# Learning about super-symmetric quantum mechanics and integrable systems

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I'm an undergraduate interested in theory. I recently asked one of my professors, a physicist in the particle theory group at my school, if he'd be willing to take on an undergraduate for a senior project. He told me to check back in at the end of the quarter, and that he might have a project related to super-symmetric quantum mechanics and integrable systems that may be feasible for me to try out then.

In the meantime, I want to familiarize myself as much as possible with the relevant math and physics so that I can impress him when I ask again. Do any of you have any books/online sources that you recommend I take a look at?

This post imported from StackExchange Physics at 2014-04-25 13:35 (UCT), posted by SE-user BRayhaun
retagged Apr 25, 2014
Possible duplicates: physics.stackexchange.com/q/26912 , physics.stackexchange.com/q/30353 and links therein.

This post imported from StackExchange Physics at 2014-04-25 13:35 (UCT), posted by SE-user Qmechanic

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regarding supersymmetric quantum mechanics:

By all means you should read and absorb the source of all supersymmetric quantum mechanics, which is still probably the deepest article on this topic, namely the original

This is one fourth of the work that won Witten the Fields medal in 1990.

When I learned this stuff back then, I was very much enchanted by the perspective of spectral geometry (i.e. Connes' spectral triples) that is highlighted in the survey

I'd say if you read that in parallel with Witten, you'll end up with a rather profound understanding of what the deep aspects here are. Because, as you will notice, the topic of susy QM has the tendency to make many people write many rather shallow articles. Stay clear of them and focus on the substantial stuff here.

Speaking of substance, of course the full conceptual impact of supersymmetric QM rests in its relation to index theory and K-theory. You may not need that for your physics project (depends), but for your own education I'd suggest to at least look at the surveys and get the basic idea. This is tremendous stuff.

Finally a fun fact to know -- in particular when somebody quizzes you on applications or is sceptical about the value of supersymmetry -- is that every fermion particle is described by relativistic supersymmetric quantum mechanics on its worldline. See here. This is in itself a somewhat trivial fact really, but drastically underappreciated. It is of course directly related to the way that Witten initially bumped into susy QM in the first place, namely by considering the point-particle limit of superstrings.

This post imported from StackExchange Physics at 2014-04-25 13:35 (UCT), posted by SE-user Urs Schreiber
answered Feb 7, 2014 by (5,885 points)
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Integrable supersymmetric models in quantum mechanics (= 1+0-dimensional field theory) are based upon shape invariance. For a comprehensive introduction see

F. Cooper, A. Khare, U. Sukhatme,
Supersymmetry and quantum mechanics,
Physics Reports 251 (1995), 267-385.

Another, very recent survey is

R. Sasaki, Exactly Solvable Quantum Mechanics, arxiv:1411.2703

answered Apr 25, 2014 by (12,640 points)
edited Nov 28, 2014

There's a book I like by Junkers, which emphasizes the stochastic equation aspect. There is an extremely simple Nicolai map, the SUSY-QM is just a stochastic equation in disguise, and this makes all the properties obvious.

The paper is also available on the arXiv: http://arxiv.org/abs/hep-th/9405029

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For an elementary introduction to supersymmetric quantum mechanics see

Supersymmetry and Quantum Mechanics by Cooper et al. available at

http://arxiv.org/abs/hep-th/9405029

This post imported from StackExchange Physics at 2014-04-25 13:35 (UCT), posted by SE-user just-learning
answered Feb 22, 2014 by (95 points)

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