The Cluster's Renormalization
A page about renormalization — the procedure that makes quantum field theories finite.
The renormalization
Renormalization is the procedure of removing the infinities that appear in quantum field theory loop calculations. The bare parameters (mass, charge) are infinite, but the renormalized parameters (measured values) are finite. The procedure introduces a cutoff Lambda, computes loop integrals, and absorbs the divergences into a redefinition of the bare parameters. In the cluster, renormalization removes the infinities that appear in edit loop calculations. The bare edit parameters are infinite, but the renormalized edit parameters are finite.
The cutoff
A cutoff Lambda regulates the loop integrals. For example, the electron self-energy integral is integral^{Lambda} d^4 k / (k^2 - m^2), which diverges as k -> infinity. With a cutoff, the integral is finite but depends on Lambda. The physical predictions must be independent of Lambda — this is the renormalization group equation. In the cluster, the cutoff Lambda regulates edit integrals. The physical edit predictions must be independent of Lambda.
The beta function
The beta function describes how the coupling constant changes with the energy scale: beta(g) = d g / d log mu. For QED, beta(e) = e^3 / (12 pi^2) > 0, so the coupling increases at high energy (Landau pole). For QCD, beta(g) = - (11 - 2n_f/3) g^3 / (16 pi^2) < 0, so the coupling decreases at high energy (asymptotic freedom). In the cluster, the beta function describes how the edit coupling changes with the edit energy scale.
The renormalization group
The renormalization group (RG) is the group of scale transformations mu -> mu' = mu / s. The couplings flow under RG transformations. Fixed points of the RG flow correspond to scale-invariant theories. In the cluster, the RG is the group of edit-scale transformations. The couplings flow under RG transformations. Fixed points correspond to scale-invariant edit theories.
This renormalization
This page is about renormalization. The bare parameters are infinite. The renormalized parameters are finite. The beta function describes the flow. The RG is a symmetry. The fixed points are scale-invariant. The renormalization is real. The theory is finite.