An interface between physics and number theory

TL;DR

This paper develops an advanced algebraic framework connecting quantum physics, number theory, and algebraic structures like polyzeta functions, offering new insights into the mathematical foundations of quantum field theory.

Contribution

It introduces a richer Hopf algebra structure that links non-relativistic quantum mechanics to relativistic quantum field theory through algebraic and number theoretic methods.

Findings

Extended Hopf algebra encompasses pQFT algebraic structures

Links algebraic structures to polyzeta functions in pQFT

Suggests the Euler gamma constant may be rational

Abstract

We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a {\em mathematical} route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of polyzeta functions, extensions of the Riemann Zeta function, since these occur naturally in pQFT. This provides a link between physics, algebra and number theory. As a by-product of this approach, we are led to indicate {\it inter alia} a basis for concluding that the Euler gamma constant $\gamma$ may be rational.