Solving the Klein-Gordon-Fock Equation Using Separation of Variables in Light-Front Coordinates

In this article, the authors present a methodological approach for solving the Klein-Gordon-Fock equation by means of the separation of variables method, with emphasis on its formulation in the light-front coordinate system. The justification for the study arises from the fact that, although plane-wave solutions are frequently presented in the literature within the context of relativistic quantum mechanics, the solving process leading to their derivation is not always developed explicitly. Initially, they revisit the Klein-Gordon-Fock equation for a free particle in Minkowski spacetime, showing how the separation between the temporal and spatial parts leads to the general solution and allows the interpretation of the components associated with particle and antiparticle propagation. Next, they rewrite the equation in the light-front coordinate system, adopting α_LF=2, and again apply the separation of variables method to the coordinates x⁺, x⁻, and x^⊥. The results show that the solution obtained in this frame preserves the plane-wave structure and recovers, under suitable choices of the superposition coefficients, the wave function expected from the covariant transformation of the scalar product pμxμ. In this way, they demonstrate the coherence between the direct approach through coordinate transformation and the explicit solution of the differential equation. Moreover, the development presented reinforces the didactic potential of separation of variables in the introductory study of relativistic quantum mechanics and indicates the usefulness of the light-front formalism in future treatments involving external potentials, interaction fields, and applications in quantum field theory.