One of the early ideas about unifying strong SU(3) and weak SU(2) was to embed them in the (tensor) product 3 x 2, rather than the (direct) sum 3 + 2. The sum gives SU(3) x SU(2) inside SU(5), which was the Georgi-Glashow idea. When this idea seemed to fail, 3 x 2 seemed like a good alternative idea. And it was – the six types of quarks (3 generations of up and down) and six types of leptons (3 electrons and three neutrinos) naturally form a 3 x 2 structure. But the devil is in the detail. The generation symmetry is not a “real” symmetry, because the three generations have different masses, so what do you do about that? No-one had a good answer to that question, so interest gradually waned.
Somehow, the symmetries have to be broken. How? The Standard Model explains how the SU(2) symmetry is broken. The way it is “explained” in the textbooks is a disaster, but the mathematics behind it is simply that SU(2) becomes SU(1,1). It is just a single sign change in the structure constants of the algebra. But what about the breaking of SU(3) symmetry? We’e not talking here about SU(3) colour symmetry, but about SU(3) flavour symmetry. In some models this means up/down/strange quark “symmetry”, in others it means down/strange/bottom generation symmetry, preserving the charge but not the mass. If you really want to talk about mass, then you have to break the symmetry down to nothing at all, which isn’t very useful.
But physical reality certainly breaks the symmetry between the first generation, of stable particles, and the second/third generations, of unstable particles. So the first step in symmetry-breaking must be to split 3 into 1+2. One could for example break SU(3) to SU(1,2). When you then construct the 3×2 tensor product, you get SU(2,4) if you use unbroken weak SU(2), or SU(3,3) if you use broken weak SU(1,1). The latter is exactly what you get if you tensor broken weak SU(1,1) with unbroken strong SU(3), so it is not clear that any real progress has been made by this step.
A more drastic type of symmetry-breaking of SU(3) into 1+2 is a restriction to U(1) x SU(2) (modulo Z_2, for the technical experts). This breaks SU(3,3) into SU(1,1) x SU(2,2), with an extra U(1), so that there is a gauge group U(1,1) commuting with the conformal group SU(2,2). This is more or less what physicists ask for – a gauge group commuting with the conformal group, as the Coleman-Mandula theorem requires. But the SU(3) does not commute with SU(2,2). Is this a deal-breaker, or is it a misunderstanding of what SU(3) does? Is SU(3) a colour symmetry group, a flavour symmetry group, or some kind of hybrid of the two? What does it mean when physicists say that the structure of spacetime breaks down on the scale of the strong force (the internal structure of the proton)? Surely it means that the gauge group SU(3) of the strong force does not commute with the Lorentz group (or the conformal group) that describes the shape of spacetime? Why then do physicists insist that SU(3) commutes with the Lorentz group? The copy of SU(3) that I’m interested in clearly doesn’t. If they’re interested in a different copy of SU(3), then I suspect that group is describing a fictitious symmetry that isn’t relevant to the real universe.
Well, this isn’t a question about the nature of physical reality, it is a question about the type of model you want to build. The reason why I can’t get physicists to listen to my ideas about models has nothing to do with whether these models are right or wrong, it is purely because they don’t want to build the type of model that I believe they have to build in order to explain the multitude of experimental “anomalies” that contradict their current models. If they don’t want to build a type of model in which our perception of spacetime depends on the matter content of the universe we observe, then there isn’t much I can do about it. Never mind that this assumption, that spacetime depends on the matter content of the universe we observe, is the fundamental tenet of General Relativity. Particle physicists simply refuse to adopt this assumption as part of their models of physical reality. Conversely, I cannot get relativists or astronomers or cosmologists to consider the possibility that the nature of the “matter” content of the universe depends on the spacetime that we assume.
Anyway, enough of that. The point is to build a model using SU(3,3) in such a way that particle physicists can see that I have modelled electro-weak interactions correctly, if we ignore mass, and relativists can see that I have modelled gravitational interactions correctly, provided we ignore charge. Only then will they have a common language in which to argue about details. As things stand at present, they have completely different mathematical languages, that are fundamentally inconsistent with each other.
So here’s my basic plan. Write U(3,3) as U(1,1) x U(3), plus 24 extra dimensions that “mix” electroweak U(1,1) with electrostrong U(3), and throw away the scalars U(1) that (appear to) have no physical meaning. Break the symmetry of U(3) “matter” into U(1) x U(2) to split bosons U(1) from fermions U(2). Tensor with U(1,1) to get the splitting U(1,1) x U(2,2) that describes electro-weak mixing exactly the way it is done in the Standard Model. Now consider the generation symmetry, which initially appeared as a continuous symmetry of the three dimensions of SU(3), but which is actually a discrete symmetry and therefore requires only one complex dimension (e.g. for leptons) or two (e.g. for quarks). This generation symmetry can therefore be written in terms of U(1) x SU(2), or U(1) or SU(2) separately.
There are many possibilities for how to describe the details, but what matters here is that the mathematics is rich enough to do it. I’ve made many suggestions, which may be wrong in detail, but which are far from exhaustive of the range of possibilities the mathematics supplies. But don’t come out with this crap that we need to invent new mathematics to deal with the complexities of fundamental physics. Yes, we need mathematics that physicists don’t know. But just because they don’t know it, that doesn’t mean that mathematicians don’t know it. We do. The mathematics they need may be “new” to them. But it is not new to us. They just need to learn a bit more mathematics, and learn how to use it properly.
The biggest mistake that was made in the beginning, was to identify the spin group SU(2) with the double cover of the rotation group SO(3). This identification leads to a whole range of unhelpful “intuitions” about quantum mechanics, all of which are refuted by experiment. The only way out of this problem is to realise that SU(2) and SO(3) are completely different ways to describe spacetime. In the context of my suggested SU(3,3) model, the quantum mechanical description of spacetime comes from breaking the symmetry to SU(1,1) x SU(2,2), and the general relativistic description comes from breaking the symmetry to SO(3,3). You can see in there the SU(2) and SO(3) respectively. You can extend SU(2) to SL(2,C). You can extend SO(3) to SO(3,1). You can call both of these groups the Lorentz group, if you like. Then you’re stuck. Really stuck. Completely stuck. For 100 years. With no prospect of ever making any progress. Utterly stuck. No way forward. Absolutely none. Until you recognise that they made a mistake.
The only way forward is back.