An Introduction to a Generalized Relativistic Wave Dierential Operator in the Quantum Field Theory

Abstract

This work presents a unified operator framework for quantum field theory based on a generalized relativistic wave differential operator defined on Minkowski space. Through the systematic selection of discrete parameters α, β, λ, m, and n ∈ N, this operator reproduces fundamental cases such as the Klein-Gordon, Dirac, and non-monogenic operators, revealing deep algebraic connections between seemingly disparate field theories. We introduce a quantum field tensor constructed from scalar, spinor, and gauge fields, where binary indices act as activation signatures. This tensor product structure naturally generates all possible field combinations and their couplings, providing a first-principles derivation of interaction terms beyond the traditional Lagrangian formulation. The framework is systematically extended to incorporate gauge interactions through a generalized minimal coupling prescription, yielding gauge-covariant operators that reproduce fundamental equations such as Proca and Maxwell. The formalism elegantly unifies gauge self-interactions, matter couplings, and free propagation within a single coherent equation, opening new avenues for exploring higher-spin fields and extensions beyond the Standard Model.