Random Thoughts on the Universe

Monday, April 10, 2006

Braids and the Standard Model

This is more than likely an interesting coincidence, but it still raises questions. (I will also apologize in advance for not providing pictures, but there are several good links at the bottom of this post)

- Take three ribbons, and write down all of the ways in which they can be simply braided (meaning one ribbon passes over a second ribbon and under the third ribbon - there are only two distinct options) with each ribbon twisted clockwise, counterclockwise, or not at all. (For this game, you also are not allowed to have a braid include two twists in opposite directions - ie if even one ribbon twists clockwise the other two cannot twist counterclockwise)

- Define three operations: applying C to any braid reverses the direction of all twists, applying P to a braid flips the whole thing from left to right, and applying T flips it from top to bottom.

- Now classify the braids:
There are two with no twists,
There are four with all twists (two all clockwise, and two all counterclockwise)
There are six with two clockwise twists and six with two counterclockwise twists
There are six with one clockwise twist and six with one counterclockwise twist
And in each case there are equal numbers of the two kinds of braids

- In the last two steps, replace twists with charge +e/3 for clockwise and -e/3 for counterclockwise twists, and replace the two types of braid with spin +1/2 and spin -1/2.
Now C changes charge to opposite charge, P reverses spins, and T flips both. In both the braids and this new form, the combination CPT leaves everything unchanged - just like these same operations do in the Standard Model.
Now look at the classified braids:
There are two particles with charge 0 (the left and right handed neutrinos)
There are two particles with charge -1 (left and right handed electrons) and two with charge +1 (the positrons)
There are six particles with charge +2/3 and six with -2/3 (down-type quarks and antiquarks, with three 'colors' as in the Standard Model)
There are six particle with charge +1/3 and six with -1/3 (up-type quarks and antiquarks with three 'colors')

So this simple game of ribbon braiding has reproduced the first generation of the Standard Model with all of the correct properties!

*There is also a more technical argument indicating that IF these ribbons actually represent particles, single and double ribbons cannot exist as particles because they cannot move through spacetime, while three ribbons braided together can.

The original (and more precise) paper is available from the arXiv. There is also a very good video of a lecture by Dr Smolin on this topic.

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