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I badly need to polish the $n$Lab entries related to path integrals. Today a student asked me how the pull-push operation in string topology is a remnant of a quantum path integral. So a started writing now
So far there is the description of the archetypical path integral for the quantum particle propagating on the line in terms of pull-tensor-push.
This is very interesting. The most rich geometric theory of quantization is so far developed for finite-dimensional mechanics. The central role are the cohomological classes related to Lagrangian geometry – most notably the Maslov class. Cohomology is higher categorical subject, so can your approach predict the appearance of the Maslov class ? That would be so interesting.
I haven’t thought much about the Maslov index for a long while. Maybe I should.
One question: in the entry you have a sentence
Lagrangean submanifold describes the phase of short-wave oscillations.
I am not sure what this means.
I must have taken this phrase from somewhere. Well, it is roughly like the role of the real submanifold in the Fourier transform. My memory is that here one takes in the sense of the eikonal approximation, which is the splitting into short wave and long wave part and assigning the coordinates to each part. I should write some time an entry on eikonal. I think this is very important for us, as in the cases in which one has topological QFT, the quasiclassical approximations are exact (by localization), hence one is likely to see the connection best by looking at eikonal-like approaches.
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