Renormalization Group - History of The Renormalization Group

History of The Renormalization Group

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, cosmology and even nanotechnology. An early article by Ernst Stueckelberg and Andre 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 counterterms. They introduced a function h(e) in 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

g(μ) = G−1( (μ/M)d G(g(M)) ) ,

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:

g(κ) = G−1( (κ/μ)d G(g(μ)) ) = G−1( (κ/M)d G(g(M)) ) .

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 conjugacy of couplings in the mathematical sense (Schröder's equation).

On the basis of this (finite) group equation, Gell-Mann and Low then focussed on infinitesimal transformations, and invented a computational method based on a mathematical flow function ψ(g) = G d/(∂G/∂g) of the coupling parameter g, which they introduced. Like the function h(e) of Stueckelberg and Petermann, their function determines the differential change of the coupling g(μ) with respect to a small change in energy scale μ through a differential equation, the renormalization group equation:

∂g / ∂ln(μ) = ψ(g) = β(g) .

The modern name is also indicated, the beta function, introduced by C. Callan and K. Symanzik in the early 1970s. 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 (cf Stueckelberg-Petermann and Gell-Mann-Low works) was confirmed 40 years later at the LEP accelerator experiments: the fine structure "constant" of QED was measured to be about 1/127 at energies close to 200 GeV, as opposed to the standard low-energy physics value of 1/137. (Early applications to quantum electrodynamics are discussed in the influential book of Nikolay Bogolyubov and Dmitry Shirkov in 1959.)

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 (although the RG exists independently of the infinities). 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 Sin-Itiro 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 (which could ultimately be taken to be infinite — infinities reflect the pileup of contributions from an infinity of degrees of freedom at infinitely high energy scales.). 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, while, infinitesimally, a tiny change in g is provided by the above RG equation given ψ(g), the self-similarity is expressed by the fact that ψ(g) depends explicitly only upon the parameter(s) of the theory, and not upon the scale μ. Consequently, the above renormalization group equation may be solved for (G and thus) g(μ).

A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional renormalizable theories, 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 contributions 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 1974, 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.

Meanwhile, the RG in particle physics had been reformulated in more practical terms by C. G. Callan and K. 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 (dilation) symmetry in a field theory. (Remarkably, quantum mechanics itself can induce mass through the trace anomaly and the running coupling.) 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 μ at which the coupling blows up (diverges). This special value is the scale of the strong interactions, μ = Λ_QCD and occurs at about 200 MeV. Conversely, the coupling becomes weak at very high energies (asymptotic freedom), 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 its success was hindered by the extensive use of perturbation theory, which prevented the theory from reaching success in strongly correlated systems. In order to study these strongly correlated systems, variational approaches are a better alternative. During the 1980s some real-space RG techniques were developed in this sense, the most successful being the density-matrix RG (DMRG), developed by S. R. White and R. M. Noack in 1992.

The 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(g) = 0. In QCD, the fixed point occurs at short distances where g → 0 and is called a (trivial) ultraviolet fixed point. For heavy quarks, such as the top quark, it is calculated that the coupling to the mass-giving Higgs boson runs toward a fixed non-zero (non-trivial) infrared fixed point.

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. It is often used in combination with the Monte Carlo method.