# Representations of the Lorentz group in an arbitrary number of space-time dimensions$.$

Let $\mathrm{SO}(1,d-1)^\uparrow$ be the connected Lorentz group in $d$ dimensions. I am looking for a book/article where its finite-dimensional projective representations are studied in detail. Rather surprisingly, I haven't been able to find anything online, so here I am.

Some topics that I'd like to see discussed in the references are:

1. Does any projective representation correspond to a regular representation of the spin group $\mathrm{Spin}(1,d-1)$?

2. Is any projective representation decomposable (i.e., it can be written, up to similarity, as the direct sum of irreducible representations)?

3. How can we classify all irreducible representations? in other words, how many labels do we need in order to specify a particular representation? are they half-integers? (in $d=4$, we have two labels, cf. the $(m,n)$ representation, which is $(2m+1)(2n+1)$-dimensional)

4. Can an arbitrary object transforming according to an irreducible representation be written as the tensor product of spinors? (in $d=4$, an element transforming according to the $(m,n)$ representation can be identified with an object carrying $m$ dotted and $n$ un-dotted spinor indices).

5. How does a large transformation ($C,P,T$) act on an arbitrary object transforming under a particular projective representation? (in $d=4$, the general formulas can be found e.g. in Wightman's PCT).

Any reference that addresses at least one of these topics will be welcome and appreciated. Ideally, the best reference would discuss all of them.

I think I know the answer for the first and second sub-questions, but I've included them anyway for completeness (the lack of results in the typical google searches suggests me that this post might end up being the first result when one googles "representations of the Lorentz in higher dimensions").

• To avoid potentially irrelevant answers, let me be (redundantly) explicit here: please note that this is about LORENTZ, not POINCARÉ. Those are very different problems. The references in the existing answer only address the latter (or the former in the particular case of $d\equiv 4$), so they do not answer any of my questions. If anyone has the intention of posting an answer, please make sure that it focuses on Lorentz instead of Poincaré. Sep 19, 2017 at 11:10
• Similar question, posted on MO: What is known about the projective representations of $\mathrm{SO}(n_1,n_2)$?. Sep 26, 2017 at 21:09
• I’ve become interested in this, and I’m disappointed to see that you didn’t get all of your questions answered either here or on MO. If you ever found a good reference, could you write an answer yourself? May 19, 2020 at 3:37
• @G.Smith Unfortunately, I still don't know the full answer to this question! I think I know the answer (well, except for q.5) in the Euclidean case SO(d), but the translation to the Lorentzian case SO(1,d-1) is complicated, and it not obvious at all to me how to do it. For example, spinors do weird things when changing the signature. May 31, 2020 at 23:46
• The Lorentz group is isomorphic to the Euclidean conformal group so the representations you are looking for are basically the highest weight representations of the conformal group. You can find a VERY detailed discussion on this in the book "Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory". Oct 20, 2021 at 16:48

The lecture notes

• Bekaert, X. and Boulanger, N., The unitary representations of the Poincaré group in any spacetime dimension [arXiv:hep-th/0611263]

are rather nice. I would say it assumes standard (physics) knowledge of QFT. Here is the abstract:

An extensive group-theoretical treatment of linear relativistic wave equations on Minkowski spacetime of arbitrary dimension $D>3$ is presented in these lecture notes. To start with, the one-to-one correspondence between linear relativistic wave equations and unitary representations of the isometry group is reviewed. In turn, the method of induced representations reduces the problem of classifying the representations of the Poincaré group $ISO(D−1,1)^\uparrow$ to the classication of the representations of the stability subgroups only. Therefore, an exhaustive treatment of the two most important classes of unitary irreducible representations, corresponding to massive and massless particles (the latter class decomposing in turn into the “helicity” and the “infinite-spin” representations) may be performed via the well-known representation theory of the orthogonal groups $O(n)$ (with $D−3\leq n\leq D−1$). Finally, covariant wave equations are given for each unitary irreducible representation of the Poincaré group with non-negative mass-squared. Tachyonic representations are also examined. All these steps are covered in many details and with examples. The present notes also include a self-contained review of the representation theory of the general linear and (in)homogeneous orthogonal group s in terms of Young diagrams.

This reference at least provides the answer to the OP's subquestion (2) [see Section 1.3], and, I think, subquestion (3) [namely: Young diagrams, see Section 4.3 onwards].

For the more mathematically inclined reader, the lecture notes

certainly deserve to be mentioned. Assuming familiarity with analysis, differential geometry and functional analysis, this reference gives a nice and detailed mathematical treatment of the topic. It includes such things as the basics of representation theory, lifting projective reps to (anti)unitary reps of the universal cover [answering subquestion (1)], the spectral-theory prerequisites, induced representations and the reverse construction, and representations of semi-direct products.

[I would expect the detailed answer to question (2) to be here somewhere too, but haven't found it upon skimming the text.]

The applications discussed towards the end, viz. Wigner's classification of unitary Poincaré irreps (see Section 12 onwards), focus on the case $d=4$, but the discussion is still instructive.

• I am aware of the first reference, which is indeed very nice, but it does not answer to any of my questions, at least not directly. That paper is concerned with the representations of $\mathrm{ISO}(1,d-1)^\uparrow$ and of $\mathrm{SO}(n)$ (which is the little group of the particles analysed therein). Here I am asking about the representations of $\mathrm{SO}(1,d-1)$, which looks like a similar problem but it is not. For one thing, the latter is semi-simple, while the former are not. I'll have a loot at the rest of references. In any case, thank you very much for your time! Sep 18, 2017 at 10:46
• (For lower dimensions see the answer at mathoverflow.net/a/210411/45956.) Sep 18, 2017 at 10:47
• Again, thanks, but that link explains the representations of Poincaré, not Lorentz. Those are very different problems, although related. Lorentz is semi-simple. Poincaré is not. The representations of the latter can be obtained, à la Wigner, by means of the representations of the orthogonal group, which is simple and compact. Thus, both Poincaré and $\mathrm{SO}(n)$ are easy to analyse, not quite so Lorentz. Sep 18, 2017 at 10:50
• @AccidentalFourierTransform All right, it was not clear to me that you do want Lorentz instead of Poincaré. Thanks for clarifying. I will leave the above asothers might still be interested in this too. Sep 18, 2017 at 10:53
• @AccidentalFourierTransform ...But I still think that Sect 3 of the final reference at least answers your subequestion (1). Sep 28, 2017 at 10:30