Towards effective Lagrangians for adelic strings

TL;DR

This paper explores the construction of effective Lagrangians for adelic strings, unifying p-adic and real string theories, resulting in nonlocal scalar field theories involving the Riemann zeta function.

Contribution

It proposes new approaches to derive effective field Lagrangians for the p-adic sector of adelic strings, incorporating nonlocality via the Riemann zeta function.

Findings

Derived Lagrangians describe higher-point scattering amplitudes.

Constructed nonlocal scalar field theories with infinite derivatives.

Linked string theory with number theory through the Riemann zeta function.

Abstract

p-Adic strings are important objects of string theory, as well as of p-adic mathematical physics and nonlocal cosmology. By a concept of adelic string one can unify and simultaneously study various aspects of ordinary and p-adic strings. By this way, one can consider adelic strings as a very useful instrument in the further investigation of modern string theory. It is remarkable that for some scalar p-adic strings exist effective Lagrangians, which are based on real instead of p-adic numbers and describe not only four-point scattering amplitudes but also all higher ones at the tree level. In this work, starting from p-adic Lagrangians, we consider some approaches to construction of effective field Lagrangians for p-adic sector of adelic strings. It yields Lagrangians for nonlinear and nonlocal scalar field theory, where spacetime nonlocality is determined by an infinite number of derivatives contained in the operator-valued Riemann zeta function. Owing to the Riemann zeta function in the dynamics of these scalar field theories, obtained Lagrangians are also interesting in themselves.