One of the big problems of theoretical/mathematical physics is to explain the chirality of elementary particles – specifically, the left-handed nature of the weak interaction. Most models find it difficult to get rid of the inherent symmetry between the ordinary particles and the so-called `mirror fermions’, for which there is precisely no experimental evidence. They therefore either make egregious claims for the existence of vast swathes of new particles and new physics, or they invoke some kind of magic to kill them off.
One of the big problems of theoretical/mathematical physics is to explain the existence of three generations of elementary fermions – in particular, the existence of the muon. Most models find it difficult to introduce a generation symmetry between the ordinary particles and the `strange’ and exotic particles, for which there is incontrovertible experimental evidence. They therefore either restrict themselves to describing a single generation, and leave out vast swathes of old particles and old physics, or they invoke some kind of magic to pull them out of a hat.
Do these two problems look like mirror images of each other to you? They do to me. The first problem is that there are too many fermions in the model, the second problem is that there are not enough fermions in the model. But they don’t cancel out, because in the first case there are twice as many fermions as needed, while in the second case there are only one-third of the number required. So to get the number of fermions that experiment tells us actually exist, we need a `magic mirror’ that gives you two reflections at once. A mirror, so to say, that shows you not yourself, but your father and your mother.
Of course, this magic mirror isn’t a physical mirror, any more than the `mirror fermions’ are images in a physical mirror. It’s just a mathematical device that behaves like a mirror. Can we make such a magic mirror in the mathematics of the Standard Model of Particle Physics? Or in a Grand Unified Theory such as an E8 model? Yes, of course we can, and I’ve already told you several times how to do it.
In the real world, a mirror is something that gives you a reflection of something, and if you reflect again you get back to where you started. In the complex world of particle physics, a mirror is something that gives you a reflection, and if you reflect again and again you must eventually get back to where you started. But there is nothing to say whether you have to reflect twice, three times or more. Well, no, that’s not true – the real universe is telling us that we have to reflect exactly three times, not twice, not four times, to get back to where we started.
The remarkable thing is that this magic mirror already exists in the standard model. But for some reason no-one has noticed that it is there. I don’t really understand why no-one has noticed, but I think it is mainly to do with the fact that physicists regard Lie algebras and Lie groups as being equivalent. Lie algebras do not contain mirrors, magic or otherwise. So that when physicists want to talk about mirror fermions they tend to invent things like “Lie superalgebras” in order to introduce the mirrors artificially. But Lie groups do contain mirrors, and some of them are magic.
For example, the Lie algebra su(3) is used to describe the strong force, in which massless gluons stick quarks together into particles like the proton and neutron. The Lie algebra doesn’t see the three generations, so it doesn’t see the mass of the particles, and it doesn’t see the magic mirror that describes the generation symmetry. But the Lie group SU(3) contains a subgroup of scalars – of order 3 – that actually is the magic mirror. So it baffles me why the Standard Model throws the magic mirror away.
The full gauge group of the Standard Model is described, by those physicists who are pedantic enough to insist on a specific group rather than just a Lie algebra, as (U(1) x SU(2) x SU(3))/Z_6. What this means is that there is a group U(1) of complex scalars of unit modulus, and inside SU(2) there are scalars of order 2 (i.e. the scalar -1), and inside SU(3) there are scalars of order 3, arranged so that the scalars in SU(2) and SU(3) are considered to be equal to the corresponding scalars in U(1). This begs two questions: first of all, when you match up the two scalars of order 3, which way round do you do it? Do you just pick the first one you think of, or do you swap one of the two scalars with its inverse? Second of all, why do you have to match them up at all, why not just use the full group U(1) x SU(2) x SU(3), instead of throwing away the information that is in the Z_6?
To take the second question first, it seems utterly bizarre to throw away a symmetry of order 3, and then complain that you can’t find the generation symmetry of order 3 in the Standard Model. Duh! The reason it’s not there is because you deliberately threw it away. Why was it thrown away? I think it probably goes back to the Georgi-Glashow model (1974) based on SU(5), since SU(5) does not contain U(1) x SU(2) x SU(3), but does contain (U(1) x SU(2) x SU(3))/Z_6. Since that time, it seems to have become accepted that the latter is the “correct” gauge group, even though it is not accepted that the Georgi-Glashow model is “correct” – since it predicts proton decay, which has never been observed. So that is the gauge group that is used in the Standard Model, but that doesn’t stop other Grand Unified Theories from using the full group.
And what about the first question? Which way round do you match up the two symmetries of order 3? In the Standard Model it matters, because changing your decision means taking the complex conjugate of one part of the model but not the rest. This has the effect of changing a left-handed theory to a right-handed theory, and is forbidden by the laws of the Standard Model. But in a bigger theory, you may not have to make a decision at all. For example, in the Manogue-Dray-W version of an E8 model, the group (U(1) x SU(3))/Z_3 is replaced by U(1) x SL(3,R), in which there is no such decision to be made. In this model, the chirality of the weak force appears in a completely different place, and the strong SU(3) has nothing to do with chirality. As you would expect.
So, you physicists who want to understand the three generations of fermions, look in the mirror. When you see both your parents staring back at you, you will understand.