In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying physical laws (codified in a quantum field theory) as the energy (or mass) scale at which physical processes occur varies. In this context, a change in scale is called a scale transformation. The renormalization group is intimately related to scale invariance and conformal invariance, symmetries in which a system appears the same at all scales (self-similarity), where under the fixed point of the renormalization group flow the field theory is conformally invariant.

As the scale varies, it is as if one is decreasing (as RG is a semi-group and doesn't have a well-defined inverse operation) the magnifying power of a notional microscope viewing the system. In renormalizable theories, systems exhibit self-similarity across different scales, with parameters that describe system components changing as the scale varies. The components, or fundamental variables, may relate to atoms, elementary particles, atomic spins, etc. The parameters of the theory typically describe the interactions of the components. These may be variable couplings which measure the strength of various forces, or mass parameters themselves. The components themselves may appear to be composed of more of the self-same components as one goes to shorter distances.

For example, in quantum electrodynamics (QED), an electron appears to be composed of electron and positron pairs and photons, as one views it at higher resolution, at very short distances. The electron at such short distances has a slightly different electric charge than does the dressed electron seen at large distances, and this change, or running, in the value of the electric charge is determined by the renormalization group equation.

The idea of scale transformations and scale invariance is old in physics: Scaling arguments were commonplace for the Pythagorean school, Euclid, and up to Galileo. They became popular again at the end of the 19th century, perhaps the first example being the idea of enhanced viscosity of Osborne Reynolds, as a way to explain turbulence.

The renormalization group was initially developed for particle physics applications but has since been applied to solid-state physics, fluid mechanics, physical cosmology, and even nanotechnology. An early article by Ernst Stueckelberg and André Petermann in 1953 anticipates the idea in quantum field theory. Stueckelberg and Petermann opened the field conceptually. They noted that renormalization exhibits a group of transformations which transfers quantities from the bare terms to the counter terms. They introduced a function h(e) in quantum electrodynamics (QED), which is now known as the beta function (see below).

Murray Gell-Mann and Francis E. Low restricted the idea to scale transformations in QED in 1954, which are the most physically significant, and focused on asymptotic forms of the photon propagator at high energies. They determined the variation of the electromagnetic coupling in QED, by appreciating the simplicity of the scaling structure of that theory. They thus discovered that the coupling parameter g(μ) at the energy scale μ is effectively given by the (one-dimensional translation) group equation $${\displaystyle g(\mu )=G^{-1}\left(\left({\frac {\mu }{M}}\right)^{d}G(g(M))\right)}$$ or equivalently, ${\displaystyle G\left(g(\mu )\right)=G(g(M))\left({\mu }/{M}\right)^{d}}$, for an arbitrary function G (known as Wegner's scaling function, after Franz Wegner) and a constant d, in terms of the coupling g(M) at a reference scale M.

Gell-Mann and Low realized in these results that the effective scale can be arbitrarily taken as μ, and can vary to define the theory at any other scale: $${\displaystyle g(\kappa )=G^{-1}\left(\left({\frac {\kappa }{\mu }}\right)^{d}G(g(\mu ))\right)=G^{-1}\left(\left({\frac {\kappa }{M}}\right)^{d}G(g(M))\right)}$$

The gist of the RG is this group property: as the scale μ varies, the theory presents a self-similar replica of itself, and any scale can be accessed similarly from any other scale, by group action, a formal transitive conjugacy of couplings in the mathematical sense (Schröder's equation).