# Contents

## Idea

There are various attempts/speculations/proposals to use p-adic numbers in place of real numbers in the description of physics.

Historically what initiated much of this development was the observation in (Volovich 87, reviewed in VVZ 95, section XIV) that the integral expression for the Veneziano amplitude of the open bosonic string naturally generalizes from an integral over the real numbers (which in this case parameterize the boundary of the open string worldsheet) to the p-adic numbers. Since this concerns the bosonic string tachyon state, such p-adic string theory has been discussed a lot in the context of tachyon condensation and Sen's conjecture (Cottrell 02).

Generally, the development of string theory has shown that its worldsheet is usefully regarded as an object in algebraic geometry (see also at number theory and physics) and mathematically the generalization from algebraic varieties over the complex numbers to more general algebraic varieties (or schemes) is often natural, if not compelling. For instance when the Witten genus (essentially the partition function of the superstring) is refined to the string orientation of tmf then the elliptic curves over the complex numbers which serve as the toroidal worldsheets over the complex numbers are generalized to elliptic curves over general rings and by the fracture theorems the computations in tmf in fact typically proceed by decomposing the general problem into that of ellitpic curves over the rational numbers and over the p-adic integers. See at p-adic string theory for more on this.

## References

### Textbooks

• Vasiliĭ Vladimirov, I. V. Volovich, E. I. Zelenov, $p$-Adic analysis and mathematical physics, World Scientific, 1995

### Reviews

• L. Brekke, P. Freund, $p$-adic numbers in physics, Phys. Rep. 233 (1993)

• Branko Dragovich, Non-Archimedean Geometry and Physics on Adelic Spaces (arXiv:math-ph/0306023)

• Branko Dragovich, $p$-Adic and Adelic Quantum Mechanics (arXiv:hep-th/0312046)

• B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, $p$-Adic Mathematical Physics (arXiv:0904.4205)

### String theory

Discussion of the Veneziano amplitude for p-adic string theory is originally due to

• I. V. Volovich, p-Адическое пространство-время и теория струн, ТМФ, 71:3 (1987)free Rus. pdf; transl. $p$-adic space-time and string theory, Theor. Math. Phys. 71, 574–576 (1987), eng doi, nonfree Eng. pdf

• William Cottrell, $p$-adic Strings and Tachyon Condensation, 2002 (pdf)
A relation of $p$-adic string theory to number-theoretic Langlands duality is hypothesized and explored somewhat it
• Michele Nardelli, On the link between the structure of A-branes observed in the homological mirror symmetry and the classical theory of automorphic forms: mathematical connections with the modular elliptic curves, $p$-adic and adelic numbers and $p$-adic and adelic strings, in p-adic and adelic physics (pdf)