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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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    http://www.dam.brown.edu/people/mumford/blog/2014/FeynmanIntegral.html2015-04-30

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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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    The lattice approach is especially useful in statistical field theory, any book such as Itzkinson's should be plentiful of ideas and facts about that.2015-07-24