Factorization of number into prime numbers viewed as decay of particle into elementary particles conserving energy

TL;DR

This paper models the factorization of integers into primes as particle decay processes in quantum and string-like models, linking number theory with energy conservation and particle physics, and analyzing symmetries and stability.

Contribution

It introduces quantum and string-inspired models where prime factorization is viewed as particle decay, incorporating energy conservation and symmetry breaking, a novel approach connecting number theory and physics.

Findings

Energy of particles proportional to log of their label

Broken modular symmetry to additive integer symmetry after interactions

Supersymmetric model achieves cancellation of zero-point energies

Abstract

Number theory is considered, by proposing quantum mechanical models and string-like models at zero and finite temperatures, where the factorization of number into prime numbers is viewed as the decay of particle into elementary particles conserving energy. In these models, energy of a particle labeled by an integer $n$ is assumed or derived to being proportional to $\ln n$. The one-loop vacuum amplitudes, the free energies and the partition functions at finite temperature of the string-like models are estimated and compared with the zeta functions. The $SL(2, {\bf Z})$ modular symmetry, being manifest in the free energies is broken down to the additive symmetry of integers, ${\bf Z}_{+}$, after interactions are turned on. In the dynamical model existing behind the zeta function, prepared are the fields labeled by prime numbers. On the other hand the fields in our models are labeled, not by prime numbers but by integers. Nevertheless, we can understand whether a number is prime or not prime by the decay rate, namely by the corresponding particle can decay or can not decay through interactions conserving energy. Among the models proposed, the supersymmetric string-like model has the merit of that the zero point energies are cancelled and the energy levels may be stable against radiative corrections.