# Path (Feynman) Integrals over Graphs

I was thinking about Feynman integrals the other day and in particular about discretizing the paths.

Does anyone know the lay of the land about what happens when you do path integrals over, say, a lattice or graph?

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I don't know, but here's another question - if you discretize the space with a different graph (for example, a honeycomb vs. a rectangular grid), do you still get the same answer? –  Nick Alger Dec 14 '10 at 0:26
Yeah ... exactly. And would that even be desirable? –  isomorphismes Dec 14 '10 at 3:02
What originally got me wondering about this is thinking about mapping $[0,1]$-weighted graphs onto $[0,1]$. "How many" different ways are there to do it and what are their characteristics? –  isomorphismes Dec 14 '10 at 3:04
@Nick Alger: Your question about different lattice geometries is investigated in lattice gauge theory; one of the main aims there is to find out whether the lattice version of the theory has a continuum limit, whether this is independent of the lattice geometry (which need not be the case), and whether it coincides with the original continuum theory. –  joriki Mar 19 '11 at 9:04

There's an entire approach to non-perturbative numerical calculations for quantum field theory, lattice gauge theory, which is based on path integrals over lattices. I did my diploma in that field, so I might be able to answer some questions, but it's been a long time :-).

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@joriki Wow! So I was thinking about how many degrees of freedom there are in a small weighted graph. Say I wanted to associate each graph to a number $\in [0,1]$. One natural way, I thought, would be to take some kind of "path integral" of the weighted graph. But aren't there many ways to do so? Possibly with different answers? –  isomorphismes Mar 19 '11 at 16:54
@Lao Tzu: I think you'll need to be a bit more specific than that. Certainly you get a different path integral for different Lagrangians -- are you thinking of some particular physical theory? Or just ways of associating numbers with graphs quite generally? –  joriki Mar 19 '11 at 16:59
@joriki I'm thinking of graphs as models of the brain / neurons. Not physics, psychology. –  isomorphismes Mar 19 '11 at 17:26
How about asking it this way. Given a weighted graph with each of the weights $\in [0,1]$, what is the variance among different possible integrals I could take of that graph? (the answer could be a "distribution") –  isomorphismes Mar 19 '11 at 17:28
@Lao Tzu: Again, I'm afraid you'll have to be more specific for those questions to have any meaningful answers. In physics, a path integral involves a phase factor determined by the Lagrangian of the theory. I don't know what sort of path integrals you're envisioning in your brain model graphs. Do you just want to add up weights along paths? Or some function of the weights? When you say "path integral", do you mean something closely analogous to path integrals in physics, or just generally something like a sum or integral along different paths in a graph? –  joriki Mar 19 '11 at 17:47