renormalizationgroup

Renormalization group (RG) is a framework for describing how the laws governing a physical system change with the length or energy scale at which the system is observed. In many theories one can integrate out short-distance or high-energy degrees of freedom to obtain an effective description valid at longer distances. The parameters of this effective theory, called couplings, depend on the scale and obey renormalization group flow equations.

Historically, the idea began with Kadanoff's block-spin construction in statistical mechanics and was developed into a quantitative method by Kenneth Wilson in the 1970s. The central picture is a flow in the space of Hamiltonians or actions produced by successive coarse-graining and rescaling. Fixed points of this flow correspond to scale-invariant theories and control critical behavior.

Key concepts include coarse graining, beta functions, and the classification of operators as relevant, irrelevant, or marginal. Under RG flow, relevant operators grow at long distances, irrelevant operators shrink, while marginal ones may stay constant. This structure explains universality: disparate microscopic details can yield the same large-scale behavior near a fixed point.

Applications span statistical mechanics and quantum field theory. In statistical physics, RG yields critical exponents for phase transitions. In quantum field theory, it explains running couplings and phenomena such as asymptotic freedom, and underpins the use of effective field theories. Method variants include real-space and momentum-space (Wilsonian) RG, and functional renormalization group approaches.

Renormalization group methods continue to be central in condensed matter physics, particle physics, and statistical physics for connecting microscopic models to macroscopic phenomena through scale changes.