One can obtain a Hilbert space from a wide variety of physical situations. Suppose we are to use a Hilbert space to model elementary particles, for instance. One the one hand, we’d like to have our model cover as many particles as possible. But to do this, we may have to assume a symmetry that is not exactly correct. For example, to get a model that includes isospin, we have a natural inclination to assume that isospin is a perfect symmetry when, in reality, it cannot be. Assuming perfect symmetry as an approximation leaves us with the need to later correct for this by including a symmetry breaking.
Recently, we’ve been discussing using Mutual Unbiased Bases (MUBs) to model the elementary particles. If we want to include very disparate objects as states in different MUB bases, we need to also include a symmetry breaking mechanism.
When elementary particle theory is built from symmetry alone, one finds that there are a nearly infinite number of ways of breaking symmetry. The freedom to do this makes it difficult to find our way. In the case of MUBs, however, we can write down some very natural ways to break symmetry if we treat the Hilbert space as the vector space of a Clifford algebra like the Dirac algebra. In this post we will show how the natural symmetry breaking in the Dirac algebra can lead to structures that are reminiscent of the structure of the elementary particles.