From $p$-Adic to Zeta Strings

TL;DR

This paper reviews methods of constructing zeta strings from p-adic strings, exploring additive and multiplicative approaches, their properties, and potential links to ordinary string theories, emphasizing collective effects over all primes.

Contribution

It provides a comprehensive review of existing approaches to derive zeta strings from p-adic strings, including new perspectives on their properties and connections to traditional string models.

Findings

Different Lagrangians for zeta strings are obtained via additive and multiplicative approaches.

Properties of zeta string potentials and equations of motion are discussed.

Potential connections between zeta strings and ordinary string theories are explored.

Abstract

This article is related to construction of zeta strings from $p$-adic ones. In addition to investigation of $p$-adic string for a particular prime number $p$, it is also interesting to study collective effects taking into account all primes $p$. An idea behind this approach is that a zeta string is a whole thing with infinitely many faces which we see as $p$-adic strings. The name zeta string has origin in the Riemann zeta function contained in related Lagrangian. The starting point in construction a zeta string is Lagrangian for a $p$-adic open string. There are two types of approaches to get a Lagrangian for zeta string from Lagrangian for $p$-adic strings: additive and multiplicative approaches, that are related to two forms of the definition of the Riemann zeta function. As a result of differences in approaches, one obtains several different Lagrangians for zeta strings. We briefly discuss some properties of these Lagrangians, related potentials, equations of motion, mass spectrum and possible connection with ordinary strings. This is a review of published papers with some new views.