Monday, October 25, 2010

Theory Update 13

In a new comment on the neutrino braids, kneemo recalls that the braid group on three strands extends the famous modular group. He matches the four neutrino braids to group generators $A$ and $B$:

Here $A$ and $B$ are related to the usual generators $S$ and $T$ by $A = T$ and $S = ABA$. If we had a copy of the modular group for each generation, we would have (roughly speaking) the M theory duality group $SL(2,Z) \times SL(2,Z) \times SL(2,Z)$, famously discussed in connection with three qubits. So as we hoped, the generations may indeed be associated with a tripling of spin. By vertically composing three copies of a neutrino braid, we obtain a (not quite the) identity braid.


  1. I feel like an early new world gold digger, returning to the east coast screaming "Gold, gold!", only to be ignored.

  2. The A and B matrices come from the Burau representation of B_3, evaluated at t=1, discussed by Baez (week 233) and in the paper by Voituriez math-ph/0103008.

  3. Oh, cool. They also generate the Stern-Brocot tree for the rational numbers ...

  4. Well, here comes the revolution :) My summary. I changed it a bit so it would not be complete nonsens (as you said). Remember, I am no physicist, but still I hope someone understand something.

    The background.

    Quite interesting.


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