Structural Algebraic Quantum Field Theory

Conventional quantum field theory is a method for studying structureless elementary particles. Non-elementary particles, on the other hand, are those with internal structure or particles that are made up of elementary constituents like the hadrons, which contain quarks and gluons. We introduce a structure-inclusive algebraic formulation of quantum field theory that could handle such particles and in which orthogonal polynomials play a central role. For simplicity, we consider non-elementary scalar particles in 3+1 Minkowski space-time and, in three appendices, we treat spinors with structure, massless vector fields, and the massive vector bosons. We show how to do scattering calculation in a nonlinear scalar-spinor coupling model where we find that loop integrals in the Feynman diagrams are remarkably finite. The aim of this short expos\'e is to motivate further studies and research using this approach.