Parallel Transport

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Latest Notes

Kartik Prabhu
replied to a post on www.math.columbia.edu with
@Skewered Cigar @Peter Woit You can construct a phase space from the Lagrangian. You start with the Lagrangian and vary it (as usual), but don’t throw away the boundary term (not as usual). This boundary term is a (pre)symplectic potential, and a second variation gives a conserved (pre)symplectic form on the space of solutions of the theory. You can try this with usual quantum mechanics viewed as a field in 1-dimensions The whole “pre” stuff is about whether this symplectic form is degenerate or not; in any gauge theory it will have degenerate directions. However, you can take functions on the phase space which are constant along the degenerate directions, and define a Poisson bracket on such functions. Quantization then be done is the usual way, along with the usual issues of operator ordering for nonlinear functions, e.g. any Hamiltonian of interest. Of course, all of this is much easier said than done, but in principle the Lagrangian gives you, at least, all the classical structure needed for quantization.
Kartik Prabhu
replied to a post on www.math.columbia.edu with
I did a quick review in https://arxiv.org/abs/1511.00388v3 Some more physics-oriented ones: Y. Choquet-Bruhat, C. Dewitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and Physics, Part I: Basics. Y. Choquet-Bruhat and C. Dewitt-Morette, Analysis, Manifolds and Physics. Part II: 92 Applications. Some more mathematical ones: S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Volume I. Interscience Tracts in Pure and Applied Mathematics. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Volume II. Interscience Tracts in Pure and Applied Mathematics.

Latest Articles

<href> in SVG

While creating an animated SVG logo for indietech.rocks we ran into a strange problem where the SVG would display in some browsers and not in others. The issue is the different ways browsers handle XML — yes SVG is XML! So here is the problem and its solution.

Placeholder Images

The size of my images changes fluidly with my responsive layout. Since the browser does not know their heights a priori, the space collapses while the images are still loading. Once the images load the entire page reflows and the rest of the content jumps around to make space for them. It would be much better if the space for the images was reserved from the start and, as a bonus, if some lower resolution version of the images displayed, while the images load. Here is how I do it.