In theoretical physics, renormalization group (RG) refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views it at different distance scales. In particle physics it reflects the changes in the underlying force laws as one varies the energy scale at which physical processes occur. A change in scale is called a "scale transformation" or "conformal transformation." The renormalization group is intimately related to "conformal invariance" or "scale invariance," a symmetry by which the system appears the same at all scales (socalled selfsimilarity).
As one varies the scale, it is as if one is changing the magnifying power of a microscope viewing the system. The system will generally make a selfsimilar copy of itself, with slightly different parameters describing the components of the system. The components, or fundamental variables, may be atoms, fundamental particles, atomic spins, etc. The parameters of the theory typically describe the interactions of the components. These may be "coupling constants" that measure the strength of various forces, or mass parameters themselves. The components themselves may appear to be composed of more of the selfsame components as one goes to shorter distances.
For example, an electron appears to be composed of electrons, antielectrons and photons as one views it at very short distances. The electron at very 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.
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The idea of scale transformations and scale invariance is old and venerable 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 within particle physics, but nowadays its applications are extended to solidstate physics, fluid mechanics, cosmology and even nanotechnology. An early article by Ernst Stueckelberg and Andre Peterman in 1953 anticipates the idea in quantum field theory.
Stueckelberg and Peterman opened the field. They noted that renormalization comes with a group of transformations which transfer quantities from the bare terms to the counterterms. Murray GellMann and Francis E. Low in 1954 restricted it to scaling transformations, which are the most interesting. They proposed the existence of a mathematical function of the coupling parameter g of a theory, ψ(g). This function determines the differential change of the coupling constant with a small change in energy scale μ by the "renormalization group equation:"
We indicate the more modern form, involving the function ψ(g) = β(g) / g introduced by Curtis Callan and Kurt Symanzik in the early 1970s. 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 field variables, which often has to deal with the problem of infinities in a quantum field theory (the RG exists independently of the infinities). This problem of dealing with the infinities of quantum field theory was solved for quantum electrodynamics by Richard Feynman, Julian Schwinger and SinItiro Tomonaga, who received the Nobel prize for their contributions. They effectively devised the theory of mass and charge renormalization in which the infinity is cutoff by an implicit ultralarge mass scale, Λ. The dependence of physical quantities, such as the electric charge or electron mass, on Λ is hidden, effectively swapped for the scales at which the physical quantities are measured.
GellMann and Low realized that the effective scale can be arbitrarily defined as μ, and can vary to define the theory at any other scale. The main point of the RG is that, as we vary the scale μ, the theory makes a selfsimilar replica of itself, with the tiny change in g given by the RG equation and ψ(g). The selfsimilarity stems from the fact that ψ(g) depends only upon the parameter(s) of the theory, not upon the scale μ.
A deeper understanding of the physical meaning of the renormalization group came from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "blockspin" 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 already the essential point and can be made exact, as one discovered through the many important contributions of Kenneth Wilson. The power of Wilson's ideas was demonstrated by a constructive and successive renormalization solution of a longstanding problem, the Kondo problem, in 1974 and the preceding seminal developments of his new method in the theory of secondorder phase transitions and critical phenomena in 1971. He was awarded the Nobel prize for this contribution in 1982.
The RG in particle physics was reformulated in 1970 in more physical terms by C. G. Callan and K. Symanzik. The ψ(g) = β(g) / g function, which describes the "running of coupling constant" with scale, is also found to be the "canonical trace anomaly" which represents the quantum mechanical breaking of scale symmetry of a field theory. Remarkably, quantum mechanics itself can induce mass through the trace anomaly and the running coupling constant. Applications of the RG to particle physics exploded in the 1970s with the canonization of the Standard Model.
In 1973 it was discovered that a theory of interacting colored quarks, called quantum chromodynamics had a negative β(g) function. This means that an initial high energy scale value of the coupling will produce 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 150 MeV. Conversely, the coupling becomes weak at very high energies, and the quarks become observable as pointlike particles, as anticipated by Bjorken scaling.
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 realspace RG techniques were developed in this sense, the most successful being the densitymatrix RG (DMRG), developed by S. R. White and R. M. Noack in 1992.
The conformal symmetry is associated with the vanishing of the β(g) 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 and is called a (trivial) ultraviolet fixed point. For heavy quarks, such as the Top quark, it is found that the coupling to the massgiving Higgs boson runs toward a fixed nonzero (nontrivial) infrared fixed point.
In string theory one requires conformal invariance of the string worldsheet as a fundamental symmetry: β = 0 is a requirement. Here β is a function of the geometry of the spacetime in which the string moves. This determines the spacetime 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 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.
This section introduces pedagogically a picture of RG which may be easiest to grasp: the block spin RG. It was devised by Leo P. Kadanoff in 1966.
