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Recently, mathematicians and theoretical physicists have been studying Quantum Field Theory (and renormalization in particular) by means of abstract geometrical objects called motives. Amongst these researchers are Marcolli, Connes, Kreimer and Konsani. You can read about Marcolli's work here*.

Now, in the wikipedia article on the Navier-Stokes Equations, there is a short paragraph on Wyld Diagrams. It is stated there that they are similar to the Feynman diagrams studied in QFT. Since motives and other algebraic approaches are currently used to study these Feynman diagrams, I was wondering if these approaches could also aid in studying the Navier-Stokes equations.

If so, how? And to what extend could they potentially help solving the (in)famous Millennium Problem? If not, why not?

Please note that I'm far from an expert in any of these fields.

Thanks in advance.

*(I would like to include more hyperlinks to the books and articles of the respective scientists, but since I am a new user I am incapable of doing so. You can find a lot more literature simply by looking up the names of these researchers.)

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Re: Millennium problem: you are looking at two completely different objects here. Motives are algebraic geometrical/categorical objects which are by definition "extremely smooth". Whereas the millennium problem concerns whether solutions to a partial differential equation satisfy certain regularity conditions. Almost by definition motives cannot have anything to say regarding the regularity theory of navier stokes. (Note that I am not ruling out application to fluids in general, I am just ruling out that particular problem.) –  Willie Wong Apr 5 '12 at 16:02
    
@WillieWong thank you for your comment. That leaves out the possibility of any application to the Navier-Stokes problem. To the AnonymousDownvoter: could you please explain your motives with regards to your down vote? –  Max Muller Apr 6 '12 at 16:20

2 Answers 2

up vote 3 down vote accepted

Although I am not an expert on either of the subjects, I think it is save to say that such an application is highly unlikely.

First of all it is my understanding that Motives came up in connection with perturbative Quantum Field Theory. That is to say in perturbative Quantum Field Theory you come to a point where you have to calculate individual Feynman diagrams and this can be related to periods of certain motives. Usually the problem can be separated into some group theoretical part, where you calculate casimirs, gamma matrix traces and so on. In the end what you are left with is a sum of terms, whose denominators are products of something like $\frac{1}{k^2 - m^2}$, the precise structure is determined by momentum conservation at each vertex and the rule is that you have to integrate over each loop momentum.

If there are enough factors, the denominator has an interesting pole structure. Basically you have a bunch of intersecting hyperboloids (that part I have not thought carefully about). Now algebraic geometers like to think in geometrical terms and have their own names for this situation: You are calculating a period of some 'motive' (it should be just related to the poles of the denominator).

Of course physicists have done these integrals long before mathematicians developed an interest for them (again?). Basic tricks are to introduce Feynman parameters or use the schwinger representation. For example Marcolli uses this representation in papers she also talks about motives. Similiarly Connes and Kreimer seem to mainly clarify constructions, that had been known to people that calculated 5-th loop order QED diagrams (i.e. how to cut up divergent diagrams). Although I probably just don't understand the more sophisticated parts of their work.

Now it is usually not emphasized, but many partial differential equations can be treated perturbatively by feynman diagram methods. Essentially one only considers tree diagrams of a corresponding QFT. I suspect that these are the Wyld diagrams.

In any case the Millenium Problem asks for existence of certain solutions, given the fact that diagram methods are mostly a calculational tool, it is highly unlikely that they can be useful to it. Since the connection of QFT with motives is calculational, it is unlikely, that they are useful.

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I am absolutely no expert in the field, but maybe this paper of Moise and Temam may be connected to your question.

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