Since I think that the standard model particles are composite, it’s natural, with Garrett Lisi’s use of E8, to wonder if E8 can arise naturally as a result of composite particles. I think it can. I will use the fact that as Baez puts it, “This means that in the context of linear algebra, E8 is most simply described as the group of symmetries of its own Lie algebra!”
This suggests that we begin with the Pauli algebra and write down its (well known) quantum states, that is, the 2×1 spinors or the pure density matrices. In order to allow ourselves to bootstrap the states, instead of thinking of the base algebra as the Pauli algebra, we think of the base algebra as these quantum states, that is, we use Schwinger’s measurement algebra. In the measurement algebra, the primitive elements correspond to elementary particle quantum states. To get the power of the Clifford algebra, you can still define the Clifford algebra canonical basis vectors in terms of the measurement algebra.
Next we somehow associate each element of the measurement algebra with a symmetry operation on the Clifford algebra (and therefore also a symmetry operation on the measurement algebra itself). Since the measurement algebra is not equivalent to E8, it will not be its own symmetry group, so there will be symmetry operations that are not included in the measurement algebra.
We next examine one of the unused symmetry operations, that is we take a symmetry of the measurement algebra that is not associated with an element of the measurement algebra. Since we somehow have a way of relating the measurement algebra elements to symmetry operations on the algebra, we somehow might be able to reverse this operation, and given a symmetry operation on the measurement algebra, turn it into the density matrix for a quantum state. Since we already used up all the single particle states, this new quantum state will be a multi-particle state.
Next we add these multi-particle states to the measurement algebra as new “primitive” states. The measurement algebra is designed to do this. The product of two primitive measurements that correspond to different numbers of particles is zero. The sum is simply the measurement that allows either of the two states. This gives us a new, bigger, measurement algebra.
Now we look at this bigger measurement algebra. If its symmetry group is E8 we are done. If it is not E8, then there are symmetries of the bigger measurement algebra that are not associated with an element of the algebra. Since we’ve already added the primitive states and the multiparticle states, these new states must be multi-multiparticle states, that is, composites made of composites. So we add them to the measurement algebra.
We continue this recursive process. Each time we add new primitive measurements to the measurement algebra its dimension gets bigger. The process cannot grow indefinitely because eventually it will reach E8 and we will have quantum states for all the states in E8. Hence Lisi’s model.
Since bound states cannot physically be equivalent to the primitive states, the E8 that we get will be broken. We will get the unbroken E8 by working in qubit space and ignoring energy differences. In effect we will be making the unphysical assumption that all particles have he same energy, whether they are the primitive ones or the composites. Correcting this inaccuracy will not change the quantum numbers, only things like masses and coupling constants. So we will get E8 quantum numbers (as Lisi did), but not an exact E8 symmetry.
Note that if we began with the Dirac algebra instead of the Pauli algebra we would get a different tower, presumably the correct one. I will work out the first stage of the tower with the Pauli algebra because it is easier. The method should work for any Clifford algebra. They will all get to E8, but the pattern of symmetry breaking should be different. This could give a clue as to the number of true dimensions of spacetime.
I’ve got most of the “somehow” details worked out. Explaining it will take some time. In short, here are the things I need to spell out for the reader:
(a) Defining the measurement algebra in terms of the Clifford algebra is a simple process that is spelled out in chapter 6 of my book on Density Operators.
(b) The “somehow” relationship between symmetry operations on the Clifford algebra is at least broadly hinted at in various things I’ve written. Let U be an element of the Clifford algebra. Let v be a small real number. Let X be an arbitrary element of the Clifford algebra. Then the map:
A -> A’ = exp(-vU) A exp(+vU),
is a symmetry of the Clifford algebra. That is, any equation that is true for U, V, W, … is also true for U’, V’, W’, … The reason is that “->” is a linear map so addition is preserved, and exp(-vU) exp(+vU) = 1, so multiplication is also preserved. This is a 1-parameter subgroup of the symmetry of the algebra.
If it happens that U squares to 1, then you can write exp(vU) = cosh(v) + sinh(v) U. As v goes to infinity, this becomes approximately M(v) = exp(v)(1 + U). Density matrices M are traditionally normalized so that MM = M. Normalizing M(v) this way we get M(v) goes to (1+U)/2. Thus the density matrices (the quantum states) are the limit points of 1-parameter subgroups of the symmetry of the Pauli algebra.
And since the measurement algebra is built from the Pauli algebra, these same symmetry groups are symmetry groups of the measurement algebra. Furthermore, you see by the exponential that we are dealing with the conversion from group to algebra so the bootstrap is fitting precisely into the self-referential definition of E8.
(c) It remains to show how to make a composite state from one of the unused symmetries of the measurement algebra, and how to add these states to the algebra. Fortunately, in pursuing the Koide relations, we’ve learned how to make composite density matrices out of density matrices. What we’ve not yet done is shown how these composite density matrices are associated (by the Lie-group / Lie-algebra relationship) with symmetries of the measurement algebra (and equivalently, of the underlying Clifford algebra). Fortunately, I figured out how to do this on the plane flight from Seattle to Denver today and will blog it when I get the chance.
(d) I’ve not yet looked at the symmetry breaking in Garrett’s method to see what (if any) Clifford algebra will generate it. But if there is such a symmetry breaking, it can be used to read off the number of hidden dimensions, and the component elements of the known particles.