Pauli matrices and Gell-Mann matrices

The Pauli matrices are 2×2 matrices introduced in the 1920s for the purpose of describing the spin of electrons. They were later used in other contexts, for example the description of weak isospin to distinguish electrons from neutrinos, and up quarks from down quarks. The Gell-Mann matrices are 3×3 matrices introduced in the 1960s for the purpose of bringing order to the particle zoo, specifically for describing the baryon octet and the meson octet (or nonet). They were later used in other contexts, for example for the description of colours and gluons in the theory of quantum chromodynamics (QCD).

There is supposed to be an analogy between the 2×2 case and the 3×3 case, as they are both sets of traceless Hermitian matrices, and so they generate the Lie algebras su(2) and su(3) respectively. (Or, if you’re a mathematician instead of a physicist, you multiply them by i first to make them anti-Hermitian, but this is just a confusing convention, which gets in the way of communication, rather than a fundamental difference.) This is true, so far as it goes. But it only allows you to treat the continuous parts of the theory analogously, using the Lie groups SU(2) and SU(3). It does not allow you to treat the discrete parts of the theory analogously, because the Pauli matrices generate a finite symmetry group, but the Gell-Mann matrices do not.

This is a fundamental difference, and is a very serious problem. So I would like to tell you how to solve this problem, and put the finite symmetry group into the Gell-Mann matrices where it belongs. First, let’s go back to the Pauli matrices, and let’s use the mathematicians’ anti-Hermitian matrices, because it’s much easier that way. The three anti-Hermitian Pauli matrices generate the quaternion group Q_8 of order 8. This group contains two scalar matrices +1 and -1, and three pairs +i/-i, +j/-j and +k/-k. One of these pairs consists of diagonal matrices that distinguish spin up from spin down, and the other pairs swap spin up with spin down. Or in the weak isospin case, you distinguish electrons from neutrinos with one pair, and swap them (with the weak force) using the other pairs.

What is important to realise is that the discrete symmetries are physically important. You cannot mix an electron with a neutrino, they are fundamentally different distinct objects, and there is no continuous symmetry that converts one to the other. So you must use the Pauli matrices as discrete symmetries, not as generators for the continuous group SU(2). The same is true for Gell-Mann matrices, because the proton, neutron, lambda baryon, three sigma baryons and two xi baryons are eight distinct particles, not an 8-dimensional continuum of particles. So we have to have a finite group generated by Gell-Mann matrices, not just a Lie algebra and a Lie group. It so happens that mathematicians know how to do this, although physicists apparently do not. So listen up, you physicists, and you will learn something both interesting and useful.

The 3×3 analogue of the quaternion group (anti-Hermitian Pauli matrices) of order 8=2x2x2 is a group of order 27=3x3x3, containing scalars of order 3, plus 6 other diagonal matrices (or 2, if you ignore the scalar multiplications), and 18 off-diagonal matrices (or 6, if you ignore the scalars. You can just write them down, using say v and w as the two primitive cube roots of 1 (that is, exp(2\pi i/3) and exp(4\pi i/3)), and putting one entry 1,v or w in each row and column, with 0 for the other six entries, and ensuring that the determinant is +1. I won’t do the exercise for you, you should do it yourself. Mathematics is easy, but it is not a spectator sport.

So there you have it, the Gell-Mann matrices done properly. But what do you notice? I’ve given you 27 matrices, Gell-Mann only gave you 8. Well, he throws away the scalars (additively), so there are 24 non-scalar matrices. And then he throws away the scalars again (multiplicatively), to reduce to 8. But what does it mean to throw away the multiplicative scalars? It means, reducing from the Lie group SU(3) to its quotient PSU(3) = SU(3)/Z_3. Can this really be true? Is the gauge group of the strong force really PSU(3), and not SU(3)? How could we tell? A mathematician can tell the difference easily, but I am rather afraid that many physicists cannot, because they do not distinguish between the Lie groups (where the difference appears) and the Lie algebras (where there is no difference.

In other words, a physicist cannot tell that I have actually done anything at all. So a physicist cannot complain that I have broken their Standard Model, because I haven’t done anything to it as far as they can see. But a mathematician can see that I have made a huge difference, and I have turned the Gell-Mann matrices from a real Lie algebra into a finite group. I have turned chalk into cheese, and I have turned coloured water into flavoured wine. Let the party begin.