Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Monday, October 23, 2006

Connes Kreimer Marcolli

There is an amazing series of papers by Connes, Marcolli and others on From Physics To Number Theory. See for example here or here or here. This goes back to work of Kreimer and Broadhurst, which is now very well known. Some of the older papers are here. I particularly recommend the paper: Broadhurst and Kreimer, Association of Multiple Zeta Values with Positive Knots via Feynman Diagrams up to 9 Loops, Phys. Lett. 393 B (1997) 403-412.

Its about turning knots into simple Feynman diagrams into Multiple Zeta Values. These MZVs satisfy all sorts of crazy relations, which the mathematicans have been studying like crazy. But really they're quite simple. They act on a set of k ordinals (yes, that's right, you should be thinking 1-ordinals) and are characterised by two numbers, namely the weight n, which is the sum of these, and k itself, the so called depth. Of course these naturally show up as special integrals of something called Mixed Tate Motives (don't even ask), so we know that the weight n is the same n of M(0,n+3). Goodness, me. The Yang-Mills problem and the Riemann hypothesis seem to be related. Well, well.

The real question, however, is how to go beyond scalars to other entities in QFT. Any guesses?


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