p-Adic description of Higgs mechanism I: p-Adic square root and p-adic light cone

TL;DR

This paper introduces a p-adic framework for the Higgs mechanism, utilizing p-adic square roots and light cones to explore particle mass spectra within Topological GeometroDynamics, emphasizing the role of p-adic topology in quantum criticality.

Contribution

It proposes a novel p-adic geometric approach to the Higgs mechanism, connecting p-adic square roots and light cones to particle mass calculations in TGD.

Findings

p-adic square root existence implies 4D algebraic extension of p-adic numbers

p-adic light cone is modeled as convergence-cubes of square root functions

Convergence cubes serve as natural quantization volumes in p-adic field theory

Abstract

This paper is the first one in the series devoted to the calculation of particle mass spectrum in Topological GeometroDynamics. TGD Universe is critical at quantum level and an attractive idea to realize criticality is via conformal invariance. Ordinary real numbers do not allow this but if one assumes that in long length scales p-adic topology replaces real topology as effective topology situation changes. The existence of square root in the vicinity of p-adic real axis implies 4-dimensional algebraic extension of p-adic numbers ($p>2$), which can be regarded as padic counterpart of light cone and consists of convergence-cubes of p-adic square root function. Later work has demonstrated that convergence cubes of square root function serve as natural quantization volumes in p-adic field theory limit of TGD.