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i got a degree on physics so my question is ?as a physicist could i learn P-adic analysis or p-adci quantum mechanics ??

  • is there any good book on the subject ? as an introductory level

  • How the p-adic schoridignuer equation is defined

  • waht are the p-adci eigenfunctions and eigenvalues ??

  • how can i define a p-adic derivative ¿are there tables for p-adic derivatives?

  • how i must define a p-adic integral ??? are there tables for it

  • what is the p-adic Melli transform and why is equal to the euler factor $ (1-p^{-s})^{-1}$

  • how the Schrodinguer equation is SOLVED for the p-adic field?

-what does it mean 'Nonarchimedean field'

  • how is the quantization carried out in the p-adic field

  • what are the adeles ? have some meaning in physics ??

  • how is the p-adic classical mechanics made ?? .. Newtonian Lagrangian and Hamiltonian mechanics

thanks in advance

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At least some of the sub-questions in there would get better response at math.stackexchange. –  Willie Wong Sep 30 '11 at 15:50

1 Answer 1

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. The p-adic numbers are a way of organising solutions to a Diophantine equation modulo higher and higher powers of $p$, with the rough idea that this is an approach to finding an integer solution, at least if you could do it for all different primes and the Archimedean (infinite) primes. 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...the derivatives are all zero.

p-adic integrals are merely finite sums, and they tend to get grouped into geometric series, that explains why the Mellin transform is merely a geometric series and turns into the Euler factor you wrote.

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