Today I want to talk about the differences between the representation theory of Lie groups over complex numbers, and the representation theory of finite groups over bits. Over complex numbers, representations (of finite or simple groups) always split as a direct sum of irreducible representations, so that one only ever needs to consider irreducible representations. Over bits, this is no longer true, so that the irreducible constituents of a representation do not have to sit side-by-side, but can be stacked on top of each other in arbitrarily complicated arrangements.

To illustrate the difference, compare SU(2) acting on a pair of complex numbers, with Sym(2) acting on a pair of bits. In the former case, there is an infinite series of irreducible representations, of dimensions 1, 2, 3, 4, …, where 1 is the trivial representation (no action), 2 is the spinor action, and 3 is the vector action, that is as SO(3). The important fact for physics is that 2×2=3+1. If, for example, this SU(2) represents the weak interaction, then the 2 represents the doublets of fermions (electron/neutrino, up/down quark, proton/neutron, etc.), while the 3 represents the Z, W^+ and W^- bosons. The challenge for physics is then to explain the “symmetry-breaking” between these three bosons, as well as the photon in the 1 representation.

Now consider instead the action of Sym(2) permuting two bits. Then the spinors are 00, 11, 10 and 01, and the symmetry breaking as 1+1+2 is obvious. So we can take 00 to represent the photon, 11 to represent the Z boson, and 10 and 01 to represent the W bosons, and Sym(2) representing charge conjugation. This representation is not irreducible, because there is an irreducible sub-representation on the even bit strings 00 and 11. But if the representation were a direct sum of irreducibles, these would both be trivial, and the whole representation would be trivial. So let us write this 2-dimensional representation as 1.1, to denote that it has a trivial sub-representation, with another trivial representation glued on top.

Physically, the representation on the bottom has mass, and the representation on the top has electric charge. This is an absolutely crucial property of charge in physics: it can never exist on its own, it always has to exist glued on top of mass.

Now let’s move up a gear, and consider the action of Sym(4) on 4 bits. we still have an “on switch” 1111 at the bottom of the representation, and we still have an “odd part” at the top of the representation. In the middle, we have a triplet of “even bits modulo complementation”, that is 0011/1100 and 1010/0101 and 1001/0110. The odd part similarly consists of four pairs of a single bit and the complementary three bits, thus 1000/0111, 0100/1011, 0010/1101 and 0001/1110. Now this representation can be used for many different things in quantum physics, depending on what four things one wants to study. For example, one could use 1111 for the Higgs boson, or some other place-holder for mass, and use the 3 even pairs for the Z and W bosons, now endowed with an extra “spin” bit to reflect the fact that they have spin 1, not spin 0. Then on top we could have four fermions, again each with a spin bit. Symmetry-breaking of the weak interaction then requires us to break the symmetry down to Sym(2) x Sym(2), so that we have two weak doublets of fermions, for example electron/neutrino and proton/neutron. (Incidentally, I leave it as an exercise for the reader to show that the sum of the masses of the even part (Z, W^+ and W^-), minus the masses of the odd part, is equal to twice the mass of the Higgs boson.)

The fun starts with 6 bits permuted by Sym(6). This group has only four irreducible representations on bits. One is the trivial 1-dimensional representation. There are two distinct representations on 4 bits, that I want to call the vector representation V and the spinor representation S. The fourth irreducible representation is the tensor product of V and S. For applications to physics, one wants the tensor product of S with itself. The structure of this tensor product happens to be a single tower with 7 floors, arranged as 1.V.1.S.1.V.1. This is the finite analogue of the Clifford Algebra used in the standard model, which has a structure like 1+V+A+V+1, where A is the anti-symmetric square of V. Indeed, in the finite case the anti-symmetric square of V is 1.S.1, so the matching of the irreducible parts of the Clifford Algebra works perfectly. But the finite case has a much more interesting structure, and permits the description of much more interesting quantum physics than is possible in the continuous case.

Since the anti-symmetric square of S is 1.V.1, we can obtain the vectors from the spinors in much the same way as is done in the standard model. But the vectors come with a scalar on the bottom (mass, remember), and a scalar on the top (charge). The vector cannot exist on its own without the mass, and the charge cannot exist without the vector and the mass. This is a good deal closer to how the real world behaves than is the standard model, in which mass and charge are optional extras.

The most astonishing thing about Sym(6), however, is its outer automorphism, that swaps V with S. Nothing like this exists for Lie groups, so that this “supersymmetry” between fermions and bosons cannot be expressed in the language of Lie groups. Indeed, as is well-known, attempts to model this supersymmetry with Lie groups led to predictions of all sorts of new and exotic particles, which experiment has not found. So it is pretty clear that Lie group supersymmetry does not describe the real universe. But I hope to convince you that Sym(6) supersymmetry does actually describe the real universe.

