Generally, p-adic numbers represent an interesting systems for arithmetics in which the numbers can continue indefinitely to the left instead of right. Ordinary numbers such as
- 3.14159265358979 ...
(I've memorized 100 digits but you won't be bothered by them here) continue indefinitely on the right side, but every politically educated scientist knows that the Left is a dead end, and the digits must eventually stop there.
In the p-adic system where "p" is a prime, things are a bit different. You may imagine that the numbers carry an infinite number of digits in the opposite direction: all left-wingers should move to the p-adic world. For example, if you want to divide "1/9", you obtain morally something like
- - ......111111111
Note that if this integer is multiplied by 9, you obtain -....9. If you subtract 1, you obtain .....0000 which is zero, and therefore the number had to be one. ;-) Of course the real p-adic numbers work a bit differently but you get the idea. Yes, it's crazy.
You should realize that the integers, as well as integer multiples of "p^{-k}", exist in the p-adic system much like they exist in the ordinary system of real numbers. However, different "more irrational" numbers outside this category are included in the p-adic system from those that we know in the system of real numbers. This is connected with a different definition of the absolute value.
The p-adic numbers are also called non-archimedean numbers because of a very interesting history. Archimedes, considered by Gauss to be one of two greatest mathematicians ever :-), was the first scientist who used infinitesimal numbers, although he did not believe their existence. The negative belief was actually more terminologically important than his positive work because "Archimedean" refers to a system of arithmetics that contains no infinitesimally small numbers "x", i.e. numbers such that the absolute value of "nx" is smaller than one for every integer "n". Yes, p-adic numbers don't satisfy this property.
In the context of string theory, one can use the set of p-adic numbers - a set that has a completely different topology than the real line - as a replacement for the real numbers parameterizing a point on the boundary of a worldsheet. One can define integrals over p-adic numbers, and there is a p-adic generalization of the Veneziano formula - namely the Beta function which was the first amplitude that started the whole field of string theory (under the obsolete name "dual models").
The Beta function is made out of ratios of Gamma functions, and one can also define a p-adic Gamma function for every "p" which is a rather simple ratio of polynomials of the argument. The interesting thing is that the product of the p-adic Veneziano amplitudes for all prime-integral values of "p" is equal to the inverse ordinary Veneziano amplitude. In other words, the product of all Veneziano amplitudes over p-adic numbers with all prime integers "p" as well as the ordinary Veneziano amplitude gives you one.
This fact about the infinite product can be easily proven if you realize that
- the Riemann zeta function can be written as an infinite product over prime integers
- the Gamma function can be written as a ratio of two zeta functions with roughly opposite arguments, up to some factors that cancel.
You can decide that it is natural to interpret the previously undefined "p=1" p-adic amplitude to be the ordinary amplitude in which the worldsheet coordinates are real. It is useful to know that the p-adic bosonic string amplitude can be derived from string field theory with a very simple and exactly known action. It is schematically equal, up to some semi-important overall factors, to the integral over ordinary spacetime of the Lagrangian density
- L = - T exp(-box/2) T + T^{p+1} / {p+1}
where "T" is the tachyon, the only field that occurs in p-adic string theory unless "p=1" in which case there are infinitely many other fields. The Feynman diagrams are trees where a single propagator is split to "p" new propagators, and such trees resemble the bizarre, left-wing p-adic worldsheet discussed above.
Debashis's proposal to prove that "p=1" p-adic string corresponds to the ordinary continuous string is based on the renormalization group: the trees are treated as discretizations of the smooth worldsheet. This naively leads to a contradiction because the trees seem to have a different volume than necessary, and Debashis seems to show that this problem actually goes away if the trees are embedded in hyperbolic spaces of constant negative curvature as opposed to Euclidean spaces which I find very elegant if it works.
The topology could also be a bit similar to the Penrose diagram for eternal inflation, but I choose not to emphasize this point. ;-)
In his picture, the effect of the renormalization group flow is to replace "p" by E-th root of "p", and the formal limit where "E" is sent to infinity is argued to reproduce the "p=1" continuous, real, right-wing string that we know and love. You will have to look at the paper yourself to decide whether it is a complete picture and whether it actually helps to explain why the product of the p-adic amplitudes for all primes "p" as well as "p=1" should give one. And, equally importantly, whether you think that the p-adic strings and p-adic numbers are going to be useful for a better understanding of the inner workings or for a better treatment of string theory.
I would also be happy if someone told me whether there is anything such as p-adic M-theory.
