P-adic Amplitudes Research Articles

Local Zeta Functions and Koba–Nielsen String Amplitudes

This article is a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions (in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. p-Adic string amplitudes share desired characteristics with their Archimedean counterparts, such as crossing symmetry and invariance under Möbius transformations. A direct connection between p-adic amplitudes and the Archimedean ones is through the limit p→1. Gerasimov and Shatashvili studied the limit p→1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov–Shatashvili Lagrangian are related to the limit p→1 of the p-adic Koba–Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba–Nielsen amplitude is just a new type of local zeta function, which can be studied using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba–Nielsen amplitudes as functions of the kinematic parameters. The Koba–Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Qp, Fp((T))), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba–Nielsen amplitudes give rise to new amplitudes which we have called Denef–Loeser amplitudes. Throughout the article, we have emphasized the explicit calculations in the cases of 4 and 5 points.

NEW P-ADIC STRING AMPLITUDES

It is shown that the branching multiplicative characters (corresponding to certain quadratic extensions) of the p-adic number fields can be used to construct the tree amplitudes in the p-adic string theory. In particular, the p-adic analogues of the tachyon amplitudes in the Neveu-Schwarz sector of the superstring and the p-adic amplitudes of the emission of the massless vector particles for the bosonic non-archimedean string are obtained.

ADELIC STRING AND SUPERSTRING AMPLITUDES

Four point amplitudes in open strings and superstrings are shown to factor in an infinite number of ways as Euler products of p-adic amplitudes. All these “adelic” formulae follow from the functional equation for Dirichlet L-functions.

P-adic string amplitude

P-adic N-point amplitudes are derived and it is shown that these amplitudes are reproduced from an effective scalar field theory. Some properties of the scalar field theory (a non-trivial vacuum solution, a new symmetry of p-adic amplitudes) are discussed. Also, comments on the validity of the adelic formula for p-adic string amplitudes are given.

New p-adic strings from old dual models.

In the simplest bosonic p-adic string, a central role is played by an ${S}_{3}$ symmetry in the four-particle amplitude. Using an ${S}_{3}$ analysis of four-point dual amplitudes, we exhibit a general class of p-adic string amplitudes. All are shown to satisfy a factorization (adelic) property for the infinite product over p-adic amplitudes. Explicit examples are worked through for a p-adic string with spin-0 and spin-2 poles and no tachyon.

P-Adic string theories

In this paper we study the p-adic string theory. Starting with the integral representation of the real string scattering amplitude, we obtain various p-adic string amplitudes. We show that it is possible to obtain a suitable p-adic theory starting with either the Koba-Nielsen or the Bardakçi-Ruegg formula. However, whereas in the real string case both representations of the amplitude correspond to the same theory, the p-adic generalizations lead to different theories. Furthermore, there is more than one p-adic generalization of each form of the amplitude. Each p-adic theory gives its own set of diagrammatic rules implying that it is describable by some nonlocal effective field theory. For completeness, in the first part of the paper we give an overview of the p-adic mathematics necessary for constructing our generalizations and for computing the resultant p-adic amplitudes.

On the adelic string amplitudes

Some remarkable properties of the adelic string amplitudes for the physical domain of the Mandelstam variables are considered. It is shown that the p-adic four-point functions are always negative. Also, a formula is obtained which expresses the product of moduli of the p-adic amplitudes and the Veneziano amplitude in terms of the zeta functions. This product is absolutely convergent unlike the divergent product of these amplitudes without moduli, recently considered by Freund and Witten. Using the zeta function representation, p-adic interpolation of the Veneziano amplitude is also considered.

Harmonic analysis and p-adic strings

A generalization of the Veneziano amplitude is considered which is a convolution of two characters on a field K. Choosing K in an appropriate way, one can obtain the usual Veneziano amplitude, the Virasoro-Shapiro amplitude, the p-adic amplitudes, and the finite Galois field amplitude. The cases when K is an algebra or a group are also discussed. These cases can be of interest in the context of the quantization of spacetime.