occasional meanderings in physics' brave new world

Name:
Location: New Zealand

Marni D. Sheppeard

## Tuesday, March 16, 2010

### Ribbon Review

Why you should love the Field with One element

In the theory of $F$, matrices are no longer interpreted as linear operators for complex number Hilbert spaces. If we can do quantum mechanics without ordinary vector spaces, using special sets instead, then we can consider the $n \times n$ quantum operators as a reduction of the larger set representing the braid group $B_n$.

For example, the two strand group $B_2$ consists of the unknot, the unlink, a series of Hopf links and the series of $(2,q)$ torus knots. These knot diagrams look like twisted ribbons. Recall that a tripling of such ribbons, using six strands, was used in the Bilson-Thompson particle scheme, as shown.