Maxwell Klein-Gordon System with General Gauge Coupling on 4-Dimensional Flat Spacetime

Abstract

In this paper, we study the Maxwell Klein Gordon system (MKG) with the addition of general gauge couplings on four-dimensional Minkowski space. This system analyzes the interaction of the electromagnetic field (photon) coupled with the complex scalar field (spin-0 particles). This research considers the multi-field interactions by adding general coupling. Our method begins from the Lagrangian of Maxwell Klein Gordon equation with potential turn on. We derive the energy of the system, then write the equation of motion in the form of a non-linear partial differential equation. By using the spherical means method, we express the solutions in an integral form of its curvature. Then, by using temporal gauge condition and conservation of energy, we prove the inequality for some general coupling. The result shows that the first and the second derivative of it bounded by the energy and IL norm. This result will become the main key to proving the global existence of Maxwell Klein-Gordon’s theory.