= Wilsonian coarse-grained statistical Hamiltonian
{c}
{title2=$e^{-\beta_TH_\Lambda[\phi_<]}=\int\mathcal D\phi_>\,e^{-\beta_TH_{\Lambda_0}[\phi_<+\phi_>]}$}
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 https://www.damtp.cam.ac.uk/user/tong/sft/sfthtml/S3.html[the Cambridge statistical field theory lecture notes].
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