In the previous post we introduced a fictitious snuark vacuum with the intention of using it to simplify calculations. In the snuark algebra, complex amplitudes show up only as a geometric result of an interaction between two mass vertices. We’d like to convert this sort of computation, which uses Pauli spin matrices, into calculations where we associate an amplitude with each mass vertex by itself. There are 36 mass vertices, 12 of which are trivial. We found that we would know the remaining 24 by symmetry if we could solve for just four of them: . In this post, we find these numbers, and then test them by comparison with explicit calculations using Pauli matrices.
A Beautiful Theorem: When visiting a strange land, it’s always nice to take in the pretty sights. For our utilitarian calculations, we will use a beautiful result for products of primitive idempotents (density matrix states) of the Pauli algebra:
Let be three real unit vectors in 3 dimensions. They define the vertices of a spherical triangle. Let A, B, and C be the projection operators for Pauli spin in the directions. Then the projection operators are related by the equation:
where is the oriented area of the spherical triangle defined by the three unit vectors.