Renormalisation

Renormalisation is a collection of techniques used in quantum field theory and statistical physics to handle infinities that appear in calculations and to understand how physical quantities depend on energy or length scale. In perturbative quantum field theory, interactions can lead to ultraviolet divergences; renormalisation introduces a regulator and absorbs these divergences into redefinitions of parameters such as masses and coupling constants, yielding finite predictions. The study of how these parameters change with energy scale is described by renormalisation group equations, yielding running coupling constants and beta functions. The concept distinguishes renormalisable theories, for which a finite number of parameters suffice to absorb all divergences, from non-renormalisable theories that require an infinite set of parameters and are treated as effective field theories at a given scale.

In statistical physics, renormalisation describes how system behaviour changes when viewed at larger length scales, via coarse-graining and rescaling. This approach explains universality: diverse microscopic details flow to the same fixed points, determining critical exponents near phase transitions. Wilson’s formulation of the renormalisation group provides a practical framework for these analyses.

Renormalisation also plays a crucial role in the Standard Model of particle physics, where precision tests rely on renormalised parameters and higher-order corrections. In effective field theories, renormalisation reflects the separation of scales: high-energy physics is encoded in a finite set of low-energy parameters, while heavier degrees of freedom are integrated out and suppressed by the cutoff scale.