Hermitian or anti-Hermitian? That is the question

There is a cultural difference between physicists and mathematicians, whereby physicists use Hermitian matrices to generate Lie algebras, whereas mathematicians use anti-Hermitian matrices. Why do they do this, and does it matter? Mathematicians use anti-Hermitian matrices, and define the Lie bracket of two matrices as [A,B]:=AB-BA (or sometimes BA-AB or (AB-BA)/2, but these differences don’t matter). This convention works because AB-BA is always anti-Hermitian, if A and B are either both anti-Hermitian, or both Hermitian. Physicists want [A,B] to be Hermitian, so they have to define [A,B]:=i(AB-BA), where i is a complex square root of -1. This converts the complex anti-Hermitian matrix AB-BA into the complex Hermitian matrix i(AB-BA). This convention works because the number of independent Hermitian matrices is equal to the number of independent anti-Hermitian matrices, that is n^2 for n x n matrices.

But this conversion between Hermitian and anti-Hermitian does not work for real (i.e. orthogonal) Lie algebras, or for quaternionic (i.e. symplectic) Lie algebras. This is because the numbers are different. In the real case, there are n(n+1)/2 independent Hermitian and n(n-1)/2 independent anti-Hermitian matrices, so more Hermitian than anti-Hermitian. In the quaternionic case, there are n(2n-1) Hermitian and n(2n+1) anti-Hermitian, so more anti-Hermitian than Hermitian. Both cause insurmountable problems for physicists, who try to create Lie algebras that are either too large or too small, by using Hermitian instead of anti-Hermitian matrices.

Let’s look first at the real case, where if you want to convert Hermitian into anti-Hermitian matrices you have to increase n to n+1. Physicists actually do this. They look at an electromagnetic field in three dimensions, and think of it as (real) Hermitian 3×3 matrices, so having 3×4/2=6 dimensions altogether, but when they calculate with it they convert to 4×4 (real) anti-Hermitian matrices (again having 4×3/2=6 dimensions altogether), and then multiply the extra dimension (time) by i to make the electric field (but not the magnetic field) Hermitian. It’s a mess, but it works, so physicists don’t care.

You can see it in the theory of gravity as well. General relativity works with symmetric (i.e. Hermitian) rank 2 tensors over spacetime, so with 4×5/2=10 dimensions. Kaluza-Klein gravity recognises that these 10 dimensions ought to form a Lie algebra, but they don’t, so extends spacetime to 5 dimensions, so that they can use the 5×4/2=10 dimensions of anti-symmetric (i.e. anti-Hermitian) rank 2 tensors instead. But they don’t correct the unnecessary and confusing multiplication of time by i, so they don’t get the correct Lie algebra, and so they don’t get a correct theory of gravity.

From a mathematical point of view, therefore, electromagnetism is best modelled with the Lie algebra so(4) of anti-symmetric real 4×4 matrices, without any of this spurious multiplication by i that makes time imaginary in the standard theory. Gravity is similarly best modelled with the Lie algebra so(5) of anti-symmetric real 5×5 matrices, not the symmetric 4×4 real matrices that Einstein used. But physicists are absolutely convinced that matrices that are not Hermitian do not exist, so they are incapable of making this leap. What they don’t see is that if you want a real differential equation to describe real physical quantities, then the Lie algebra that describes the infinitesimals actually does contain anti-Hermitian matrices.

Now let’s look at the quaternionic case. Here the situation is even worse, because you cannot rescue the situation by inventing an extra dimension of space or time. In one dimension, you have one Hermitian matrix, and three anti-Hermitian. If you work with Hermitian matrices, as physicists do, then you only see one generation of electrons. If you work with anti-Hermitian matrices, as mathematicians do, you see all three generations. If you are not prepared to make the leap to anti-Hermitian matrices, then you will never find a theory that deals adequately with the three generations. And when I say never, I mean NEVER.

In two dimensions, there are 6 Hermitian and 10 anti-Hermitian (independent) quaternion matrices. The Lie algebra you get from the anti-Hermitian matrices is in fact the same as the one you get from the real 5×5 anti-Hermitian matrices, so you can translate from one notation to the other if you like. This enables you to translate between quantum mechanical notation (2×2 quaternions) and gravitational notation (5×5 reals) and eventually (hopefully) unify the two approaches. At any rate, you’ve got the same Lie algebra in both cases, and therefore the same differential equations in both cases. But if you are not prepared to make the leap to anti-Hermitian matrices, then you will never find a theory that deals adequately with the unification of gravity and particle physics. And when I say never, I mean NEVER.

Of course, you need to go up to three dimensions to understand how space and the strong force work. Here there are 15 Hermitian and 21 anti-Hermitian matrices. The anti-Hermitian ones are the ones you need to generalise strong su(3) to three generations. Naively, you might expect to need 8 x 3 =24 dimensions, but it seems that only 21 are available. So there are 24-21=3 parameters over which you have no control, corresponding to the three quark-generation-mixing parameters of the standard model.

And finally, in order to make the theory relativistic, you have to go to 4 dimensions, where there are 28 Hermitian and 36 anti-Hermitian matrices. People who are interested in GUTs like the number 28, because it is the dimension of so(8), and they love so(8) because of its exotic properties like triality. But unfortunately these 28 matrices are Hermitian, and therefore they do not generate a Lie algebra, so they do not generate so(8). Any resemblance of so(8) to real physics is purely coincidental. The Lie algebra you actually need to describe the whole of fundamental physics is sp(4), and the Lie algebra you need to make this model relativistic is sl(4,H). The compact part, sp(4), is generated by the anti-Hermitian matrices. The Hermitian matrices extend to gl(4,H), and deal with relativity.

The situation is analogous to Dirac’s relativistic spin group SL(2,C), or rather its Lie algebra sl(2,C). The non-relativistic spin group su(2) is generated by the anti-Hermitian 2×2 complex matrices, which also generate a scalar u(1), and the extension to sl(2,C) is generated by Hermitian 2×2 complex matrices, which also generate a real scalar. But extending from Dirac’s 6 real dimensions, or his 16 complex dimensions in the Dirac algebra, to 16 quaternion dimensions, allows us to include three independent generations, three independent directions of spin, and three independent directions of quantised gravity, all in a 36-dimensional compact Lie group.

So which is it to be, Hermitian or anti-Hermitian? Over to you, Prince of Denmark.