In
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, the
renormalization group (RG) refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales. In particle physics, it reflects the changes in the underlying force laws (codified in a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and manybody systems. It is the natural and quantitative language of particle physics and...
) as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the
uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
(cf.
Compton wavelengthThe Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons...
).
A change in scale is called a "
scale transformationIn physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...
". The renormalization group is intimately related to "scale invariance" and "conformal invariance", symmetries in which a system appears the same at all scales (socalled
selfsimilarityIn mathematics, a selfsimilar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically selfsimilar: parts of them show the same statistical properties at many scales...
). (However, note that
scale transformationsIn physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...
are included in
conformal transformationsIn theoretical physics, conformal symmetry is a symmetry under dilatation and under the special conformal transformations...
, in general: the latter including additional symmetry generators associated with
special conformal transformations.)
As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system. In socalled renormalizable theories, the system at one scale will generally be seen to consist of selfsimilar 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 "
couplingsIn physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
" which 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, in
quantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
(QED), an electron appears to be composed of electrons, positrons (antielectrons) 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 "
dressedIn theoretical physics, the term dressed particle refers to a bare particle together with some excitations of other quantum fields that are physically inseparable from the bare particle...
electron" seen at large distances, and this change, or "running," in the value of the electric charge is determined by the renormalization group equation.
History of the renormalization group
The idea of scale transformations and scale invariance is old in physics. Scaling arguments were commonplace for the
Pythagorean schoolPythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism...
,
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
and up to Galileo. They became popular again at the end of the 19th century, perhaps the first example being the idea of enhanced
viscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
of
Osborne ReynoldsOsborne Reynolds FRS was a prominent innovator in the understanding of fluid dynamics. Separately, his studies of heat transfer between solids and fluids brought improvements in boiler and condenser design.Life:...
, as a way to explain turbulence.
The renormalization group was initially devised in particle physics, but nowadays its applications extend to
solidstate physicsSolidstate physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solidstate physics studies how the largescale properties of solid materials result from...
,
fluid mechanicsFluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
,
cosmologyCosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...
and even
nanotechnologyNanotechnology is the study of manipulating matter on an atomic and molecular scale. Generally, nanotechnology deals with developing materials, devices, or other structures possessing at least one dimension sized from 1 to 100 nanometres...
. An early article by
Ernst StueckelbergErnst Carl Gerlach Stueckelberg was a Swiss mathematician and physicist. Career :In 1927 Stueckelberg got his Ph. D. at the University of Basel under August Hagenbach...
and Andre Petermann in 1953 anticipates the idea in
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and manybody systems. It is the natural and quantitative language of particle physics and...
. Stueckelberg and Petermann opened the field conceptually. They noted that
renormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of selfsimilar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
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 GellMannMurray GellMann is an American physicist and linguist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles...
and
Francis E. LowFrancis Eugene Low was an American theoretical physicist. He was an Institute Professor at MIT, and served as provost there from 1980 to 1985.Early career:...
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 and a constant d, in terms of
the coupling at a reference scale M. GellMann 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 selfsimilar 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, GellMann 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. CallanCurtis Callan is a theoretical physicist and a professor at Princeton University. He has conducted research in gauge theory, string theory, instantons, black holes, strong interactions, and many other topics...
and
K. SymanzikKurt Symanzik was a German physicist working in quantum field theory. Life :Symanzik was born in Lyck , East Prussia, and spent his childhood in Königsberg. He started studying physics in 1946 at Universität München but after a short time moved to Werner Heisenberg at Göttingen...
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 StueckelbergPetermann and GellMannLow 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 lowenergy physics value of 1/137. (Early applications to
quantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
are discussed in the influential book of
Nikolay BogolyubovNikolay Nikolaevich Bogolyubov was a Russian and Ukrainian Soviet mathematician and theoretical physicist known for a significant contribution to quantum field theory, classical and quantum statistical mechanics, and to the theory of dynamical systems; a recipient of the Dirac Prize...
and
Dmitry ShirkovDmitry Vasil'evich Shirkov is a Russian theoretical physicist known for his contribution to quantum field theory and to the development of the renormalization group method.Biography:...
in 1959.)
The renormalization group emerges from the
renormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of selfsimilar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
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 FeynmanRichard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
,
Julian SchwingerJulian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...
and
SinItiro Tomonagawas a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with Richard Feynman and Julian Schwinger.Biography:...
, 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
cutoffIn theoretical physics, cutoff is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation...
by an ultralarge
regulatorIntroduction:In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and nonsensical expressions by introducing an auxiliary concept of a 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 longerdistance 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. GellMann 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 selfsimilarity 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 "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 the conceptual point and was given full computational substance in the extensive important contributions of
Kenneth WilsonKenneth Geddes Wilson is an American theoretical physicist and Nobel Prize winner.As an undergraduate at Harvard, he was a Putnam Fellow. He earned his PhD from Caltech in 1961, studying under Murray GellMann....
. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a longstanding problem, the
Kondo problemIn physics, the Kondo effect describes the scattering of conduction electrons in a metal due to magnetic impurities. It is a measure of how electrical resistivity changes with temperature....
, in 1974, as well as the preceding seminal developments of his new method in the theory of secondorder phase transitions and
critical phenomenaIn physics, critical phenomena is the collective name associated with thephysics of critical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down...
in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.
(The connection between the StueckelbergPetermann and the Wilson renormalization groups has also been discussed by M. Duetsch, arXiv:1012.5604.)
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 quantummechanical 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 ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
.
In 1973, it was discovered that a theory of interacting colored quarks, called
quantum chromodynamicsIn theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of colorcharged fermions...
had a
negative beta function. This means that an initial highenergy 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 freedomIn physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...
), and the quarks become observable as pointlike particles, in
deep inelastic scatteringDeep inelastic scattering is the name given to a process used to probe the insides of hadrons , using electrons, muons and neutrinos. It provided the first convincing evidence of the reality of quarks, which up until that point had been considered by many to be a purely mathematical phenomenon...
, as anticipated by FeynmanBjorken 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 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
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 quarkThe top quark, also known as the t quark or truth quark, is an elementary particle and a fundamental constituent of matter. Like all quarks, the top quark is an elementary fermion with spin, and experiences all four fundamental interactions: gravitation, electromagnetism, weak interactions, and...
, it is calculated that the coupling to the
massgiving
Higgs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
runs toward a fixed nonzero (nontrivial)
infrared fixed pointIn physics, an infrared fixed point is a set of coupling constants, or other parameters that evolve from initial values at very high energies , to fixed stable values, usually predictable, at low energies...
.
In
string theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
conformal invariance of the string
worldsheet is 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 relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
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 phenomenaIn physics, critical phenomena is the collective name associated with thephysics of critical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down...
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 methodMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
.
Block spin renormalization group
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
. The strength of their
interaction is measured by a certain
coupling constantIn physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
. The
physics of the system will be described by a certain formula, say
.
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 variables which describe the
average behavior 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
and
:
. (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
, 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 modelThe Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
), in which the J coupling constant denotes the
trend of neighbour
spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
s to be parallel. The configuration of the system is the result of
the tradeoff between the ordering J term and the disordering
effect of temperature. For many models of this kind there are three
fixed points:
 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.
 and . Exactly the opposite, temperature dominates, and the system is disordered at large scales.
 A nontrivial point between them, and . In this point, changing the scale does not change the physics, because the system is in a fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reducedsize copy of the whole," a property called selfsimilarity...
state. It corresponds to the CurieIn physics and materials science, the Curie temperature , or Curie point, is the temperature at which a ferromagnetic or a ferrimagnetic material becomes paramagnetic on heating; the effect is reversible. A magnet will lose its magnetism if heated above the Curie temperature...
phase transitionA phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
, and is also called a critical pointIn physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions at which a phase boundary ceases to exist...
.
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
of the state variables
and a certain set of coupling constants
. This function may be a
partition functionIn quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral:...
,
an
actionIn physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
, a
HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
, 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
. Now let us try to rewrite the
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 electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
,
quantum chromodynamicsIn theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of colorcharged fermions...
and electroweak interaction, but not gravity, are exactly
renormalizable. Also, most theories in condensed matter physics are
approximately renormalizable, from
superconductivitySuperconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
to fluid
turbulence.
The change in the parameters is implemented by a certain
beta function:
, which is said to induce a
renormalization flow (or RG flow) on the
space. The values of
under the flow are
called
running couplings.
As 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 in such systems are
lossy (i.e.: the number of
variables decreases  see as an example in a different context,
Lossy data compressionIn information technology, "lossy" compression is a data encoding method that compresses data by discarding some of it. The procedure aims to minimize the amount of data that need to be held, handled, and/or transmitted by a computer...
), there need not be an inverse for a given RG
transformation. Thus, in such lossy systems, the renormalization group is, in fact, a
semigroupIn mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
.
Relevant and irrelevant operators, universality classes
Let us consider a certain observable
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 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; an irrelevant observable is not. Marginal observables
may or may not need to be taken into account. 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 transitionThe λ universality class is probably the most important group in condensed matter physics. It regroups several systems possessing strong analogies, namely, superfluids, superconductors and smectics...
), alloy physics, etc. This was called
universalityIn statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together...
