Effective Nonlocal Lagrangian Origin of Acceleration-Squared Vacuum Forces

This work develops an effective nonlocal action framework for describing vacuum-induced forces in accelerated systems. The model incorporates memory effects through a temporal kernel coupling accelerations at different times, representing the vacuum as an effective medium with finite relaxation time. In the adiabatic regime, where the system evolves slowly compared to the memory timescale, the nonlocal dynamics reduces to an effective local contribution proportional to the square of the acceleration. The resulting force scales as ħ a² / c³, suggesting a quantum origin related to vacuum response rather than particle-specific properties. The formulation provides a phenomenological effective field theory perspective on acceleration-dependent effects, highlighting the role of nonlocality, dissipation, and temporal correlations in vacuum dynamics.