p-Adic TGD: Mathematical Ideas

TL;DR

This paper explores the mathematical foundations of p-adic numbers and their extensions, applying them to physical concepts like the Higgs mechanism, and revealing implications for space-time dimensions in p-adic models.

Contribution

It develops a p-adic mathematical framework that generalizes key physical and geometric concepts, including topology, probability, and symmetry, with implications for space-time dimensionality.

Findings

p-adic numbers induce a topology and differentiable structure on the real axis.

Canonical identification allows generalization of probability, Hilbert spaces, and metrics to p-adic context.

Unique algebraic extensions determine space-time dimensions as multiples of 4 or 8.

Abstract

The mathematical basis of p-adic Higgs mechanism discussed in papers hep-th@xxx.lanl.gov 9410058-62 is considered in this paper. The basic properties of p-adic numbers, of their algebraic extensions and the so called canonical identification between positive real numbers and p-adic numbers are described. Canonical identification induces p-adic topology and differentiable structure on real axis and allows definition of definite integral with physically desired properties. p-Adic numbers together with canonical identification provide analytic tool to produce fractals. Canonical identification makes it possible to generalize probability concept, Hilbert space concept, Riemannian metric and Lie groups to p-adic context. Conformal invariance generalizes to arbitrary dimensions since p-adic numbers allow algebraic extensions of arbitrary dimension. The central theme of all developments is the existence of square root, which forces unique algebraic extension with dimension $D=4$ and $D=8$ for $p>2$ and $p=2$ respectively. This in turn implies that the dimensions of p-adic Riemann spaces are multiples of $4$ in $p>2$ case and of $8$ in $p=2$ case.