Jason (jcreed) wrote,
Jason
jcreed

This blog post about diagrammatic reasoning in physics and mathematics is interesting. It led me back to Yves Lafont's work with higher-dimensional rewriting, and also to this paper titled "Kindergarten Quantum Mechanics.

On the same note I think I recently accidently reinvented (a small part of) the index-free graph notation for tensor contraction due to Penrose, who came up with it quite a few decades ago, and which I think is the thing being described above. It's one of those kinda-obvious-in-hindsight things, but it's really nice in some ways. The basic idea is that a tensor is a little blob like a circuit element with one wire going up for every (let's say) contravariant index, and one wire going down for every covariant index. Multiplying tensors amounts to connecting an up-wire from one to a down-wire from another. The way Lafont presents it seems to have to do with little finite axiomatizations of associativity properties of these wirings, but my guts tell me that doing that and then proving coherence theorems isn't as good as just starting from an "unbiased" presentation (using the terminology of the higher-dimensional algebra folks) where you just state associativity all at once in an n-ary kind of way.

Here is another Penrose-y flavored paper that goes into some graph theory, by Peter Zograf.
Tags: diagrams, penrose, physics
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