I was interested to read the answers to the other Many-Body Problem questions on this site and was left with one nagging question of my own. What does the Many-Body Problem reveal about reality and our attempt to understand it? To explain: if the solutions we find to this problem are in the form of lengthy iterative expansions may this not signify in some way that nature operates along discrete iterative paths? What does this say about our theories if these problems are so difficult/time-consuming for them but so quickly solved by the real world around us? Is this problematic? Will more complete theories-ie possibly string theory- address this or do they even need to?
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2$\begingroup$ This might be off topic, but even if not, it's not clear to me what you're asking here. What do you mean about nature operating along "discrete iterative paths"? $\endgroup$– David ZCommented Feb 26, 2011 at 2:49
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2$\begingroup$ @David, I think what @jaskey13 is speaking of is that a many-body system appears to "explore" large parts of the phase-space far more efficiently than any of our algorithms designed to simulate such a system. $\endgroup$– user346Commented Feb 26, 2011 at 3:04
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$\begingroup$ @Deepak @David Exactly- though phrased much better than I could have done myself. $\endgroup$– user1567Commented Feb 26, 2011 at 3:15
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$\begingroup$ hm, ok. @Deepak, feel free to have a go at editing the question to make it clearer. $\endgroup$– David ZCommented Feb 26, 2011 at 4:27
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$\begingroup$ @Deepak If you do please preserve that I am asking about the implications of this concerning current, developing, and future theories. $\endgroup$– user1567Commented Feb 26, 2011 at 4:56
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The core of the question is about whether Infinite sums and similar expressions which appear in many attempts at an analytical solution to equations of motion can represent "how Nature does it". Well Feynman once noted something similar in an analysis which ended with the observation that "down at the electron there seemed to be a lot of computation going on in a very small region".
Part of the answer for classical physics is that we dont always represent the problem that Nature is solving locally: particles move in their local field according to local potentials. In analysing a larger system we find difficult analytics, but Nature never has to solve these.
In Quantum Mechanics (and its relatives) we reach the wavefunction ontology question. That is "is the wavefunction just our representation of the physical reality" or "is the wavefunction actually physical reality"?
Clearly if the wavefunction is just our representation then it is not telling us what Nature actually does; and the analytic issues we have with it (e.g. in the N body case) are just our problem, not Nature's. What Nature actually does would then be a theory which reduces to Quantum Mechanics. The compututational world might describe such a theory as "a theory which Implements Quantum Mechanics".
If the wavefunction corresponds directly to reality, then there probably is still some need to find a good "computational engine" for making it all work. The Quantum Computational school around David Deutsch seem to be in this area.
As far as N-body issues are concerned I am becoming impressed by the "emergent properties" found in that area by the Condensed Matter Physicists. Perhaps this is telling us that Nature has found "higher level descriptions" with which to work with complex systems.
I am still learning about String Theory, but I am getting the impression that it is quite "conventional" in terms of these types of fundamental issues with QM.
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$\begingroup$ So it seems, in the classical sense at least, that nature has a massive parallel processing advantage. And if Q.M. is correct- or on the road to being correct- then it may be that we need a quantum computer to allow this model to predict physics in a timely/effective manner? A) Is this a problem of parallel processing as well and us solving it by computing using qubits? B) Or is it that if QM is correct then the only proper way to compute is on a quantum computer as a classical computer can't properly represent superpositions? And/or is B) yes because of A) yes $\endgroup$– user1567Commented Feb 27, 2011 at 2:55
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$\begingroup$ and condensed matter--> very interesting- thank you $\endgroup$– user1567Commented Feb 27, 2011 at 3:19
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$\begingroup$ (for above) as I'm not clear as to whether comments on an answer automatically go to your inbox? $\endgroup$– user1567Commented Feb 27, 2011 at 16:42
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$\begingroup$ @jaskey13 : Yes to all questions so far! $\endgroup$ Commented Feb 27, 2011 at 17:18