Harmonic analysis and p-adic strings
This is one of the series of the author's papers advocating the following point of view: at the Planck length scale the usual geometrical notions are no longer applicable. In particular, a non-Archimedean geometry may emerge; this kind of geometry can, in turn, be realized on the p-adic numbers. Therefore in the process of building the physical models the p- adic numbers should be used instead of real ones which would only emerge in the macroscopic limit (in the reviewer's opinion at least a part of arguments, leading to such conclusions, are of the form: ``Why not?''). The theory to be considered from such point of view is the string theory, which is widely believed to be the proper microscopic theory at Planck scale. The starting point is the Veneziano amplitude. According to its two well known representations, one in terms of gamma functions and the integral one, there are two ways of generalizing the Veneziano amplitude. Firstly, one may use the p-adic interpolation of gamma functions. The second way is to generalize the integral representations to the form of the convolution of two characters \[ A(\gamma_ a,\gamma_ b)=\int_{K}\gamma_ a(x)\gamma_ b(1-x)dx, \] where K is some field (or a group), \(\gamma_ a\) is a multiplicative character on K with values in some field while dx is a measure on K. Various possible choices of K are considered: \({\mathbb{R}}\), \({\mathbb{C}}\), \({\mathbb{Q}}_ p\), finite Galois field. The general conclusion is that due to the problems with the analyticity properties connected with locality the approach using the p- adic interpolation seems to be more promissing.
scientific article; zbMATH DE number 4094122