# Renormalization group

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, but nowadays its applications are extended to solid state
physics, fluid mechanics and even cosmology.

## Contents

## 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}**
. The strength of their
interaction is measured by a certain coupling constant **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J}**
. The
physics of the system will be described by certain **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle formula}**
, say
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(T,J)}**
.

Now we proceed to divide the solid into **blocks** of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J}**
: **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(T',J')}**
.

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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (T,J)\to (T',J')}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 neigbour 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) **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=0}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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) **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\to\infty}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=T_c}**
and
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z}**
of the state variables
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{s_i\}}**
and a certain set of coupling constants
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{s_i\}\to \{\tilde s_i\}}**
,
the number of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde s_i}**
must be lower than the number of
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i}**
. Now let us try to rewrite the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z}**
function *only* in terms of the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde s_i}**
, but
keeping it invariant under the change. If this is achievable by a
certain change in the parameters, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{J_k\}\to \{\tilde J_k\}}**
, the the theory is said to be
**renormalizable**.

By some reason, all fundamental theories of physics but gravity (QED, QCD and electro-weak interaction) 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
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta}**
-function: **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\tilde J_k\}=\beta(\{ J_k \})}**
, which is said to induce a
**renormalization flow** (or RG flow) on the
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J}**
-space. The values of **running coupling constants**.

As it was stated in the previous paragraph, 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}**
of a physical
system undergoing a RG transformation. The magnitude of the observable
may be (a) always increasing, (b) always decreasing or (c) have
fluctuations, but no definite average trend. 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
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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, 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.

## Real and 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**. The technique which has a longer history, although is harder
to grasp at a first attempt, is momentum-space RG. Within this
framework, the degrees of freedom under consideration are the
Fourier modes of a given field, and a RG transformation proceeds
by *integrating out* a certain set of high momentum. Since high
momentum is related to short length scales, the main picture is the
same as that of real space RG.

Momentum space RG usually starts out with a perturbative series, i.e.: the idea that the true physics of our system is close to that of a free system, so we may find out the differences between the values of observables by summing a series of terms with decreasing magnitude, classified according to powers of some of the coupling constant. This approach has proved very successful for some theories, including most of particle physics, but fails for systems whose physics is very far away 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 practicioners 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 R.P. Feynman, J. Schwinger and S. Tomonaga, who obtained 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.

## 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
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.
- J. Rodrguez-Laguna (2002): Real space renormalization group techniques and

applications. Ph.D. thesis, UCM. [1].

es:Grupo de Renormalización ja:繰り込み群 ru:Ренормализационная группа sl:Renormalizacijska grupa