Renormalisation group

The renormalisation group is a procedure for handling physics in which fluctuations occur on every length scale at once — the situation at a critical point, where the correlation length has diverged and no single scale dominates. Rather than treating all scales together, one integrates out the shortest-wavelength fluctuations, replacing blocks of degrees of freedom by effective single ones, and asks what theory describes the coarser system that remains. The result has the same form with different couplings, and repeating the step traces a flow — not of a particle through space, but of a theory through the space of all theories.

The structure of that flow explains criticality. A system exactly at its critical point has no characteristic length, so coarse-graining leaves it unchanged: it sits at a fixed point. Directions that shrink under the flow are irrelevant, and nearly all microscopic detail turns out to be irrelevant, which is why different systems flow to the same fixed point and share critical behaviour. The few directions that grow are relevant, and in a simple fluid or magnet there is essentially one — temperature — which is why criticality must be tuned to rather than found over a range.

The critical exponents are the eigenvalues of the flow linearised about the fixed point; the correlation-length exponent, for instance, is ν = ln b / ln λ₁ for a rescaling factor b and relevant eigenvalue λ₁. A number measured with a densitometer in a laboratory is thus an eigenvalue of a map on the space of theories. The same machinery, applied to quantum field theory, explains how running couplings work and why effective theories are predictive despite ignoring unknown short-distance physics.

Renormalisation grew out of 1940s quantum electrodynamics; the group structure was noted by Gell-Mann and Low in 1954. Its use for critical phenomena is due to Kenneth Wilson, 1971–1974, building on Leo Kadanoff's block-spin picture of 1966. Wilson received the 1982 Nobel Prize in Physics for it.