Renormalization group

In theoretical physics, the renormalization group (RG) refers to a mathematical 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 force laws (codified in a quantum field theory) as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the uncertainty principle. g ( μ ) = G − 1 ( ( μ M ) d G ( g ( M ) ) ) {displaystyle g(mu )=G^{-1}left(left({frac {mu }{M}} ight)^{d}G(g(M)) ight)} , g ( κ ) = G − 1 ( ( κ μ ) d G ( g ( μ ) ) ) = G − 1 ( ( κ M ) d G ( g ( M ) ) ) {displaystyle g(kappa )=G^{-1}left(left({frac {kappa }{mu }} ight)^{d}G(g(mu )) ight)=G^{-1}left(left({frac {kappa }{M}} ight)^{d}G(g(M)) ight)} . ∂ g ∂ ln ⁡ μ = ψ ( g ) = β ( g ) {displaystyle displaystyle {frac {partial g}{partial ln mu }}=psi (g)=eta (g)} . In theoretical physics, the renormalization group (RG) refers to a mathematical 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 force laws (codified in a quantum field theory) as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the uncertainty principle. 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 (so-called self-similarity). As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system. In so-called renormalizable theories, the system at one scale will generally be seen to consist of self-similar copies of itself when viewed at a smaller scale, with different parameters describing the components of the system. 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 electrons, positrons (anti-electrons) 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 devised in particle physics, but nowadays its applications extend 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 transfer quantities from the bare terms to the counter terms. They introduced a function h(e) in quantum electrodynamics (QED), which is now called the beta function (see below). Murray Gell-Mann and Francis E. Low in 1954 restricted the idea to scale transformations in QED, 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 group equation for some function G (unspecified—nowadays called Wegner's scaling function) 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: