Renormalization

Renormalization is a collection of techniques in quantum field theory and statistical physics that aims to remove or absorb infinities or scale dependence into redefined parameters so that predictions for physical quantities are finite and regulator-independent.

In perturbative quantum field theory, calculations often produce ultraviolet divergences. Renormalization introduces counterterms and renormalization constants to relate bare parameters to renormalized ones measured at a chosen momentum scale. Physical observables are required to be independent of the regulator and of the specific regularization scheme used.

Renormalization group concepts describe how a theory changes with energy scale. Couplings become scale-dependent, known as running couplings, and are governed by beta functions. Wilson’s approach interprets renormalization as coarse-graining: integrating out short-distance degrees of freedom, rescaling to restore the cutoff, and following the flow in parameter space. This flow captures how theories evolve when viewed at different resolutions.

Fixed points of the renormalization group flow correspond to scale-invariant theories and determine critical behavior and universality classes in statistical mechanics and phase transitions.

In quantum field theory, renormalizability means a finite set of parameters suffices to absorb divergences; nonrenormalizable terms are suppressed at low energies and can be organized within effective field theories, which describe physics up to a given cutoff.

Applications include critical phenomena, phase transitions, lattice gauge theory, and the understanding of asymptotic freedom in quantum chromodynamics. While intermediate steps depend on the renormalization scheme and regulator, physical predictions remain scheme-independent.