Welcome to my Humble Unified Theory

My HUT is your HUT. Beiti beitak. Welcome. Ahlan wa sahlan. Make yourself at home. Let me show you around. There isn’t much to see, but I think you’ll be impressed at how much I have managed to fit into a finite space.

First let me show you the framework: the 8 Gell-Mann matrices in the 8 corners of the room. I wrote them down explicitly for you in three dimensions in the previous post. Each of them is multiplied by three scalars represented by the beams and joists that hold the 8 corners of the HUT together. Add in my identity, and you get a group of order 27 that is the scaled-down essence of the Standard Model SU(3). But there is no need for all that S+M, just have a cup of tea: 27 tea-leaves should make a nice refreshing drink.

Next let me show you the walls. Standing in the front door, you can see three of them: the Pauli matrices. But they are mathematicians’ Pauli matrices, not physicists’ – they generate the quaternion group of order 8: three walls, top and bottom, and my identity, here at the front door to welcome you. This is the scaled-down essence of the Standard Model SU(2). But there is no need for all that S+M,just have a cup of coffee: 8 coffee beans should make a nice refreshing drink.

Have you noticed how the 8 corners (Gell-Mann matrices) are related to the 3 walls (Pauli matrices)? Isn’t it really simple and functional? Do you see how it makes the HUT stand up, instead of being a pile of bits and pieces, like the Standard Model, looking like an earthquake has hit it? Do you see how, instead of a direct product SU(2) x SU(3) as in the Standard Model, the frame and the walls act together to create a structure, a model HUT that one can live in?

Now look at the roof. A single structure, with three layers, to keep out the rain. No frills, just one matrix with three scales. In the Standard Model it’s called U(1), but in essence what is important is the three layers. Without the three layers, it is very hard to understand the three generations of fermions in the Standard Model. But no need for all that S+M, just have a glass of whisky: a triple shot of single malt will do nicely, thank you very much.

Have you noticed how the roof is joined to the walls? Not SU(2) x U(1) as in the Standard Model, but fixed together to make a structural whole? Not mixed together at a ridiculous `mixing angle’ as in the Standard Model, that changes as the wind blows, but fixed together properly so it is rigid and doesn’t move. So can you now see the beautiful simplicity of my HUT, that has everything you require, and nothing more? Frame, walls and roof. That is all.

It has the essence of SU(3) and SU(2) and U(1), triple-distilled and triple-filtered into a finite group, of order 27 x 8 x 3 = 648. It has a group of order 27 generated by (slightly modified) Gell-Mann matrices, a group of order 8 generated by (slightly modified) Pauli matrices, and a group of order 3 generated by (slightly modified) scalars on top. The Pauli matrices act on the Gell-Mann matrices to create a quark-mixing matrix and a lepton-mixing matrix, and the (unnamed) matrices on the top act on everything so that three generations can live under one roof.

For those who are technically minded, and want the full spec, this group of order 648 is the full unitary group of 3×3 unitary matrices written over the field of order 4. But the generators I have given you can also be interpreted as unitary matrices over the complex numbers, by mapping the four elements of the finite field to 0, 1 and the two primitive cube roots of 1. So now you can measure everything with real and complex numbers, work out the dimensions, cut everything to size, weigh everything, measure the angles, and build you own Standard Model to your own specifications.

But don’t forget – it is the finite symmetry group that tells you how to fit the walls to the frame, and the roof to the walls and the frame. You can’t do that with SU(3) x SU(2) x U(1).