Let us consider a 2D solid, a set of atoms in a perfect square array, as depicted in the figure. Let us assume that atoms interact among themselves only with their nearest neighbours, and that the system is at a given temperature T. The strength of their interaction is measured by a certain coupling constant J. The physics of the system will be described by a certain formula, say H(T,J).
Now we proceed to divide the solid into blocks of squares; we attempt to describe the system in terms of block variables, i.e.: some magnitudes which describe the average behaviour of the block. Also, let us assume that, due to a lucky coincidence, the physics of block variables is described by a formula of the same kind, but with different values for T and J: H(T',J'). (This isn't exactly true, of course, but it is often approximately true in practice, and that is good enough, to a first approximation)
Perhaps the initial problem was too hard to solve, since there were too many atoms. Now, in the renormalized problem we have only one fourth of them. But why should we stop now? Another iteration of the same kind leads to H(T'',J''), and only one sixteenth of the atoms. We are increasing the observation scale with each RG step.
Of course, the best idea is to iterate until there is only one very big block. Since the number of atoms in any real sample of material is very large, this is more or less equivalent to finding the long term behaviour of the RG transformation which took and . Usually, when iterated many times, this RG transformation leads to a certain number of fixed points.
Let us be more concrete and consider a magnetic system (e.g.: the Ising model), in which the J coupling constant denotes the trend of neighbour spins to be parallel. Physics is dominated by the tradeoff between the ordering J term and the disordering effect of temperature. For many models of this kind there are three fixed points:
(a) T = 0 and . This means that, at the largest size, temperature becomes unimportant, i.e.: the disordering factor vanishes. Thus, in large scales, the system appears to be ordered. We are in a ferromagnetic phase.
(b) and . Exactly the opposite, temperature has its victory, and the system is disordered at large scales.
(c) A nontrivial point between them, T = T_{c} and J = J_{c}. In this point, changing the scale does not change the physics, because the system is in a fractal state. It corresponds to the Curie phase transition, and is also called a critical point.
So, if we are given a certain material with given values of T and J, all we have to do in order to find out the large scale behaviour of the system is to iterate the pair until we find the corresponding fixed point.
In more technical terms, let us assume that we have a theory described by a certain function Z of the state variables {s_{i}} and a certain set of coupling constants {J_{k}}. This function may be a partition function, an action, a Hamiltonian, etc. It must contain the whole description of the physics of the system.
Now we consider a certain blocking transformation of the state variables , the number of must be lower than the number of s_{i}. Now let us try to rewrite the Z function only in terms of the . If this is achievable by a certain change in the parameters, , then the theory is said to be renormalizable.
For some reason, most fundamental theories of physics such as quantum electrodynamics, quantum chromodynamics and electroweak interaction, but not gravity, are exactly renormalizable. Also, most theories in condensed matter physics are approximately renormalizable, from superconductivity to fluid turbulence.
The change in the parameters is implemented by a certain βfunction: , which is said to induce a renormalization flow (or RG flow) on the Jspace. The values of J under the flow are called running coupling constants.
As it was stated in the previous section, the most important information in the RG flow are its fixed points. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points.
Since the RG transformations 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, the renormalization group is, in practice, a semigroup.
Let us consider a certain observable A of a physical system undergoing an RG transformation. The magnitude of the observable as the scale of the system goes from small to large may be (a) always increasing, (b) always decreasing or (c) other. In the first case, the observable is said to be a relevant observable; in the second, irrelevant and in the third, marginal.
A relevant operator is needed to describe the macroscopic behaviour of the system, but not an irrelevant observable. Marginal observables always give trouble when deciding whether to take them into account or not. A remarkable fact is that most observables are irrelevant, i.e.: the macroscopic physics is dominated by only a few observables in most systems. In other terms: microscopic physics contains (Avogadro's number) variables, and macroscopic physics only a few.
Before the RG, there was an astonishing empirical fact to explain: the coincidence of the critical exponents (i.e.: the behaviour near a second order phase transition) in very different phenomena, such as magnetic systems, superfluid transition (Lambda transition), alloy physics... This was called universality and is successfully explained by RG, just showing that the differences between all those phenomena are related to irrelevant observables.
Thus, many macroscopic phenomena may be grouped into a small set of universality classes, described by the set of relevant observables.
See also dangerously irrelevant operator.
RG, in practice, comes in two main flavours. The Kadanoff picture explained above refers mainly to the socalled realspace RG. Momentumspace 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 (high spatial frequency) modes. Since high spatial frequency is related to short length scales, the momentumspace RG results in an essentially similar coarsegraining effect as with realspace RG.
