Over the course of the past year or so I seem to have converged on the idea that the fundamental algebra that is required for a theory of “everything” is the tensor product of three copies of the quaternions. As you know, Hamilton introduced the quaternion algebra in 1843 for the purpose of describing physical space, and the classical mechanics that takes place within physical space. Newtonian mechanics can be, and was, written entirely using one copy of the quaternions. Quantum mechanics requires two copies, as does Einstein’s theory of gravity. But you can’t use the same two copies for both theories at the same time – Einstein knew that. Hence the need for three.
The picture is somewhat complicated by the fact that Maxwell’s theory of electromagnetism (or, equivalently, the Special Theory of Relativity) requires one and a half, which when quantised extends to two and a half. The Standard Model of Particle Physics, therefore, is based on two and a half copies of the quaternion algebra. It has been clear for a long time that in order to accommodate the three generations of electrons into the Standard Model, it is necessary to extend a half copy of the quaternions (i.e. a copy of the complex numbers) to a whole copy. But it has not been clear (and probably still isn’t clear) exactly how to do this. And it certainly has not been clear (and still isn’t) whether this is enough.
To recap briefly, Hamilton’s quaternions were invented/discovered for the purpose of multiplying space by itself. He discovered that if you wanted an algebra of space coordinates x, y, z then you were forced to have xy = -yx, and xx = -1. In particular, you were forced to introduce a fourth coordinate, which is related to time. So we may as well write our quaternions with coordinates (x,y,z,t), and define xx=-t.
Now quantise the theory, so that x,y,z,t become infinitesimals, and xx=-t becomes a differential equation – differentiate by x twice on the left hand side, and once by t on the right hand side. This is the one-dimensional Schroedinger equation, which is the basis of quantum mechanics. The reason you need a second copy of the quaternions at this point is that you need something to differentiate. You can call that something a “wave-function” if you like, but it doesn’t really exist, so this way madness lies.
The madness arises from watching the film of 24 still frames per second (or whatever the current standard is), and insisting that the picture on the screen is physically moving. Of course it isn’t – we know how “moving pictures” work, and we know that the movement is an illusion. Unfortunately, we don’t apply that knowledge where it is most important – in the interpretation of quantum mechanics.
Be that as it may, the Einstein picture of gravity suffers from the same defect: as the film runs, the shapes of things appear to change. This is an illusion, of course. But not according to the textbook interpretations of General Relativity – apparently, the shape of spacetime itself actually does change, or bend, or “curve”. How on earth is it possible to come up with such nonsense? We know how movies work, why don’t we apply that knowledge where it is most important – in the interpretation of gravity?
Fundamentally, GR is based on the tensor product of two copies of spacetime, or, equivalently, two copies of the quaternions. You won’t find the quaternionic formalism in the textbooks, but the quaternions are really there. Let’s take two copies of spacetime, say x,y,z,t and X,Y,Z,T, and remember that t and T are real (square to +) and x,y,z,X,Y,Z are imaginary (square to -). Hence the “imaginary” terms in the tensor product are tX, tY, tZ, Tx, Ty, Tz (product of one “real” and one “imaginary” term). The other ten are real: tT, xX, xY, xZ, yX, yY, yZ, zX, zY, zZ. The ten real terms are the ones that Einstein uses to write down the ten Einstein field equations.
So is it not clear that if you want to quantise gravity, you have to introduce a third copy of the quaternions? Is it not clear, moreover, that that is all you have to do? We need, therefore, the algebra H x H x H, where H (for Hamilton) denotes the quaternion algebra, and x denotes the (real) tensor product. Since the quaternion algebra is 4-dimensional (the clue is in the name), the whole algebra has dimension 4x4x4=64.
