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Unitary Irreducible Representations of the Poincaré Group

Most particle physicists will recognise this title immediately, but to non-specialists it will be just gibberish.

But even if you need to skip the technical bits, you might find the observations about the sociology of physics interesting.

When, as a second-year graduate student in Oxford, I submitted a not-particularly-significant contribution to this subject to the journals in 1983, the reaction seemed to be one of boundless rage. I still do not understand why. Outside of Penrose's group in Oxford (which I did not belong to, having been attached to the physics rather than mathematics department) the only qualified person in any way supportive was Professor Lochlainn O'Raifeartaigh of the Dublin Institute of Advanced Studies.

Even if you do not understand the title, you may be prepared to believe that it (the title) is mathematically precise.

As one enrolled on one of the less mathematical and more practical physics courses as an undergraduate (Oxford), I found more-abstract mathematics a huge boon. I had the devil’s own job getting to understand quantum mechanics on the basis of plausibility arguments and classical analogies that were all the rage at the time. But when I discovered infinite-dimensional complex vector spaces (from Dirac’s book), my eyes were opened. A good strategy when working with quantum mechanics is this: do not try to understand it by relating it to the everyday world, just try to understand it through the mathematics. It opened up a whole world of operators and operator algebras. Indeed, one of the neater things that the physics undergrad ever does is to derive the spectrum of eigenstates of the angular momentum operators. I loved the power and generality of this, especially given how ad hoc and arbitrary so much of the other stuff we had been taught seemed to be. I now know that all one is doing here is classifying the Unitary Irreducible Representations of the Three-Dimensional Rotation Group (or, more accurately, its double cover SU(2)), but it seemed like a lot of output for little input: any complex vector space that has three operators Jx, Jy and Jz with the SU(2) algebra acting non-trivially on it must be expressible as linear combinations of states characterised by a spin (a non-negative half integer) and a spin z-component (a half-integer which steps from minus the spin to plus the spin in whole steps). What is more, having more than one set of operators does not invalidate the analysis - it merely makes it more interesting, and it is crucial to Atomic Physics that one may, using so-called Clebsch-Gordon coefficients, switch between different classifications of quantum states with different behaviour under the rotation group.

Traditionally, one represents spin states with the pair (s, ms), with ms being allowed to roam over the 2s+1 different values as shown in the table below:

Spin (s) Spin z-component (ms)
0 0
1/2 -1/2, 1/2
1 -1, 0, 1
3/2 -3/2, -1/2, 1/2, 3/2
2 -2, -1, 0, 1, 2
etc. etc.

However  (s, ms) is not the only way to represent spin: one may also use SU(2) tensors (something, by the way, that is rarely taught in quantum mechanics courses). The basic building block here is a two-component vector in a complex vector space known as a spinor. It can be labelled Ta, where a = 1 or 2. Higher spins are then just formed as symmetrical tensor products, and the table above becomes 

Spin (s) Tensor
0 T
1/2 Ta
1 Tab
3/2 Tabc
2 Tabcd
etc. etc.

That these tensors are symmetrical follows from irreducibility: for example a rank two tensor that is not symmetric can be written as Tab + U εab where εab is the alternating tensor (a preserved constant tensor of SU(2)), showing that the tensor here is reducible as it mixes both the symmetric spin one Tab  and the scalar (spin zero) U. Note that the number of components match: for example in spin 3/2 the independent components are T111,  T112,  T122,  and T222 ( T121, for example is not independent as symmetrisation gives T112) - in one-to-one correspondence with the ms values of -3/2, -1/2, 1/2 and 3/2.

It would be very inconvenient to do all of angular momentum in atomic physics in terms of SU(2) tensors, but if one did at least one would have no need to calculate any Clebsch-Gordon coefficients as here one need only follow the requirements of symmetry and covariance. Coupling two spin 1/2's, for example, leads to spin 1 and spin 0. Calculating the Clebsch-Gordon coefficients in this case is simple enough, but with SU(2) tensors it is even simpler: the symmetrization Xa Yb +Xb Ya is spin 1 and the antisymmetrization Xa Yb -Xb Ya , being proportional to the alternating tensor, is spin zero.

The rotation group is a symmetry that comes with the physical principle that space is isotropic. As is the case in classical mechanics, symmetries result in conservation laws: angular momentum conservation follows from rotational invariance, energy conservation from time translation invariance and momentum conservation from space translation invariance. But in quantum mechanics symmetry also tells one the size and shape of one's building blocks.

In Special Relativity, one also has symmetry with respect to different inertial frames of reference - in other words, the physical laws experienced by two inertial frames of reference in motion with respect to each other must be the same. The symmetry group of Special Relativity is known as the Poincaré Group and has ten dimensions: one for time translation, three for space translation, three for rotation and three for invariance under change of velocity of the reference frame.

