For over 50 years, since the establishment of the Standard Model of Particle Physics as the accepted description of the quantum forces of electromagnetism and the nuclear forces, physicists have been looking for a theory that unifies these forces into a larger but simpler model. Typically, the idea is to find a large simple Lie group that contains the small complicated Lie groups of the Standard Model. There are two flavours of unification: first, the Grand Unified Theory (GUT) paradigm, which unifies the gauge groups into a compact real Lie group, and second, the Theory of Everything (TOE) paradigm, which unifies a GUT with the Lorentz group into a non-compact real Lie group.
Now there are not very many simple Lie groups to choose from, so we can examine them all. Simple Lie groups are classified into (a) orthogonal/spin, (b) unitary, (c) symplectic and (d) exceptional. Early GUTs were either orthogonal (such as the Georgi SO(10) model – really Spin(10), of course) or unitary (such as the Georgi–Glashow SU(5) model). Later GUTs and TOEs have included exceptional groups, such as E6 or E8. As for symplectic models – I don’t recall ever seeing one, although someone must have thought of this idea before. The orthogonal/spin case is popular, because it is easy to include Spin(3,1) as the Lorentz group, but on the other hand it is hard in this case to explain the strong force gauge group SU(3), which is not a spin group. The exceptional case is really just an extension of the spin case, since G_2 is too small, and F_4, E_6, E_7 and E_8 contain Spin(9), Spin(10), Spin(12) and Spin(16) respectively, and these spin groups are fundamental because they define the splitting between fermions and bosons.
The unitary case has also been popular, because the Standard Model gauge groups are unitary, and the Lorentz group Spin(3,1) can be extended to the “conformal group” Spin(4,2)=SU(2,2), which is unitary. This was Penrose’s idea, and forms the basis for my SU(7,2) model, splitting to SU(3) x SU(2) x SU(2,2) plus a couple of copies of U(1), and embedded in E8 also. But SU(7,2) suffers from the same problems as the Georgi-Glashow SU(5) model. That leaves the symplectic groups, and my arguments in favour of Sp(3,1) that I have put forward in recent posts.
So now let us look at the groups systematically. The three families of Lie groups are the simple/special orthogonal group SO(n), with dimension n(n-1)/2, the simple/special unitary groups SU(n) with dimension (n+1)(n-1), and the symplectic groups Sp(n) with dimension n(2n+1). The symplectic groups are already simple, and the orthogonal groups are almost simple (missing only a factor of 2 or 4), but the full unitary groups U(n) have an extra dimension, taking them up to n^2. The smallest simple Lie group has dimension 3, and has three names: Spin(3) = SU(2) = Sp(1). The only other pairs of aliases are Spin(5) = Sp(2) and Spin(6)=SU(4), and we must also note that Spin(4) = SU(2) x SU(2) is not simple.
Have a look at these lists of dimensions:
- SO(n): 0,1,3,6,10,15,21,28,36,45,…
- SU(n): 0,3,8,15,24,35,…
- Sp(n): 3,10,21,36,…
The three occurrences of 3 are the same group. We have 6=3+3 in the cases Spin(4) and Spin(2,2), but not for Spin(3,1). The two 10s are the same, as are the two 15s. Other than that, any coincidence of numbers is pure numerology, and does not extend to an isomorphism of groups.
A traditional GUT is a compact simple Lie group that contains U(1) x SU(2) x SU(3). If it is orthogonal, then we need at least SO(9), compared to SO(10) that Georgi proposed in 1974. If it is unitary, then we need SU(5), as proposed by Georgi and Glashow in 1974. If it is symplectic, then we need Sp(4). The consensus is that SU(5) fails, because it predicts proton decay, and therefore SO(10), and E6, also fails for the same reason. I’m not sure about SO(9), and its extension to F4. I’m also not sure that Sp(4) has ever been considered seriously enough to actually be ruled out. So let’s keep Sp(4) and SO(9) – really Spin(9) – in consideration for now.
If we want a TOE, we should add the Lorentz group to these, which extends SO(9) to either SO(12,1) or SO(10,3). These both have 78 dimensions, the same as E6 – a curious coincidence, but probably nothing more. Often people propose SO(12,4), because that embeds in E8 and looks nice. But we have severe issues of interpretation to try to avoid the proton decay problem. It may be possible to find a good interpretation, but in my opinion, nobody has found one yet. In the unitary case, we extend SU(5) to SU(7,2), of dimension 80, which I have looked at in detail, and although it looks promising in many ways, it still doesn’t avoid the proton decay problem – although I have provided an interpretation that might work. Finally, Sp(4) would have to extend to Sp(5,1), which has 78 dimensions. It is curious that this is again the same as the dimension of E6.
Other types of models include (a) emergent spacetime models, in which the Lorentz group emerges from the GUT without having to be added in separately; (b) mixed models, in which the gauge groups do not commute with each other, but still commute with the Lorentz group; and (c) combined emergent/mixed models, in which the gauge groups fail to commute with each other, and fail to commute with the Lorentz group.
