Quantization and Cohomology (Week 12)
Posted by John Baez
This week’s class on Quantization and Cohomology introduced the theme of ‘rigs’ (rings without negatives), foreshadowed last week:

Week 12 (Jan. 30)  Classical, quantum and statistical mechanics as ‘matrix mechanics’. In
quantum mechanics we use linear algebra over the ring $\mathbb{C}$; in classical mechanics everything is formally the same, but we instead use the rig $\mathbb{R}^{min} = \mathbb{R} \cup \{+\infty\}$, where the addition is min and the multiplication is +. As a warmup for bringing statistical mechanics into the picture  and linear algebra over yet another rig  we recall how the dynamics of particles becomes the statics of strings after Wick rotation.
 Supplementary reading: Grigori L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction. Also see the longer version here.
Last week’s notes are here; next week’s notes are here.
Posted at January 31, 2007 1:56 AM UTC
HamiltonJacobi Equation
So, where do I go to learn about the HamiltonJacobi equation? Assume that I am mathematically sophisticated, and I understand on a mathematical level what people mean when they describe mechanics through symplectic forms and Hamiltonians, but my formal training in mechanics was mostly directly from Newton’s laws with a quick reference to Lagrangians at the very end of the course.
I ask because I’ve been doing tropical math for five years now, so I understand very well the degeneration of (C,+,*) to (R,min,+). I’ve always know that people describe the reverse process as quantization, but I had the impression that this was just in the vague way that mathematicians like to refer to any situation where you perturb an algebraic object into something more complicated and useful as “quantization”. This is the first time that anyone laid out for me a real physical analogy, so now I want to learn more.
Thanks!