This supersymmetry allows us to construct the spinors from the vectors in the same way that we constructed the vectors from the spinors. That is, the anti-symmetric square of V has the structure 1.S.1. This is extraordinary. Nothing like this exists in the standard model. It means that in the foundations of quantum mechanics it is not necessary to dream up an abstract concept of spin that is superimposed on spacetime. The concept of spin emerges from the quantum nature of spacetime itself. Or vice versa, however you want to look at it. Moreover, the “spin” representation, like the “vector” representation, is not really 4-dimensional, but 6-dimensional. It has an extra on/off bit at the bottom, and an extra even/odd bit at the top. These extra bits allow the inclusion of the concepts of left- and right-handed spin, but as a single integrated concept, not as two completely divorced types of spin.

Both 1.V.1 and 1.S.1 are described by permutation representations of Sym(6). One of them can be taken as the usual representation on 6 letters A,B,C,D,E,F, and then the other one is the image under the outer automorphism. That makes for some difficult notation, but one can also switch to a different set of six letters U,V,W,X,Y,Z for example, and match them up later. There is loads of stuff written on this, that I don’t want to go into just now.

Let’s just study the details of one of these representations at a time. First there is the overall on/off switch 111111. Then there are 15 pairs and their complements 110000/001111 etc. If you add together two pairs that overlap, you get another pair (remembering that 1+1=0), but if you add together two pairs that don’t overlap, you get the complement of a pair. That is why you cannot separate the pairs from their complements. The odd bit strings and their complements are of two types, 6 of type 1/5 such as 100000/011111, and 10 of type 3/3 such as 111000/000111. (Another exercise for the reader: choose a physical interpretation of this model to show that the total mass of three generations of electrons and neutrinos, plus three protons, is equal to the mass of five neutrons.)

To get the standard model 1+4+1 from this 1.4.1, we need to break the symmetry in some way. In particular, we need to fix one of the odd pairs. In fact, the standard model fixes one odd pair of each type, so breaking the symmetry simultaneously to 1+5 and 3+3, in other words to 1+2+3. Well, I suppose this is the symmetry-breaking on the bosons (vectors). The corresponding symmetry-breaking on the other 1.4.1 representation is then 1x2x3, which therefore gives us the three generations of fermions. Even this is not enough symmetry-breaking to get the standard model, because the 2 still has to be broken to 1+1 in order to make 1+1+1+3 in the vector representation, which is then subjected to Lorentz transformations to make 1+1+4. By this time, I am afraid, so much violence has been done to the mathematics that it no longer makes any sense at all. By all means continue to calculate with the standard model as before. Just don’t try to pretend that all that hocus-pocus has anything to do with how things really are.

There are various grand unified theories that attempt to model fundamental particles by doing less symmetry-breaking. The Georgi-Glashow model, for example unifies 1+2+3 into 1+(2+3), while the Pati-Salam model unifies 2+1+3 into 2+(1+3). But they do this with Lie groups, SU(5) in the former case, and SU(2) x SU(2) x SU(4) in the latter. There are also models that attempt the full unification of (1+2+3) with SU(6). But, as I have shown, a Lie group model can never reproduce the structure of the actual quantum world, in which representations are stacked on top of each other.

To see how these unification ought to work, we need to look at both the vector and the spinor representations, not just one of them. First look at Georgi-Glashow 1+5. The group reduces from Sym(6) to Sym(5), and the actions on one copy of 6+10+15 are (1+5)+(10)+(5+10), while the actions on the other are then (6)+(10)+(15), i.e without any splitting. A Lie group model can only look at one of these, and Georgi-Glashow uses (1+5)+(10)+(5+10). They then throw away the scalar (which contains the mass), and identify the odd and even parts as real and imaginary parts of a complex number, thereby throwing away the charge also. They are left with a complex 5-space and a complex 10-space, and assume that the latter is the anti-symmetric cube of the former (which it is, in fact).

The Pati-Salam model, on the other hand, reduces the symmetry to Sym(2) x Sym(4), so that one copy of 6+10+15 breaks as (2+4) + (4+6) + (1+6+8), while the other breaks as (6) + (4+6) + (3+12).

A third possibility is to split as 3+3, in which case one can also include the symmetry that swaps the two 3s. In this scenario we get (6) + (1+9) + (6+9). I do not know if there is a grand unified theory that does this, but general relativity certainly does. So this is the one to study if you want to quantise GR.

Well, there is so much more to say about six bits, but I think this is enough for one day. I have shown you how 6 bits are enough to encode all the fundamental forces of nature separately. Now we need to unify them. See you tomorrow for the quantum theory of everything.