The first step in seeing how a broken E8 can arise by treating composite particles as if they were elementary consists in understanding how it is that a quantum state can be interpreted as a symmetry operator. That is the subject of this post. The remaining steps are outlined in the previous post.
We begin with the Pauli algebra. Since this is an introductory post, we will use the notation the majority of my readers have already learned, that of the Pauli spin matrices and Pauil spinors. I have found that students learn best by example, so we will turn the spinor for spin in the +x direction into a symmetry operator on the quantum states of the Pauli algebra:
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.