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 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 (large wavenumber) modes. Since large wavenumbers are 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
perturbationPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
expansion. The validity of such an expansion is predicated upon the true physics of our system being close to that of
a
free fieldIn classical physics, a free field is a field whose equations of motion are given by linear partial differential equations. Such linear PDE's have a unique solution for a given initial condition....
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.
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 momentumspace RG practitioners sometimes declaim to integrate out high momenta or high energy from their theories.
Appendix: Exact Renormalization Group Equations
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 transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
into
momentum space after
Wick rotatingIn physics, Wick rotation, named after GianCarlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginarynumber variable for a realnumber variable...
into
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. Insist upon a hard momentum
cutoffIn theoretical physics, cutoff is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation...
,
so that the only degrees of freedom are those with momenta less than Λ. The
partition functionIn quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral:...
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
smoothSmooth means having a texture that lacks friction. Not rough.Smooth may also refer to:In mathematics:* Smooth function, a function that is infinitely differentiable; used in calculus and topology...
UV
regulatorIntroduction:In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and nonsensical expressions by introducing an auxiliary concept of a regulator...
cutoffIn theoretical physics, cutoff is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation...
. 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 equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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 functionalIn quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral:...
s 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 renormalizationIn quantum field theory, wave function renormalization is a rescaling, or renormalization, of quantum fields to take into account the effects of interactions. For a noninteracting or free field, the field operator creates or annihilates a single particle with probability 1. Once interactions are...
, 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
smoothSmooth means having a texture that lacks friction. Not rough.Smooth may also refer to:In mathematics:* Smooth function, a function that is infinitely differentiable; used in calculus and topology...
. 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 DistlerJacques Distler is a physicist currently working in string theory. He has been a professor of physics at the University of Texas at Austin since 1994.Early life and education:...
claimed
http://golem.ph.utexas.edu/~distler/blog/archives/000648.html without proof that this ERGE isn't correct nonperturbatively.
The
Effective average action ERGE involves a
smoothSmooth means having a texture that lacks friction. Not rough.Smooth may also refer to:In mathematics:* Smooth function, a function that is infinitely differentiable; used in calculus and topology...
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 actionIn quantum field theory, the effective action is a modified expression for the action, which takes into account quantummechanical corrections, in the following sense:...
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 actionIn quantum field theory, the effective action is a modified expression for the action, which takes into account quantummechanical corrections, in the following sense:...
.
For a real
scalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinateindependent, meaning that any two observers using the same units will agree on the...
, we add an IR cutoff
to the
actionIn physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
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
smoothSmooth means having a texture that lacks friction. Not rough.Smooth may also refer to:In mathematics:* Smooth function, a function that is infinitely differentiable; used in calculus and topology...
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 actionIn quantum field theory, the effective action is a modified expression for the action, which takes into account quantummechanical corrections, in the following sense:...
. 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.
Threshold effect
In
particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, a threshold effect is small corrections to rough calculations based on the renormalization group that arise from the detailed behavior near the scale where new physics takes place. In the context of renormalization group, we often "integrate out" modes of quantum fields with frequencies exceeding a certain energy scale (cutoff). If the cutoff is very close to the energy scale that we want to study, the threshold effects become important and contribute small terms to formulae such as those for the
beta functionIn theoretical physics, specifically quantum field theory, a beta function β encodes the dependence of a coupling parameter, g, on the energy scale, \mu of a given physical process....
s.
See also
 Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of selfsimilar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
with reference to perturbation theory, associated to momentumspace RG.
 Scale invariance
In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...
 Schröder's equation
 Regularization (physics)
Introduction:In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and nonsensical expressions by introducing an auxiliary concept of a regulator...
 Density matrix renormalization group
The density matrix renormalization group is a numerical variational technique devised to obtain the low energy physics of quantum manybody systems with high accuracy. It was invented in 1992 by Steven R...
 Functional renormalization group
In theoretical physics, functional renormalization group is an implementation of the renormalization group concept which is used in quantum and statistical field theory, especially when dealing with strongly interacting systems. The method combines functional methods of quantum field theory with...
 Critical phenomena
In physics, critical phenomena is the collective name associated with thephysics of critical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down...
Historical papers
 E.C.G. Stueckelberg
Ernst Carl Gerlach Stueckelberg was a Swiss mathematician and physicist. Career :In 1927 Stueckelberg got his Ph. D. at the University of Basel under August Hagenbach...
, A. Petermann (1953): Helv. Phys. Acta, 26, 499.
 M. GellMann
Murray GellMann is an American physicist and linguist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles...
, F.E. LowFrancis Eugene Low was an American theoretical physicist. He was an Institute Professor at MIT, and served as provost there from 1980 to 1985.Early career:...
