## Saturday, October 23, 2010

### Theory Update 12

After much faffing, to borrow a phrase from English climbing friends, I have decided that the best name and phase convention is this one.

That puts ordinary matter at $+ 2/9$ and mirror matter at $- 2/9$. It lets the $\overline{\nu}_{R}$ antineutrino annihilate via composition with the mirror particle $\overline{\nu}_{L}$. It gives left handed neutral particles the $+ \pi /12$ phase. Phew.

1. Interesting use of the braid group B_3 on 3 strands. Using the homomorphism from B_3 to SL(2,Z) gives a nice matrix description of your particles in terms of SL(2,Z) generators:

A=
[1 1]
[0 1]

B=
[1 0]
[-1 1]

This gives the following correspondences:

nu_L = (B^-1)A
nu*_L = B(A^-1)
nu_R = (A^-1)B
nu*_R = A(B^-1)

so that, for example

nu*_L(nu*_R)=B(A^-1)A(B^-1)=I

as expected.

2. Yeah, cool kneemo. Your last result comes from drawing the picture, of course. Good to see we are getting those M theory qubit dualities sorted out, slowly.

3. Similar diagrams are found on page 280
of Schiller's text on http://www.motionmountain.net/fun.html
Are they related?

Frank