A Map from Scalar Field Theory to Integer Polynomial Solutions

Abstract

The terms in the quantum scattering in scalar field theory models is parameterized by the invariants $\prod s_{ij}^{n_{ij}}$. The $s_{ij}$ are kinematic two-particle invariants, and the $n_{ij}$ are integers. The coefficients of these terms are computed via integrating all Feynman diagrams, or within the derivative expansion by solving the iteration equations. The latter has been provided recently; the functions which are prefactors of the individual terms $\prod s_{ij}^{n_{ij}}$ can be interpreted as terms in the expansions of L-series, which may be specified by collections of their zeroes. Once finding the appropriate elliptic curve coefficients, these quantum field solutions provide an algorithm to determining all of the mod p zeros to the algebraic curves. The latter is presumably determined by 'experimental' computer modeling or by the appropriate determination of the quantum prefactors.