Derivation of a 4D Force for the 4D Quantum Potential

In this paper, we derive a four-force associated with the four-dimensional (4D) quantum potential introduced in Ref. [1]. This force, defined as F μ = −m(ημν − uμuν )∂ν Q(x), generalizes the de Broglie-Bohm quantum force to relativistic contexts and ensures consistency with special relativity. We present a rigorous derivation of this force from first principles, starting with the Klein-Gordon equation and systematically transitioning to a classical scalar field through a mean-field approximation. The effective action for this field is derived using the Wentzel-Kramers-Brillouin (WKB) method, ensuring a well-motivated bridge between quantum and classical regimes. We discuss how this four-force can drive particle dynamics in flat Minkowski space, reproducing gravitational effects such as post-Newtonian corrections in General Relativity (GR).