  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.
