Renormalization group

In theoretical physics, the renormalization group 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 as the energy 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, 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 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, 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.

History

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 in quantum electrodynamics, which is now known as the beta function.

Beginnings

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 at the energy scale μ is effectively given by the group equation
or equivalently,, for an arbitrary function G and a constant d, in terms of the coupling g 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:
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.
On the basis of this group equation and its scaling property, Gell-Mann and Low could then focus on infinitesimal transformations, and invented a computational method based on a mathematical flow function of the coupling parameter g, which they introduced. Like the function h of Stueckelberg–Petermann, their function determines the differential change of the coupling g with respect to a small change in energy scale μ through a differential equation, the renormalization group equation:
The modern name is also indicated, the beta function, introduced by Curtis Callan and Kurt Symanzik in 1970. Since it is a mere function of g, integration in g of a perturbative estimate of it permits specification of the renormalization trajectory of the coupling, that is, its variation with energy, effectively the function G in this perturbative approximation. The renormalization group prediction was confirmed 40 years later at the Large Electron–Positron Collider experiments: the fine structure "constant" of QED was measured to be about at energies close to 200 GeV, as opposed to the standard low-energy physics value of.

Deeper understanding

The renormalization group emerges from the renormalization of the quantum field variables, which normally has to address the problem of infinities in a quantum field theory. This problem of systematically handling the infinities of quantum field theory to obtain finite physical quantities was solved for QED by Richard Feynman, Julian Schwinger and Shin'ichirō Tomonaga, who received the 1965 Nobel prize for these contributions. They effectively devised the theory of mass and charge renormalization, in which the infinity in the momentum scale is cut off by an ultra-large regulator, Λ.
The dependence of physical quantities, such as the electric charge or electron mass, on the scale Λ is hidden, effectively swapped for the longer-distance scales at which the physical quantities are measured, and, as a result, all observable quantities end up being finite instead, even for an infinite Λ. Gell-Mann and Low thus realized in these results that, infinitesimally, while a tiny change in g is provided by the above RG equation given ψ, the self-similarity is expressed by the fact that ψ depends explicitly only upon the parameter of the theory, and not upon the scale μ. Consequently, the above renormalization group equation may be solved for g.
A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilation group of conventional renormalizable theories, considers methods where widely different scales of lengths appear simultaneously. It came from condensed matter physics: Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group. The "blocking idea" is a way to define the components of the theory at large distances as aggregates of components at shorter distances.
This approach covered the conceptual point and was given full computational substance in the extensive important work of Kenneth Wilson. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem, in 1975, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.

Reformulation

Meanwhile, the RG in particle physics had been reformulated in more practical terms by Callan and Symanzik in 1970. The above beta function, which describes the "running of the coupling" parameter with scale, was also found to amount to the "canonical trace anomaly", which represents the quantum-mechanical breaking of scale symmetry in a field theory. Applications of the RG to particle physics exploded in number in the 1970s with the establishment of the Standard Model.
In 1973, it was discovered that a theory of interacting colored quarks, called quantum chromodynamics, had a negative beta function. This means that an initial high-energy value of the coupling will eventuate a special value of at which the coupling blows up. This special value is the scale of the strong interactions, = and occurs at about 200 MeV. Conversely, the coupling becomes weak at very high energies, and the quarks become observable as point-like particles, in deep inelastic scattering, as anticipated by Feynman–Bjorken scaling. QCD was thereby established as the quantum field theory controlling the strong interactions of particles.
Momentum space RG also became a highly developed tool in solid state physics, but was hindered by the extensive use of perturbation theory, which prevented the theory from succeeding in strongly correlated systems.

Conformal symmetry

Conformal symmetry is associated with the vanishing of the beta function. This can occur naturally if a coupling constant is attracted, by running, toward a fixed point at which β = 0. In QCD, the fixed point occurs at short distances where g → 0 and is called a ultraviolet fixed point. For heavy quarks, such as the top quark, the coupling to the mass-giving Higgs boson runs toward a fixed non-zero infrared fixed point, first predicted by Pendleton and Ross, and C. T. Hill.
The top quark Yukawa coupling lies slightly below the infrared fixed point of the Standard Model suggesting the possibility of additional new physics, such as sequential heavy Higgs bosons.
In string theory, conformal invariance of the string world-sheet is a fundamental symmetry: β = 0 is a requirement. Here, β is a function of the geometry of the space-time in which the string moves. This determines the space-time dimensionality of the string theory and enforces Einstein's equations of general relativity on the geometry. The RG is of fundamental importance to string theory and theories of grand unification.
It is also the modern key idea underlying critical phenomena in condensed matter physics. Indeed, the RG has become one of the most important tools of modern physics.