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I am a graduate student just started learning dispersive PDE in MSRI's summer program. I roughly finished reading the paper by Klainerman and Machedon "ON THE UNIQUENESS OF SOLUTIONS TO THE GROSS-PITAEVSKII HIERARCHY". However I do not really know much about the many body system problem in general. May I ask what should I read to understand the general context of their paper and other related topics? I briefly searched on arxiv and found there are quite a few (+30) papers on this topic, but they all seem rather dense. So I do not really know where to start. I am also trying to read Erdos, Schlein, H-T Yau's paper on this topic, but I found myself getting lost after a while as I do not really have enough background. I know basic quantum mechanics and basic PDE, and it seems these are vastly insufficient to understand the huge topic.

I am reading Tao's article (https://terrytao.wordpress.com/2009/11/26/from-bose-einstein-condensates-to-the-nonlinear-schrodinger-equation/#more-3135) and I found it really helpful. However, I am rather lost with questions like "What are the open questions in this topic?" "Why these questions are important?" "Does physicists care about these results?" "What is a meaningful improvement of ths paper?", etc. I also found this phenomenon that we understand the mathematical statement but having poor insight of its physical background (or the real motivation behind the work) is quite common in my study group. So I venture to ask.

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3 Answers 3

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Mean field theory is one of the most active subject concerning many body systems, and the papers you mentioned fit in this context. I think you should read this review by Golse. However there are at least four different approaches to the problem. One, called coherent states method, introduced by Hepp and developed by many (especially Ginibre and Velo and recently Rodnianski and Schlein); another are the BBGKY hierarchies (look at the Golse review and also the references you already mentioned); there are also works by Pickl; and finally a method using semiclassical analysis in infinite dimensions by Ammari and Nier. I am not allowed to post all the links, sorry. However they should constitute a quite large and clarifying amount of literature on the subject, if you look for them.

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Thanks! This is very helpful. –  Bombyx mori Jul 12 at 1:34

A question along these lines was asked recently to two eminent mathematical physicists, Mel Levy and Elliott Lieb, and here is their wish list of open problems in many-electron theory.

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Thanks! I did not know this. –  Bombyx mori Jun 25 at 0:21

The main thing to know about the quantum many-body problem is that (exaggerating only a little) nothing is known about it. In fact, virtually nothing is known about how to attack the much easier classical many-body problem. For example, we don't know whether our solar system will eject one of its planets in the distant future.

All we really have are some special cases that happen to be more tractable. Apparently techniques are known for doing quite a bit with atoms. Nuclei are harder, and there are no high-precision techniques as there are for atoms. Part of what makes the problem hard for nuclei is that they exhibit collective behavior like rotation and vibration. You don't get this with atoms, although you do with molecules. Qualitatively different techniques are used for different types of nuclei, such as stiff spherical nuclei, vibrational nuclei, and deformed nuclei (which can collectively rotate).

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