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In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. It was initially devised within particle physics (in the guise of the beta function and the Callan-Symanzik equations), but nowadays its applications are extended to solid state physics, fluid mechanics and even cosmology.

Kadanoff's blocking picture

This section introduces pedagogically the picture of RG which may be easiest to grasp: Kadanoff's blocks. It was devised by Leo P. Kadanoff in 1966, when RG already had a long history behind it.

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 ${\displaystyle T}$. The strength of their interaction is measured by a certain coupling constant ${\displaystyle J}$. The physics of the system will be described by certain formula, say ${\displaystyle H(T,J)}$.

Now we proceed to divide the solid into blocks of ${\displaystyle 2\times 2}$ squares. Now 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 ${\displaystyle T}$ and ${\displaystyle J}$: ${\displaystyle H(T',J')}$. (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 ${\displaystyle H(T'',J'')}$, 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 ${\displaystyle (T,J)\to (T',J')}$ and ${\displaystyle (T',J')\to (T'',J'')}$. 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) ${\displaystyle T=0}$ and ${\displaystyle J\to \infty }$. 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) ${\displaystyle T\to \infty }$ and ${\displaystyle J\to 0}$. Exactly the opposite, temperature has its victory, and the system is disordered at large scales.

(c) A nontrivial point between them, ${\displaystyle T=T_{c}}$ and ${\displaystyle J=J_{c}}$. 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.

Elements of RG theory

In more technical terms, let us assume that we have a theory described by a certain function ${\displaystyle Z}$ of the state variables ${\displaystyle \{s_{i}\}}$ and a certain set of coupling constants ${\displaystyle \{J_{k}\}}$. 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 ${\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}}$, the number of ${\displaystyle {\tilde {s}}_{i}}$ must be lower than the number of ${\displaystyle s_{i}}$. Now let us try to rewrite the ${\displaystyle Z}$ function only in terms of the ${\displaystyle {\tilde {s}}_{i}}$. If this is achievable by a certain change in the parameters, ${\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}}$, then the theory is said to be renormalizable.

For some reason, most fundamental theories of physics such as quantum electrodynamics, quantum chromodynamics and electro-weak 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 ${\displaystyle \beta }$-function: ${\displaystyle \{{\tilde {J}}_{k}\}=\beta (\{J_{k}\})}$, which is said to induce a renormalization flow (or RG flow) on the ${\displaystyle J}$-space. The values of ${\displaystyle J}$ 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.

Relevant and irrelevant operators, universality classes

Let us consider a certain observable ${\displaystyle A}$ 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 ${\displaystyle \approx 10^{23}}$ 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.

Momentum space RG

RG, in practice, comes in two main flavours. 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 (high spatial frequency) modes. Since high spatial frequency is related to short length scales, the momentum-space RG results in an essentially similar coarse-graining effect as with real-space RG.

Momentum-space 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.) electrons-positrons, 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 charged will depend on how close to it our probes 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 momentum-space RG practitioners sometimes claim to integrate out high momenta or high energy from their theories.

History of the renormalization group

Of course, the idea of 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.

RG made its appearance in physics in very different guise. An article by E.C.G. Stueckelberg and A. Peterman in 1953 and another one by M. Gell-Mann and F.E. Low in 1954 opened the field, but as a mathematical trick to get rid of the infinities in quantum field theory. As a pure technique, it obtained maturity with the book by N.N. Bogoliubov and D.V. Shirkov in 1959. The RG term was inherited from this time and, although most people agree that it is incorrect, no other alternative has been proposed so far.

The technique was developed further by Richard Feynman, Julian Schwinger and Sin-Itiro Tomonaga, who received the Nobel prize for their contributions to quantum electrodynamics. They devised the theory of mass and charge renormalization.

But real understanding of the physical meaning of the technique came with Leo P. Kadanoff's paper in 1966. The new blocking idea reached maturity with Kenneth Wilson's solution of the Kondo problem in 1974. He was awarded the Nobel prize of this contribution in 1982. The old-style RG in particle physics was reformulated in 1970 in more physical terms by C.G. Callan and K. Symanzik. In this field, momentum space RG is a very mature tool, its only failure being the non-renormalizability of gravity. 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, as we have stated before, 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 80's some real space RG techniques were developed in this sense, being the most successful the density matrix RG (DMRG), developed by S.R. White and R.M. Noack in 1992.

See also

• Renormalized perturbation theory is the main technique associated to

momentum-space RG.

• Density matrix renormalization group is the most successful variational

real-space RG technique up to date.

• Exact renormalization group equation
• Critical phenomena
• Fractals

References

Historical papers

• E.C.G. Stueckelberg, A. Peterman (1953): Helv. Phys. Acta, 26, 499. M. Gell-Mann, F.E. Low (1954): Phys. Rev. 95, 5, 1300. The origin of renormalization group
• N.N. Bogoliubov, D.V. Shirkov (1959): The theory of quantized fields, Interscience. The first text-book on RG.
• L.P. Kadanoff (1966): Scaling laws for Ising models near ${\displaystyle T_{c}}$, Physica 2, 263. The new blocking picture.
• C.G. Callan (1970): Phys. Rev. D 2, 1541. K. Symanzik (1970): Comm. Math. Phys. 18, 227. The new view on momentum-space RG.
• K.G. Wilson (1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 4, 773. The main success of the new picture.
• S.R. White (1992): Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863. The most successful variational RG method.

Didactical reviews

• N. Goldenfeld (1993): Lectures on phase transitions and the renormalization group. Addison-Wesley.
• D.V. Shirkov (1999): Evolution of the Bogoliubov Renormalization Group. arXiv.org:hep-th/9909024. A mathematical introduction and historical overview with a stress on group theory and the application in high-energy physics.
• B. Delamotte (2004): A hint of renormalization. American Journal of Physics, Vol. 72, No. 2, pp. 170\u2013184, February 2004. A pedestrian introduction to renormalization and the renormalization group. For non subscribers see arXiv.org:hep-th/0212049
• H.J. Maris, L.P. Kadanoff (1978): Teaching the renormalization group. American Journal of Physics, June 1978, Volume 46, Issue 6, pp. 652-657. A pedestrian introduction to the renormalization group as applied in condensed matter physics.

External links

• Renormalization group on arxiv.org

• Quantum field theory
• Statistical mechanics
• Renormalization group
• Fixed points