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

In this article, we present a methodological and systematic approach to solving the Klein–Gordon–Fock equation using the separation of variables method, with particular emphasis on its formulation in light-front coordinates. Although the plane-wave solution is well known in relativistic quantum mechanics, the explicit procedure leading to this solution is not always developed in detail, especially when the equation is written in light-front variables. We first revisit the Klein–Gordon–Fock equation for a free particle in Minkowski spacetime, showing how the usual separation between temporal and spatial variables leads to the expected plane-wave form. This treatment is used as a reference for the corresponding analysis in light-front coordinates. We then rewrite the equation in light-front coordinates, adopting αLF=2, and apply the separation of variables method to the coordinates x+, x−, and x⊥. In this formulation, x+ and x− appear coupled through the mixed derivative term ∂+∂−, with the separation process requiring an additional decoupling step involving an inverse relation and a nonzero constant λ. We show that an appropriate choice of this constant, together with a suitable choice of the superposition coefficients, allows the separated solution to recover the plane-wave structure obtained from the covariant transformation of the scalar product pμxμ. Thus, the results clarify the consistency between the direct coordinate-transformation approach and the explicit solution of the differential equation in light-front coordinates, while also highlighting the usefulness of separation of variables as a methodological tool in the study of relativistic wave equations.