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- Smirnov : Calculation of general $p$-adic Feynman amplitude.
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I feel that this topic is a waste of time. Stanislaw Ulam wrote a paper long ago on whether p-adic stuff could be used in Physics, I never heard of any useful follow-up. This is great in number theory, as Weil and Langlands showed, and it helped to prove Fermat's last theorem via base change Saito, Shintani, Langlands, Tunnell as Wiles was able to show, but there is no reason to think it will help in Physics.
All of this p-adic harmonic analysis works on the space of test functions which are locally constant, so it makes no sense to study derivatives There is a vast physical literature on the subject which cite Reason 1 as justification, this is sometimes called the Vladimirov Hypothesis. It is a seductive idea, but there is no evidence for it. Moreover, this would beg the question of why would Nature choose a particular prime number.
A perhaps better hypothesis is the Adelic one advocated by Manin, i. In any case, this is quite speculative at present. When studying complex multiscale phenomena it is often important to decompose functions into time-frequency atoms which live on a tree, e.
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Unfortunately for most questions of interest the metric which governs how these atoms interact with each other is not the natural from the tree point of view ultrametric distance, but the Euclidean metric of the underlying continuum. Hierarchical models in physics amount to changing the model so it is the ultrametric distance which defines atomic correlations. The same idea also appears in mathematics where such toy models are often called "dyadic models". See this wonderful post by Tao for a nice discussion of this circle of ideas. Given a problem in Euclidean space, there are lots of ways of setting up a simplified hierarchical model for it.
To go back to Wilson and the RG, he also used a hierarchical model and gave it yet another name "the approximate recursion". The importance of this toy model in Wilson's path to discovery is clear from his quote:. Knowing this it is then trivial, given my field theoretic training, to construct the beginning of the epsilon expansion for critical exponents. Sign up to join this community.
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