected) discussions: the unitary representations of the (universal cover of) Poincaré group and the non-unitary finite dimensional representations of the Lorentz group. The first classifies one-particle states, the second classifies fields. – Gold Nov 22 '21 at 18:48
  • The first one is discussed in Chapter 2 of Weinberg's book, the other (as well as the connection to the first) is discussed in Chapter 5. Weinberg's presentation is though (and I really understood it after I have taken a course which used it as textbook), but it is not that much unrigorous as it seems. In the end what worked for me was to bridge what he is talking about to "standard" representation theory, then what he discusses makes a lot of sense, and I started to view it as the nicest discussion. – Gold Nov 22 '21 at 18:50
  • Nevertheless, I would like to suggest you take a look on "Quantum Field Theory for Mathematicians" by Robin Ticciati, specially Chapter 6. This is a quite detailed account of the finite-dimensional non-unitary representations of the Lorentz group (with all relevant definitions of rep theory included). If I recall correctly he just does not discuss thoroughly the Poincaré case. But then there's this nice paper https://arxiv.org/abs/0809.4942 which discusses both the Weinberg approach, the mathematically rigorous approach, and the connection between them. – Gold Nov 22 '21 at 18:52
  • Thanks! These look pretty interesting. I should go through Weinberg more carefully. But ultimately, I think I'm still dissatisfied by his use of infinitesimals and sketchy procedure of neglecting second-order infinitesimals. – WillG Nov 22 '21 at 19:32
  • I agree that is most of the time ill justified, but it can be done better, in fact. The key aspect is the relation between a Lie group representation and a Lie algebra representation. You can go from the former to the later by the derived representation construction. Going backwards is more subtle and involves the exponential map. For the unitary irreps of Poincaré you don't actually need this, the construction is not through exponentiation directly, but rather from the induced representation method. – Gold Nov 22 '21 at 19:49

1 Answers1

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A classical and fairly complete (even if it is considered somewhat outdated by someone) reference is [1]: instead of describing its content, let me report the first sentences of the "Phisics Today" review of the book by J. E. Mansfield

Professor Gel'fand has performed another service to physicists with this study. There are a few place where one can find a treatment of the Lorentz group that is both coherent and complete.

Perhaps, even if it is a quite old reference, it is a nice place to start, even considering that a quite inexpensive Dover paperback reprint has been recently published.

Addendum. In the preface of of [1], the Authors recommend the monograph [2] (see here for a review published in the Journal of the London Mathematical Society) "... to the reader wishing to study the representation of the Lorenz group more thoroughly and at a greater length", in their own exact words ([1] p. xviii).

References

[1] I. M. Gel’fand, R. A. Minlos and Z. Ya. Shapiro, Representations of the rotation and Lorentz groups and their applications. Translated from the Russian by G. Cummins and T. Boddington (English), Oxford-London-New York-Paris: Pergamon Press. xviii, 366 pp. (1963), MR0114876, Zbl 0108.22005.

[2] M. A. Najmark, Linear representations of the Lorentz group. Translated by A.Swinfen and O.J.Marstrand. (English) International Series of Monographs in Pure and Applied Mathematics. 63. Oxford etc.: Pergamon Press. XIV, 447 p. (1964), MR0170977, Zbl 0116.32904.

Daniele Tampieri
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