Momentumspace RG is usually performed on a perturbation expansion (i.e., approximation). The validity of such an expansion is predicated upon the true physics of our system being close to that of a free field system. In this case, we may calculate observables by summing the leading terms in the expansion. This approach has proved very 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 we will give a short description of charge renormalization in quantum electrodynamics (QED). Let us 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 pairs of (e.g.) electronspositrons, which will be annihilated very quickly. But in their short life, the electron will be attracted by the charge, and the positron will be repelled. Since this happens continuously, these pairs are effectively screening the charge from abroad. Therefore, the measured strength of the charge will depend on how close to our probes it may enter. We have a dependence of a certain coupling constant (the electric charge) with distance.
Energy, momentum and length scales are related, according to Heisenberg's uncertainty principle. The higher the energy or momentum scale we may reach, the lower the length scale we may probe. Therefore, the momentumspace RG practitioners sometimes claim to integrate out high momenta or high energy from their theories.
An exact renormalization group equation (ERGE) is one that takes irrelevant couplings into account. There are several formulations.
The Wilson ERGE is the simplest conceptually, but is practically impossible to implement. Fourier transform into momentum space after Wick rotating into Euclidean space. Insist upon a hard momentum cutoff, so that the only degrees of freedom are those with momenta less than Λ. The partition function is
For any positive Λ′ less than Λ, define S_{Λ′} (a functional over field configurations φ whose Fourier transform has momentum support within ) as
Obviously,
In fact, this transformation is transitive. If you compute S_{Λ′} from S_{Λ} and then compute S_{Λ″} from S_{Λ′}, this gives you the same Wilsonian action as computing S_{Λ″} directly from S_{Λ}.
The Polchinski ERGE involves a smooth UV regulator cutoff. Basically, the idea is an improvement over the Wilson ERGE. Instead of a sharp momentum cutoff, it uses a smooth cutoff. Essentially, we suppress contributions from momenta greater than Λ heavily. The smoothness of the cutoff, however, allows us to derive a functional differential equation in the cutoff scale Λ. As in Wilson's approach, we have a different action functional for each cutoff energy scale Λ. Each of these actions are supposed to describe exactly the same model which means that their partition functionals have to match exactly.
In other words, (for a real scalar field; generalizations to other fields are obvious)
and Z_{Λ} is really independent of Λ! We have used the condensed deWitt notation here. We have also split the bare action S_{Λ} into a quadratic kinetic part and an interacting part S_{int Λ}. This split most certainly isn't clean. The "interacting" part can very well also contain quadratic kinetic terms. In fact, if there is any wave function renormalization, it most certainly will. This can be somewhat reduced by introducing field rescalings. R_{Λ} is a function of the momentum p and the second term in the exponent is
when expanded. When , R_{Λ}(p)/p^2 is essentially 1. When , R_{Λ}(p)/p^2 becomes very very huge and approaches infinity. R_{Λ}(p)/p^2 is always greater than or equal to 1 and is smooth. Basically, what this does is to leave the fluctuations with momenta less than the cutoff Λ unaffected but heavily suppresses contributions from fluctuations with momenta greater than the cutoff. This is obviously a huge improvement over Wilson.
The condition that
can be satisfied by (but not only by)
Jacques Distler claimed [1] without proof that this ERGE isn't correct nonperturbatively.
The Effective average action ERGE involves a smooth IR regulator cutoff. The idea is to take all fluctuations right up to a IR scale k into account. The effective average action will be accurate for fluctuations with momenta larger than k. As the parameter k is lowered, the effective average action approaches the effective action which includes all quantum and classical fluctuations. In contrast, for large k the effective average action is close to the "bare action". So, the effective average action interpolates between the "bare action" and the effective action.
For a real scalar field, we add an IR cutoff
to the action S where R_{k} is a function of both k and p such that for , R_{k}(p) is very tiny and approaches 0 and for , . R_{k} is both smooth and nonnegative. Its large value for small momenta leads to a suppression of their contribution to the partition function which is effectively the same thing as neglecting large scale fluctuations. We will use the condensed deWitt notation
for this IR regulator.
So,
where J is the source field. The Legendre transform of W_{k} ordinarily gives the effective action. However, the action that we started off with is really S[φ]+1/2 φ⋅R_{k}⋅φ and so, to get the effective average action, we subtract off 1/2 φ⋅R_{k}⋅φ. In other words,
can be inverted to give J_{k}[φ] and we define the effective average action Γ_{k} as
Hence,
thus
is the ERGE which is also known as the Wetterich equation.
As there are infinitely many choices of R_{k}, there are also infinitely many different interpolating ERGEs. Generalization to other fields like spinorial fields is straightforward.
Although the Polchinski ERGE and the effective average action ERGE look similar, they are based upon very different philosophies. In the effective average action ERGE, the bare action is left unchanged (and the UV cutoff scale  if there is one  is also left unchanged) but we suppress the IR contributions to the effective action whereas in the Polchinski ERGE, we fix the QFT once and for all but vary the "bare action" at different energy scales to reproduce the prespecified model. Polchinski's version is certainly much closer to Wilson's idea in spirit. Note that one uses "bare actions" whereas the other uses effective (average) actions.
