If a Hilbert space is d-dimensional, we expect that the number of elementary particles we can describe with it is d. For example, a Pauli spinor is 2-dimensional and electrons come in two states, spin-up and spin-down. The Dirac spinors are 4-dimensional and the related theory describes electrons plus their antiparticles, and therefore 4 states.
So long as we think of elementary particles as the things that can only be represented by spinors (i.e. the 2×1 or 4×1 vectors of the Pauli or Dirac theory, respectively), each Hilbert space can only provide a home for a number of particles given by the dimension of the vectors of that space. There are d complex degrees of freedom for each particle and therefore, since quantum mechanics is more elegant with complex wave functions, there is room for only d particles represented by that spinor. That’s all there is, and that’s how elementary particles has been done for many decades.
In the usual theory, the Clifford algebra acts on the states. When we discuss a “basis” here it will be in the context of the d-dimensional Hilbert space, that is, we will mean a basis for the states that the Clifford algebra acts upon. A base defines the quantum states that we can consider to be different particles, or different aspects of the same particle. For the Dirac algebra, one basis has the four particles: {spin up electron, spin down electron, spin up positron, spin down positron}. But this is not the only possible basis for the Dirac algebra quantum states. We could instead pick {right handed electron, left handed electron, right handed positron, left handed positron}.
From a quantum information point of view, the splitting of the Hilbert space into a particular basis is a somewhat inadequate description of the information contained in a quantum state on the space. Suppose we have a large number of identical states and we wish to determine what state it is. If we make measurements with respect to just one basis, we will get the right answer if the state is entirely within that basis, but this won’t necessarily be all the information about the state.
For example, suppose the state is a spin-1/2 state. If it is polarized in the y-direction, then anytime we measure it in the z-direction we will get the value + or – with equal probabilities. This will not allow us to distinguish between, an +y oriented state and a -y oriented state. And vice versa. To determine what state we really have we need to measure it with respect to more than just the usual single direction. And this is where MUBs come in. It turns out that if you measure an arbitrary quantum state (even a mixed density matrix state) with respect to a “complete set of MUBs”, you will get just barely enough information to completely determine the quantum state.
The idea we will discuss here is to use MUBs to describe elementary particles, a subject of current research. We will briefly discuss the motivation, the history, and some of the advantages of this idea, and finally show how it can be used to derive some of the features of Garrett Lisi’s E8 model
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