A year ago, I explained how to quantise this algebra, with a finite group that makes everything discrete. It is this finite group that essentially proves there is no alternative. In order for the strong force (described by the Gell-Mann matrices) to act on elementary particles in spacetime (described by the Dirac matrices), the Gell-Mann and Dirac matrices must be unified into a single finite group, and the only group available for this purpose is the one I described. Along the way, the argument proves that the two and a half copies of the quaternions that make up the Dirac algebra H x H x C must be extended to three.
Therefore I have two completely independent arguments, one from gravity, one from quantum mechanics, both proving that the algebra required for quantum gravity is H x H x H. Of course, physicists don’t like mathematics, and they really don’t like quaternions. But what they really really hate more than anything is the Gruppenpest. So any argument that starts out from the Gruppenpest is rigorously ignored, and simply does not exist as far as they are concerned. Therefore the strategy adopted by physicists who want a new model of physics is to guess the answer, instead of using mathematics.
To be fair, many physicists do know that the Dirac algebra, that they use for electro-weak theory, has the structure H x H x C. They know also that they can’t fit the strong force, with gauge group SU(3), into the same algebra. But physicists who know about quaternions, and understand that the electro-weak gauge group U(1) x SU(2) fits into C x H, also know about octonions. Hence they want to fit the strong gauge group SU(3) into octonions. Never mind that the octonions are non-associative, so do not contain any groups, apart from the ones that already fit inside the quaternions.
The idea of extending H x H x C to O x H x C seems to have occurred first to Geoffrey Dixon, 30 or 40 years ago. Today, Nichol Furey is busy re-inventing this particular (square) wheel. Why? I have no idea. It has already been tried, and it really doesn’t work. The idea of using O x O instead, and using E8 as a receptacle for O x O, is of roughly the same vintage, or perhaps even older. Many varieties of E8 theories have been proposed over the decades, but they really don’t work. Even string theory uses E8, and that really really doesn’t work.
You remember Benson’s Algorithm? This is a universal algorithm that can be applied to absolutely any problem whatsoever:
- Try something at random.
- If it doesn’t work, go back to Step 1.
This is the algorithm that I use to design my models of physics. I long ago lost count of how many times I have been around this loop, but I tend to reach Step 2 every few months at most. I’ve been back to Step 1 so often, that I’ve tried the same (random) thing more than once. Most certified crackpots, and a lot of certified physicists too, never reach Step 2 at all. The standard algorithm that is applied (for example in string theory) is
- Try something at random.
- If it doesn’t work, carry on regardless.
- Go back to Step 2.
It doesn’t take a genius to realise that this algorithm doesn’t terminate. Unlike Benson’s algorithm, which will eventually find a solution, if a solution exists. Notice, however, that Benson’s algorithm does not test to see if the idea has been tried before. This is not a bug, it is a feature. It is absolutely vital that you do not dismiss an idea just because it’s been tried before and didn’t work. If you do, you get another algorithm that doesn’t terminate. That is why I keep coming back to E8, even though I “know” that E8 doesn’t work. Actually what I know is, that E8 didn’t work last time. I do not know, and can never know, that it won’t work next time.
Nowadays, Benson’s algorithm is called “Artificial Intelligence”. We can be fairly sure, then, that AI will eventually find the ultimate theory of everything. Just as long as the programmers don’t make the mistake of making the programme learn from its mistakes. Every theory that has ever been useful in physics has always been proved wrong before it even started. It is therefore absolutely essential to take no notice whatsoever of any such “proof”. If you eliminate all the theories that have already been proved wrong, then you are left only with the theories that can never be proved wrong (like string theory). A theory that can never be proved wrong is, as Karl Popper explained, not a scientific theory.
But I digress. What I’ve really been trying to explain today is that three is the magic number. Not two (as in Einstein’s theory of gravity) or even two and a half (as in Dirac’s theory of relativistic quantum mechanics), but three. The tensor product of three quaternions, H x H x H, is as simple as possible, but no simpler. It contains everything physical, as I have already explained, and there is nothing smaller that contains everything.