The issue with (a) is that the Lorentz group is not compact. Hence we have to change the signature in some way. Assuming the individual gauge groups are still compact, this allows us to use SO(6,3) or SU(3,2) or Sp(3,1). All these have various copies of the Lorentz group in them, that might allow for an emergent spacetime. It is easy to see that they all contain SO(3,1), and the unitary and symplectic cases also contain a (completely different) copy of Spin(3,1) inside SU(2,2) or Sp(1,1) respectively. In the orthogonal case, we are really looking at Spin(6,3), and therefore at Spin(3,3)=SL(4,R), where again we have both SO(3,1) and Spin(3,1)=SL(2,C) subgroups. So potentially any one of these cases might support an emergent spacetime – we use Spin(3,1) for the Dirac equation, and SO(3,1) separately for relativity. I have described many of these emergent spacetime models over the years. They all have the property that they have two different versions of the Lorentz group – the macroscopic SO(3,1) is related only mathematically, and not physically, to the microscopic Spin(3,1). The orthogonal and unitary cases both fit inside SU(3,3), and thence into E6, E7 or E8. The symplectic case also fits inside E8. But the people who are interested in E8 are not interested in emergent spacetime, and vice versa, so this idea falls on deaf ears. And worse, the fact that two separate Lorentz groups are required turns people off, despite the fact that this is a mathematical necessity for any unification of relativity with quantum mechanics, as I proved already ten years ago.
So let’s consider (b), where the gauge groups fail to commute with each other. I fail to understand why people don’t consider this idea seriously, because we actually know that the U(1) gauge group of electrodynamics fails to commute with the SU(2) gauge group of the weak force. So why does the standard model invent a new gauge group called “hypercharge” that does commute with SU(2)? It seems an unnecessary complication. Anyway, if we allow SU(2) and SU(3) also to fail to commute then we can reduce SU(5) to SU(4), which has two advantages over SU(5): first, it removes the proton decay problem; second, it extends the gauge group from 12 dimensions to 15, so we can use the extra 3 dimensions for mixing angles between the weak and strong forces. If we apply the same idea in the orthogonal case, we can squeeze things down to SO(6), also 15 dimensions, and in fact the same group, since Spin(6) = SU(4). Finally, in the symplectic case, we can squeeze down to Sp(3), with 21 dimensions. In this case we get 21-12=9 mixing angles, which is the total number of mixing angles in the Standard Model. This fact argues strongly for Sp(3) as the correct unified gauge group in the paradigm (b).
Finally, let’s consider (c). Here we have the problem that if we change the signature of the unified gauge group, we inevitably change the signature of one of the Standard Model gauge groups. This is usually considered to be a fatal flaw, but if we consider the possibility for a moment, then we can change SO(6) to SO(3,3), or SU(4) to SU(3,1), or Sp(3) to Sp(2,1). The first of these is a model I have advocated for years, but nowadays I don’t think it is big enough. The other two cases only have one type of Lorentz group: SU(3,1) contains SO(3,1) but not Spin(3,1), while Sp(2,1) contains Spin(3,1) but not SO(3,1). In particular, SU(3,1) cannot support a Dirac equation, as well as being too small to contain all the necessary mixing angles. Even Sp(2,1), with 21 degrees of freedom, seems to be too small. So all in all, (c) seems to be too good to be true.
To summarise, then, in paradigm (a) we have options Spin(6,3), SU(3,2) and Sp(3,1); in paradigm (b) we have Spin(6) = SU(4) and Sp(3); and in paradigm (c)=(a)+(b) there is nothing plausible. Let me introduce a fourth paradigm (d) in which we combine (a) and (b) without changing the signature of (b). In paradigm (d) we have two options: Spin(6,3) and Sp(3,1). Both have 36 dimensions. They are of Lie type B4 and C4 respectively – they have the same Dynkin diagram, apart from swapping the short roots with the long roots. They are difficult to tell apart locally – as an aside, the computational problem of distinguishing B_n from C_n in the finite group case is hard, and was solved by reducing to the centralizer of an involution – which in physics terms means distinguishing between fermions and bosons. So why do I think that C4 is a better option than B4 for a model of physics?
The difference appears when we embed the Lorentz group Spin(3,1). In both cases the centralizer of Spin(3,1) is Spin(5)=Sp(2), but then the centralizer of the latter is Spin(3,1) in the B4 case, but Spin(4,1) in the C4 case. Hence the C4 case permits a Dirac equation that commutes with Spin(5), whereas the B4 case only allows a Dirac equation if we break the symmetry of Spin(5) to Spin(4). Does this argue in favour of C4? I’m not sure. What if we look at the SO(3,1) version of the Lorentz group? This embeds in Spin(6,3) via Spin(3,3), so centralizes Spin(3) out of Spin(5), as well as Spin(1,1) in Spin(3,3). Similarly, it embeds in Sp(3,1) in the natural way, so commutes with the quaternionic scalars Sp(1)=Spin(3), but not with any real scalar Spin(1,1). So the question is, do we need this real scalar or not? It seems to appear in the Standard Model as gamma_5. Does this argue in favour of B4? I’m not sure. So we are left with my original argument for the paradigm (b), in which compact C3 is the only option that has room for all 9 mixing angles from the Standard Model. This means that only C4 is capable of covering the whole of the Standard Model, with mixing and emergent spacetime.
Take home message: having considered all the finite-dimensional real simple Lie groups as possible gauge groups for a unified model, we have eliminated all possibilities except Sp(3,1). Having eliminated the impossible, whatever is left, however implausible, must be the truth.