# Daily Archives: February 13, 2008

## Koide formulas and Qubit / Qutrit MUBs

Recently we’ve been discussing Mutually Unbiased Bases or MUBs on this blog. This has not been for any particular interest in pure mathematics, but instead in their application to a preon theory of the elementary particles, and the E8 quantum numbers. The relationship is rather long and difficult to explain. All the major pieces are already on the web in various places, but it would be rather difficult for someone to piece together the details as they are spread around, and use various notation. The full explanation is too long for a blog post, but what I can do is give an overview of the explanation, along with links to resources. I will begin with the end, the Koide formula, as it hasn’t been discussed here in a few weeks:

Koide Formulas: The Koide formula for the charged leptons that we will use here is:
$\sqrt{m_n} = \mu_1(1+\sqrt{2}\cos(2in\pi/3 + 2/9) )$
where $\mu_1$ is a constant with units of square root of mass, n is the generation number and $m_n$ is the mass of the generation n charged lepton, that is, the masses of the electron, muon, and tau. The formula is accurate to a part in a million.

There is a similar formula for the neutrinos (i.e. the neutral leptons):
$\sqrt{m_n} = \mu_0(1+\sqrt{2}\cos(2in\pi/3 + \pi/12 + 2/9 ) ).$
The accuracy of the neutrino formula is unknown, it accounts for the differences in masses of the neutrinos, but those masses are rather poorly measured.