Daily Archives: November 23, 2007

Non Hermitian Density Matrices

Recently Garrett Lisi’s used a broken E8 symmetry to classify the known elementary particles. I believe that E8 arises naturally from composite particles. The outline of the argument (as covered in the first post) amounts to noticing that E8 is unique among the simple Lie algebras in that “in the context of linear algebra, E8 is most simply described as the group of symmetries of its own Lie algebra!”

In the second post in this E8 series, I showed how particle states could be used as symmetry generators for certain symmetries of the particle states. This was done using pure density matrices. We began with a certain symmetry of the Clifford algebra of the Pauli algebra, and turned that symmetry into one of the pure density matrices of the Pauli algebra. In effect, we got the quantum states to operate on themselves to make other quantum states, exactly the sort of thing that is necessary to build E8.

We want to add composite or “bound” states to the “free” states that are represented by the usual pure density matrices. We will then add these bound states to the usual pure density matrices to create a larger algebra. (In that algebra, multiplication of states with differing numbers of particles will give zero.) The bound states will also be represented as density matrices. In this post we will lay the foundations for describing bound states by very simple Feynman diagrams, it will be collections of these Feynman diagrams, formed into matrices, that will create the density matrices for the bound states.
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