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wrote the first part of a discussion of prequantized Lagrangian correspondences, showing how traditional Hamiltonian and Lagrangian mechanics are naturally absorbed into the context of “local prequantum field theory” and “motivic quantization”.
Simple as it is, but does anyone know if the proposition in the section The classical action functional prequantizes Hamiltonian correspondences has been made explicit in the literature before? I can’t find it, but it should have been discussed before. If anyone has a citation, please let me know. Of course all the ingredients of the little proof are simple classical steps, but I am wondering if this has been observed as a statement, simple as it may be, on the prequantum lift of the more famous Lagrangian correspondences.
I have added a bit more to the entry today and started cross-linking more with Legendre transform. But then I got interrupted before completing. Will continue tomorrow.
(This here just in case anyone is watching the “recently revised”-list and is wondering…)
With this interpretation of the Legendre transform, does it cast any light on old Cafe conversations, e.g, here John had been talking about the Legendre transform as switching between the Lagrangian approach via the tangent space to the Hamiltonian approach via the cotangent space? Also there, there’s the issue of the Legendre transform as low temperature limit of the Laplace transform.
When we move to higher local quantum field theory, what is the equivalent of the Legendre transform?
So that’s the usual way of presenting the Legendre transform. Here the point is: both the Lagrangian description and the Hamiltonian description of classical mechanics are unified in the notion of corresponences in the slice topos of smooth spaces over the moduli space of prequantum bundles. A morphism in the slice, hence a diagram of the form
$\array{ X && \longrightarrow && Y \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}U(1)_{conn} }$has two components: 1. that top horizontal 1-morphism and 2. the 2-morphism filling the diagram.
The statement is: the first expresses the Hamiltonian evolution (its infinitesimal version is a Hamiltonian vector field corresponding to a Hamiltonian function $H$), the second is the Legendre transformed Lagrangian information (in that the $U(1)$-component of this natural equivalence is the exponentiated action $\exp(i S)$ where $S = \int_{t_0}^t L \mathbf{d}t$ is the action and $L$ is the Legendre transform of $H$).
I’ll explain all this in detail in the entry. Am working on it. But it turned out that I used up a good part of the morning with working on a section previous to the one that gets to Legendre transforms, namely the one on Trajectories and Lagrangian correspondences.
It’s all getting a good bit more expositional now than I had origianlly intended. But I guess that’s just bad for my time management, not bad in general.
Okay, I have edited a good bit more at prequantized Lagrangian correspondences and, among other things, tried to highlight the appearance of Legendre transformations a bit more.
I am out of steam for the time being and also need to be concentrating on other things now. But if anyone pushes me, I’ll further expand and/or polsih or the like.
I am pasting this also as a new section into geometry of physics.
I have been expanding and polishing at prequantized Lagrangian correspondence. Especially the central section Hamiltonian flows, the Legendre transform and the Hamilton-Jacobi action I have streamlined a good bit.
There’d be more polishing to do, but I am out of resources now.
Will be using this tomorrow (or rather today, in a few ours) as the script for the second session of a lecture series.
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