Кванттық өріс теорияларында масштаб өзгерістерінің қолданылуы

Method for using scale changes to understand physical theories such as quantum field theories

In theoretical physics, the term renormalization group (RG) refers to 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 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 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 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 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).

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(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 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 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 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.

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 (dilation) 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 μ at which the coupling blows up (diverges). This special value is the scale of the strong interactions, μ = Λ 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 was hindered by the extensive use of perturbation theory, which prevented the theory from succeeding in strongly correlated systems.

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, the coupling to the mass giving Higgs boson runs toward a fixed non zero (non trivial) infrared fixed point, first predicted by Pendleton and Ross (1981), 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. It is often used in combination with the Monte Carlo method.

This section introduces pedagogically a picture of RG which may be easiest to grasp: the block spin RG, devised by Leo P. Kadanoff in 1966. For D = 4, the triviality has yet to be proven rigorously, but lattice computations have provided strong evidence for this. This fact is important as quantum triviality can be used to bound or even predict parameters such as the Higgs boson mass in asymptotic safety scenarios. Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question. Since the RG transformations in such systems are lossy (i. e.: the number of variables decreases see as an example in a different context, Lossy data compression), there need not be an inverse for a given RG transformation. Thus, in such lossy systems, the renormalization group is, in fact, a semigroup, as lossiness implies that there is no unique inverse for each element.

Consider a certain observable A of a physical system undergoing an RG transformation. The magnitude of the observable as the length scale of the system goes from small to large determines the importance of the observable(s) for the scaling law:
If its magnitude then the observable is always increases relevant always decreases irrelevant other marginal
A relevant observable is needed to describe the macroscopic behaviour of the system; irrelevant observables are not needed. Marginal observables may or may not need to be taken into account. A remarkable broad fact is that most observables are irrelevant, i. e., the macroscopic physics is dominated by only a few observables in most systems. As an example, in microscopic physics, to describe a system consisting of a mole of carbon 12 atoms we need of the order of 10 (the Avogadro number) variables, while to describe it as a macroscopic system (12 grams of carbon 12) we only need a few. Before Wilson's RG approach, there was an astonishing empirical fact to explain: The coincidence of the critical exponents (i. e., the exponents of the reduced temperature dependence of several quantities near a second order phase transition) in very disparate phenomena, such as magnetic systems, superfluid transition (Lambda transition), alloy physics, etc. So in general, thermodynamic features of a system near a phase transition depend only on a small number of variables, such as the dimensionality and symmetry, but are insensitive to details of the underlying microscopic properties of the system. This coincidence of critical exponents for ostensibly quite different physical systems, called universality, is easily explained using the renormalization group, by demonstrating that the differences in phenomena among the individual fine scale components are determined by irrelevant observables, while the relevant observables are shared in common. Hence many macroscopic phenomena may be grouped into a small set of universality classes, specified by the shared sets of relevant observables.

Renormalization groups, in practice, come in two main "flavors". The Kadanoff picture explained above refers mainly to the so called real space RG. Momentum space RG on the other hand, has a longer history despite its relative subtlety. It can be used for systems where the degrees of freedom can be cast in terms of the Fourier modes of a given field. The RG transformation proceeds by integrating out a certain set of high momentum (large wavenumber) modes. Since large wavenumbers are related to short length scales, the momentum space RG results in an essentially analogous coarse graining effect as with real space RG. Momentum space RG is usually performed on a perturbation expansion. The validity of such an expansion is predicated upon the actual physics of a system being close to that of a free field system. In this case, one may calculate observables by summing the leading terms in the expansion. This approach has proved successful for many theories, including most of particle physics, but fails for systems whose physics is very far from any free system, i. e., systems with strong correlations. As an example of the physical meaning of RG in particle physics, consider an overview of charge renormalization in quantum electrodynamics (QED). Suppose we have a point positive charge of a certain true (or bare) magnitude. The electromagnetic field around it has a certain energy, and thus may produce some virtual electron positron pairs (for example). Although virtual particles annihilate very quickly, during their short lives the electron will be attracted by the charge, and the positron will be repelled. Since this happens uniformly everywhere near the point charge, where its electric field is sufficiently strong, these pairs effectively create a screen around the charge when viewed from far away. The measured strength of the charge will depend on how close our measuring probe can approach the point charge, bypassing more of the screen of virtual particles the closer it gets. Hence a dependence of a certain coupling constant (here, the electric charge) with distance scale. Momentum and length scales are related inversely, according to the de Broglie relation: The higher the energy or momentum scale we may reach, the lower the length scale we may probe and resolve. Therefore, the momentum space RG practitioners sometimes claim to integrate out high momenta or high energy from their theories.

The most successful variational RG method. A mathematical introduction and historical overview with a stress on group theory and the application in high energy physics. A pedestrian introduction to renormalization and the renormalization group. A pedestrian introduction to the renormalization group as applied in condensed matter physics.