Prime numbers, quantum field theory and the Goldbach conjecture

TL;DR

This paper explores a quantum field theory model where a scalar field is constructed from prime-numbered fermion modes, linking number theory and quantum physics, and discusses implications for the Goldbach conjecture.

Contribution

It introduces a novel quantum field model based on prime modes and analyzes its renormalization properties under the Riemann hypothesis.

Findings

The prime-mode quantum field is not renormalizable.

The model's properties depend on the Riemann hypothesis.

Potential implications for the Goldbach conjecture are discussed.

Abstract

Motivated by the Goldbach conjecture in Number Theory and the abelian bosonization mechanism on a cylindrical two-dimensional spacetime we study the reconstruction of a real scalar field as a product of two real fermion (so-called \textit{prime}) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators $b_{p}^{\dag}$ --labeled by prime numbers $p$-- acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory and the assumption of the Riemann hypothesis, allow us to prove that the theory is not renormalizable. We also comment on the potential consequences of this result concerning the validity or breakdown of the Goldbach conjecture for large integer numbers.