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. Notably, we show that this force is necessarily anti-aligned with the four-velocity, enforcing an orthogonal constraint on motion. As a result, particle trajectories do not simply follow geodesics but instead reorganize into an emergent 3D+1 structure, revealing a fundamental link between the quantum potential and spacetime geometry.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 demonstrate that the four-force, derived from the quantum potential in flat Minkowski space, naturally incor-porates post-Newtonian corrections, revealing relativistic ravitational effects as an intrinsic feature of its dynamics.