In both cases, the RG reveals that the low-energy physics is determined by the flow of couplings, not the bare microscopic values.
Start with a microscopic Hamiltonian. Integrate out short-wavelength (high-energy) fluctuations, then rescale lengths and fields. This generates a new Hamiltonian with modified coupling constants. The set of transformations forms a semi-group—the Renormalization Group.
The RG approach reveals a deep unity:
Kondo found that the third-order scattering term produces a contribution to the resistivity proportional to (\ln(D/T)), where (D) is the bandwidth and (T) the temperature. This logarithm was not small—it diverges as (T \to 0). Perturbation theory fails. The problem became infamous: how to compute physical quantities when the coupling grows at low energies?
In both cases, the RG reveals that the low-energy physics is determined by the flow of couplings, not the bare microscopic values.
Start with a microscopic Hamiltonian. Integrate out short-wavelength (high-energy) fluctuations, then rescale lengths and fields. This generates a new Hamiltonian with modified coupling constants. The set of transformations forms a semi-group—the Renormalization Group.
The RG approach reveals a deep unity:
Kondo found that the third-order scattering term produces a contribution to the resistivity proportional to (\ln(D/T)), where (D) is the bandwidth and (T) the temperature. This logarithm was not small—it diverges as (T \to 0). Perturbation theory fails. The problem became infamous: how to compute physical quantities when the coupling grows at low energies?