Renormalization group

The renormalization group is a framework for tracking how a physical theory’s parameters and effective description change as the scale of observation changes.

Coarse-graining: A procedure that replaces fine-grained degrees of freedom with fewer variables describing larger-scale behavior. Renormalization group transformations often integrate out short-distance details through coarse-graining.
Universality class: A collection of systems sharing the same critical behavior despite differences in microscopic structure. Systems flowing to the same fixed point exhibit the same critical exponents.
Kadanoff block-spin transformation: A coarse-graining method that groups neighboring spins into larger blocks in a statistical model. Kadanoff made scale changes concrete by transforming spin systems block by block.
Dimensional regularization: A method for regulating divergent integrals by analytically continuing the number of spacetime dimensions. It handles divergences algebraically, whereas renormalization group analysis tracks scale dependence.
Renormalization flow: The trajectory of a theory’s parameters as the scale of observation changes. It records how couplings move under successive scale transformations.
Critical exponent: A power-law exponent describing how a physical quantity behaves near a continuous phase transition. Scaling dimensions at a fixed point determine critical exponents.
Kenneth Wilson: An American theoretical physicist who developed the modern renormalization group approach to critical phenomena and quantum field theory. Wilson unified scale transformations, fixed points, and critical behavior into a practical framework.
Minimal subtraction: A renormalization scheme that removes only divergent poles in dimensional regularization. It is a choice of scheme for defining parameters, not the scale-flow framework itself.
Renormalization-group fixed point: A theory unchanged by a renormalization group transformation, apart from possible rescaling. Fixed points identify scale-invariant behavior and organize nearby flows.
Wilson–Fisher fixed point: An interacting fixed point governing critical behavior in many systems below four spatial dimensions. It provides a central example of how an interacting theory predicts universality.

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