Why Ramified Primes Are So Special Physically

Abstract

Ramified primes are special in the sense that their expression as a product of primes P_i of extension contains higher than first powers and the number P_i is smaller than the maximal number n defined by the dimension of the extension. The proposed interpretation of ramified primes is as p-adic primes characterizing space-time sheets assignable to elementary particles and even more general systems. It is not quite clear why ramified primes appear as preferred p-adic primes and in the following Dedekind zeta functions and what I call ramified zeta functions inspired by the interpretation of zeta function as analog of partition function are used in attempt to understand why ramified primes could be physically special. The intuitive feeling is that quantum criticality is what makes ramified primes so special. In O(p)=0 approximation the irreducible polynomial defining the extension of rationals indeed reduces to a polynomial in finite field F_p and has multiple roots for ramified prime, and one can deduce a concrete geometric interpretation for ramification as quantum criticality using M₈-H duality.