(1954): Phys. Rev. 95, 5, 1300. The origin of the renormalization group.
 N.N. Bogoliubov
Nikolay Nikolaevich Bogolyubov was a Russian and Ukrainian Soviet mathematician and theoretical physicist known for a significant contribution to quantum field theory, classical and quantum statistical mechanics, and to the theory of dynamical systems; a recipient of the Dirac Prize...
, D.V. ShirkovDmitry Vasil'evich Shirkov is a Russian theoretical physicist known for his contribution to quantum field theory and to the development of the renormalization group method.Biography:...
(1959): The Theory of Quantized Fields. New York, Interscience. The first textbook on the renormalization group method.
 L.P. Kadanoff
Leo Philip Kadanoff is an American physicist. He is a professor of physics at the University of Chicago and a former President of the American Physical Society . He has contributed to the fields of statistical physics, chaos theory, and theoretical condensed matter physics.Biography:Kadanoff...
(1966): "Scaling laws for Ising models near ", Physics (Long Island City, N.Y.) 2, 263. The new blocking picture.
 C.G. Callan
Curtis Callan is a theoretical physicist and a professor at Princeton University. He has conducted research in gauge theory, string theory, instantons, black holes, strong interactions, and many other topics...
(1970): Phys. Rev. D 2, 1541.http://prola.aps.org/abstract/PRD/v2/i8/p1541_1 K. Symanzik (1970): Comm. Math. Phys. 18, 227.http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103842537 The new view on momentumspace RG.
 K.G. Wilson
Kenneth Geddes Wilson is an American theoretical physicist and Nobel Prize winner.As an undergraduate at Harvard, he was a Putnam Fellow. He earned his PhD from Caltech in 1961, studying under Murray GellMann....
(1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 4, 773.http://prola.aps.org/abstract/RMP/v47/i4/p773_1 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.
Pedagogical reviews
 N. Goldenfeld (1993): Lectures on phase transitions and the renormalization group. AddisonWesley.
 D.V. Shirkov
Dmitry Vasil'evich Shirkov is a Russian theoretical physicist known for his contribution to quantum field theory and to the development of the renormalization group method.Biography:...
(1999): Evolution of the Bogoliubov Renormalization Group. arXiv.org:hepth/9909024. A mathematical introduction and historical overview with a stress on group theory and the application in highenergy 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:hepth/0212049
 H.J. Maris, L.P. Kadanoff (1978): Teaching the renormalization group. American Journal of Physics, June 1978, Volume 46, Issue 6, pp. 652657. A pedestrian introduction to the renormalization group as applied in condensed matter physics.
 R. Shankar (1994): Renormalizationgroup approach to interacting fermions. Reviews of Modern Physics, Vol. 66, pp. 129–192 (1994). An excellent review to momentum space RG as applied in condensed matter physics. For non subscribers see arXiv:condmat/9307009
Books
 T. D. Lee
TsungDao Lee is a Chinese bornAmerican physicist, well known for his work on parity violation, the Lee Model, particle physics, relativistic heavy ion physics, nontopological solitons and soliton stars....
; Particle physics and introduction to field theory, Harwood academic publishers , 1981, [ISBN 3718600331]. Contains a Concise, simple, and trenchant summary of the group structure, in whose discovery he was also involved, as acknowledged in GellMann and Low's paper.
 L.Ts.Adzhemyan, N.V.Antonov and A.N.Vasiliev; The Field Theoretic Renormalization Group in Fully Developed Turbulence; Gordon and Breach, 1999. [ISBN 9056991450].
 Vasil'ev, A.N.; The field theoretic renormalization group in critical behavior theory and stochastic dynamics; Chapman & Hall/CRC, 2004. [ISBN 9780415310024] (Selfcontained treatment of renormalization group applications with complete computations);
 ZinnJustin, Jean ; Quantum field theory and critical phenomena, Oxford, Clarendon Press (2002), ISBN 0198509235 (a very thorough presentation of both topics);
 The same author: Renormalization and renormalization group: From the discovery of UV divergences to the concept of effective field theories, in: de WittMorette C., Zuber J.B. (eds), Proceedings of the NATO ASI on Quantum Field Theory: Perspective and Prospective, June 15–26, 1998, Les Houches, France, Kluwer Academic Publishers, NATO ASI Series C 530, 375388 (1999) [ISBN ]. Full text available in PostScript.
 Kleinert, H.
Hagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany , at theWest University of Timişoara, at thein Bishkek. He is also of the...
and Schulte Frohlinde, V; Critical Properties of φ^{4}Theories, World Scientific (Singapore, 2001); Paperback ISBN 9810246587. Full text available in PDF.
External links