Dan Piponi on Nostr: nprofile1q…ze5g0 There's a long exercise I did years ago that goes something like ...

nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqky223zcc4q69d8t0me4vg5uw8mw0yxeukjgvz6h92laqnenr0ajsgze5g0 (nprofile…e5g0) There's a long exercise I did years ago that goes something like this:

You usually think of classical mechanics as the classical limit of quantum mechanics but this isn't quite true in the form classical mechanics is written.

In QM the field is exp(i∫L(x,∂x)Dx) where the integral is a path integral over all paths that satisfy some constraints at the start end end point of the dynamics you're considering. If you hold the start point constant this field is a function of the end point and is the thing we usually solve for in the Schrodinger equation.

You can also write classical (point particle) mechanics in terms of fields, but this time the field is given by the inf of L(x,∂x) over all paths. You can extract actual motions from these fields or you can study these "fields" in their own right. Usually we discard the actual value of L. You can mimic much of the theory of QM in this classical setting.

If you're like me you have better intuition about Fourier transforms than Legendre transforms but this alternative classical frameworks allows you to use knowledge about Fourier transforms to reason using Legendre transforms. Partly it comes down to looking closely at what the boundary conditions are - eg. you have to work with a different field if your boundary specifies momentum rather than position and there are mixtures like known momentum at the start and known position at the end. It's all very analogous to switching from position to momentum in QM by taking the Fourier transform.

I know of one paper that does something like this. It's in an optics journal and could take a long time to find again...