The unitary irreducible representations of the Poincaré group were first classified by Wigner in 1939. This is an extremely interesting piece of mathematics, and one that has far-reaching consequences, as it shows that, in much the same way that quantum states in atomic physics are classified by their angular momentum quantum numbers, quantum states in relativistic quantum theory are comprised of entities classified by mass and spin.

 The classification by mass implies the Klein-Gordon equation, an equation for a free particle that reduces to the Schrödinger equation in the non-relativistic limit, and it is highly satisfying that it can be obtained from such general considerations and not merely classical analogy. Similarly, the source-free Maxwell equations follow just from the stipulation that the photon has mass zero and spin one (helicity one, actually, but there is no need to get into that here).

One thing that is not so nice about the Wigner analysis, though, is the way that he handles spin. He introduces a thing called (by everyone else) the Wigner rotation which, although covariant under the rotation group, is not covariant under the larger Lorentz group. This is unfortunate as things that are easy to demonstrate with the Lorentz group - or, more accurately, the group SL(2,C), which is its double cover - are very hard when one uses the Wigner rotation. For example, the proof of the spin-statistics theorem in Weinberg's Quantum Field Theory, Volume 1, Chapter 5, using this notation, is sufficiently complex that he advises his readers to skip it on the first reading. With SL(2,C), though, the proof is just a few lines. The difference between the Wigner rotation and SL(2,C) here is very much like the difference between the two tables of spin states for the rotation group above. Both are correct: it is just that for the more theoretical work, the second is easier to work with.

I had learned my SL(2,C) from the Penrose group, this being a pre-requisite for Twistor theory. Physicists, also, were having to learn it at the time (1982) as it was needed for supersymmetry. Realising that, unlike the careless physicists, Penrose had thought the subject out properly, I quickly switched to his conventions, and then worked out the tensor structures of the irreducible representations of the Poincaré group as an alternative to the Wigner rotation. The massless case was the most interesting, as the analysis gave an equation which was the massless Dirac equation for spin 1/2, the source-free Maxwell equations for spin 1 and linearised, source-free General Relativity for spin 2. This equation was well known to the Penrose group, and was so simple that it was hard to believe that papers in theoretical physics deriving Lagrangians for massless particles of arbitrary spin (by Fang and Fronsdal) were solving the same problem. These papers had ferociously complex formulae whose connection with the underlying principles was all-but-impossible to discern. Nonetheless, it was these Lagrangians that were the jumping-off point for higher-spin studies in particle physics.

So I decided to see if they did amount to the same thing. They did, and to see the Fang-Fronsdal formulae drop out after weeks of swimming in index soup was the second most satisfying moment of my brief academic career. The Fang-Fronsdal Lagrangian was in terms of the gauge potential whereas the Penrose one was in terms of the field strength. The field strength was the physical thing, but it could be derived from the gauge potential. Certain changes, known as gauge transformations, could be applied to the gauge potential without changing the field strength. The Fang-Fronsdal Lagrangians followed if one required that the Lagrangian was invariant under the largest possible class of gauge transformations. The Lagrangians were not unique, though. Just as gauge fixing terms can be added to the Maxwell Lagrangian without affecting any dynamics, so with higher spin, gauge fixing is possible by, amongst other things, leaving out auxiliary fields.

Anyway, feeling pleased with myself, I submitted my work to a journal. I am not sure what I was expecting, but what I got back was that I was a complete idiot who knew nothing. Apparently, the bits of my paper that were not merely repetitions of text books were just wrong. When I got them to be specific, it was easy to show that the criticisms were invalid, but that made no difference: whatever happened, they were not going to publish the paper. Looking back, I think that the mentality was just that there was only one way of doing this - their way, the right way, and if anyone tried anything different then they would do everything they could to prevent this being seen. They certainly were not going to consider the possibility that something as fundamental as the Wigner rotation could be improved on - and all this in spite of of the evidence from Penrose and others. These journal referees were anonymous, and I doubt that they would have made some of their more intemperate remarks if they had had to take responsibility. But when criticisms are sufficiently ignorant, they become like water off a duck's back: I ignored them and the work became the core of a D.Phil. thesis on arbitrary-spin field theory.

I revisited a lot of the standard ground of so-called "axiomatic" field theory in the process, including the spin-statistics theorem. Luckily one of the experts in the field who was prepared to accept that the last word had not necessarily been written on Wigner's Unitary Irreducible Representations of the Poincaré Group was Professor O'Raifeartaigh, who was my external examiner, and I was passed in June 1984.

I do see papers on higher spin appearing on ArXiv on a fairly regular basis: mostly generalisations of one kind or another, but, given that so far we can only be sure that fundamental particles of spin 1/2 and 1 exist (no, I do not believe in the Higgs), I question the value. On the other hand the fact that one can build interacting field theory entirely from free particle states means that the Wigner work, instead of being interesting but ultimately irrelevant - as many seem to imagine - is essential.