Integrating out high-momentum modes reveals effective low-energy physics
Coarse-graining: Integrate out modes with k ∈ [Λ/b, Λ], then rescale. UV fluctuations dress lower-energy couplings.
Shown: Field φ(x) in momentum space. High-k modes (red) are integrated out each RG step, leaving an effective theory with renormalized couplings.
Key idea: Physics at scale μ knows nothing about modes above Λ(μ). Different UV completions give same IR if they flow to the same fixed point.
Fixed points: Gaussian (g=0, trivial), Wilson-Fisher (g*, interacting) — controls critical phenomena.