Adaptive Constraint Cascade (ACC): A Unified Framework for Quantum Field Theory – Case Study ACC_3: Nonrelativistic QM → Quantum Field Theory

We extend the Adaptive Constraint Cascade (ACC) framework to the transition from nonrelativistic quantum mechanics (NRQM) to quantum field theory (QFT). The control parameter is the energy cutoff Λ, which determines the number of resolved degrees of freedom. At low energies (Λ → 0), the system has a finite number of particles and obeys the Schrödinger equation (NRQM). At high energies (Λ → ∞), infinitely many degrees of freedom emerge, and the system obeys field commutation relations and the equations of QFT. The ACC_3 framework equation is presented as a deformed commutation relation that smoothly interpolates between these two regimes, with finite-Λ corrections encoded in a smoothed delta function and non-local terms. We illustrate the cascade using a one-dimensional crystal lattice: as the cutoff is lowered, the continuous phonon field coarse-grains into discrete atomic oscillations, recovering NRQM in the deep infrared limit. This work is the third and final part of the ACC trilogy, following ACC_1 (quantum–classical transition) and ACC_2 (Newton → SR → GR).