p-adic quantum mechanic [closed]

Question

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

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.

Answer 2

I think there are mainly two reasons for motivating the introduction of $p$-adic models in physics.

1. They could exist in nature.
2. They provide insightful toy models for physical phenomena.

There is a vast physical literature on the subject which cite Reason 1 as justification, this is sometimes called the Vladimirov Hypothesis. Namely, we do not know the texture of spacetime at the Planck scale therefore it is possible that it might look more like $\mathbb{Q}_p^d$ than $\mathbb{R}^d$. 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.e., all primes should occur democratically. In any case, this is quite speculative at present.

The use of $p$-adic numbers in physics is not a "waste of time" because of Reason 2. By exaggerating only a little, one could argue that $p$-adic toy models is what guided Kenneth Wilson when making his great discoveries in the theory of the renormalization group which entirely revolutionized physics in the early seventies.

When studying complex multiscale phenomena it is often important to decompose functions into time-frequency atoms which live on a tree, e.g., when using a wavelet decomposition. 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. The $p$-adics, in some sense provide the most canonical, stuctured and principled way of doing this.

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:

"Then, at Michael's urging, I work out what happens near four dimensions for the approximate recursion formula, and find that d-4 acts as a small parameter. Knowing this it is then trivial, given my field theoretic training, to construct the beginning of the epsilon expansion for critical exponents."

which can be found here (about one third from the bottom of the page).

As for references, a good entry point in the subject is the two articles:

1. "Tree-like structure of eternal inflation: A solvable model" by Harlow, Shenker, Stanford, and Susskind, Phys. Rev. D 85, 063516 – 2012.
2. "Nonarchimedean Conformal Field Theories" by Melzer, Int. J. Mod. Phys. A, 04, 4877 (1989).

Then for learning $p$-adic analysis there are various books or review articles, e.g.,

1. "P-Adic Analysis and Mathematical Physics" by V.S. Vladimirov, I.V. Volovich, and E.I. Zelenov, World Scientific Publishing Company, 1994.
2. "Theory of P-adic Distributions: Linear and Nonlinear Models" by S. Albeverio, A. Khrennikov, and V. M. Shelkovich, Cambridge University Press, 2010.
3. "An Introduction to p-adic Fields, Harmonic Analysis and the Representation Theory of SL2" by Sally in Lett. Math. Phys. 1998.