P-adic strings

p-adic quantum mechanics is a relatively recent approach to understanding the nature of fundamental physics. It is the application of p-adic analysis to quantum mechanics. The p-adic numbers are a counterintuitive arithmetic system that was discovered by the German mathematician Kurt Hensel in about 1899. The closely related adeles and ideles were introduced in the 1930s by Claude Chevalley and André Weil. Their study has now transformed into a major branch of mathematics. They were occasionally applied to the physical sciences, but it wasn't until a publication by the Russian mathematician Igor Volovich in 1987 that the subject was taken seriously. There are now hundreds of research articles on the subject, along with international journals as well.

This article provides an introduction to the subject, followed by a review of the mathematical concepts involved. It then considers modern research on the subject, from Schrödinger-like equations to more exploratory ideas. Finally it lists some precise examples that have been considered.

Many studies of nature deal with questions that occur at the Planck length, in which ordinary reality doesn't seem to exist. In some ways, the experimental apparatus and experimenter become indistinguishable, so that no experiments can be done. The unification of the immensity of cosmology with the Hilbert space formalism of quantum mechanics presents a formidable challenge. Most researchers feel that the geometry and topology of the sub-Planck lengths need not have any relation whatever to ordinary geometry and topology. Instead the latter are believed to emerge from the former, just as the color of flowers emerges from atoms. Currently many frameworks have been proposed, and p-adic analysis is a reasonable candidate, having several accomplishments in its favor.

Another motivation for applying p-adic analysis to science is that the divergences that plague quantum field theory remain problematic as well. It is felt that by exploring different approaches, such inelegant techniques as renormalization might become unnecessary. Another consideration is that since no primes have any special status in p-adic analysis, it might be more natural and instructive to work with adeles.

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