Motivated by the Goldbach and Polignac conjectures in Number Theory, we propose the factorization of a classical non-interacting real scalar field (on a two-cylindrical spacetime) as a product of either two or three (so-called primer) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such primer fields and construct the corresponding Fock space by introducing creation operators $a_p^{\dag}$ (labeled by prime numbers $p$) acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory, suggests intriguing connections between different topics in Number Theory, notably the Riemann hypothesis and the Goldbach and Polignac conjectures. Our analysis also suggests that the (non) renormalizability properties of the proposed model could be linked to the possible validity or breakdown of the Goldbach conjecture for large integer numbers.
Monday, April 23, 2012
Prime numbers, quantum field theory and the Goldbach conjecture
Motivated by the Goldbach and Polignac conjectures in Number Theory, we propose the factorization of a classical non-interacting real scalar field (on a two-cylindrical spacetime) as a product of either two or three (so-called primer) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such primer fields and construct the corresponding Fock space by introducing creation operators $a_p^{\dag}$ (labeled by prime numbers $p$) acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory, suggests intriguing connections between different topics in Number Theory, notably the Riemann hypothesis and the Goldbach and Polignac conjectures. Our analysis also suggests that the (non) renormalizability properties of the proposed model could be linked to the possible validity or breakdown of the Goldbach conjecture for large integer numbers.
