Generalization of Riemann Zeta to Dedekind Zeta & Adelic Physics

Abstract

Adelic physics postulates that a hierarchy of extensions of rationals defines an evolutionary hierarchy of physics of matter and cognition. The Galois groups of the extension play a key role in the proposal and define number theoretical symmetries. The so called L-functions, in particular,  Riemann zeta coding information about ordinary primes and its generalizations to extensions of rationals, are expected to play key role in adelic physics. The so called radial conformal weights for the representations of super-symplectic algebra at boundaries of causal diamonds are proposed to correspond to zeros of ζ, and the discrete p-adic coupling constant evolution for the inverse of electroweak U(1) coupling is proposed to correspond to the zeros of  ζ associated to certain p-adic length scales for primes p near prime powers of 2. If this picture generalizes to all extensions K/Q of rationals, the zeta functions ζ_K would give quite an impressive quantitative grasp to adelic physics. In this article this possibility will be considered in more detail and it is found that extensions of rationals have a nice physical interpretation.