Observe how conjugation by the Pauli group operations recovers all $B_3$ generators from a single one.

Recall that the three Pauli operations give a set of MUBs for qubits. They are also used to create a simple unknot quandle. In M theory, triality places $2 \times 2$ matrices inside $3 \times 3$ matrices in three (or six) distinct ways. This is a tripling of the massless particle spectrum of the standard model (allowing for mirror neutrinos). Recall that the braid group $B_3$ extends the $2 \times 2$ modular group, notoriously important in $S$ duality.

8 years ago

Category people will recall that a braiding is associated to an extra product for our category. Quantum arithmetic requires one or two extra products, to distinguish it from classical sets.

ReplyDeleteIs there a typo in the picture ?

ReplyDeleteThe way I look at it, the matrices in the middle column on the RHS should all be the same, but the one in the Z row is different, there is a 1 out of place at the bottom left position.

But then again, I do have a naive and uninformed persperctive.

Oops, thanks a lot! Will fix.

ReplyDeleteHmm. More fairy fields excluded at galactic scale:

ReplyDeletehttp://arxiv.org/abs/1108.2914

Daniel, I use the term fairy field chiefly for the Higgs boson. Stringy dark matter is called zombie stuff.

ReplyDeleteShouldn't be stringy dark matter, or any stringy field, be called zombie stuff? The more you kill them, the more they appear.

ReplyDeleteYes, they also have the scientifically notable property of non existence.

ReplyDelete