 Wilsonian coarse-grained statistical Hamiltonian

Integrate field modes with momenta between a fixed microscopic ultraviolet cutoff ${\Lambda }_0$ and a lower cutoff $\Lambda$, retaining the modes below $\Lambda$ as backgrounds. The exact statistical Hamiltonian includes all generated interactions and field-independent constants. Integrating its retained modes reproduces the original partition function, so long-distance observables do not depend on the arbitrary intermediate cutoff. With sources retained, the corresponding coarse observables reproduce long-distance response and connected correlation functions. A local gradient expansion or a finite-coupling truncation is an additional approximation, rather than a consequence of cutoff independence. The momentum-shell construction is also explained in the Cambridge statistical field theory lecture notes.