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| {{Renormalization and regularization}}
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| In [[theoretical physics]], the '''renormalization group (RG)''' refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different ''[[Length scale|distance scales]]''. In particle physics, it reflects the changes in the underlying force laws (codified in a [[quantum field theory]]) as the ''energy scale'' at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the [[uncertainty principle]] (cf. [[Compton wavelength]]).
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| A change in scale is called a "[[Scale invariance|scale transformation]]". The renormalization group is intimately related to "scale invariance" and "conformal invariance", symmetries in which a system appears the same at all scales (so-called [[self-similarity]]). (However, note that [[Scale invariance|scale transformations]] are included in [[Conformal symmetry|conformal transformations]], in general: the latter including additional symmetry generators associated with [[special conformal transformation]]s.)
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| As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system. In so-called renormalizable theories, the system at one scale will generally be seen to consist of self-similar copies of itself when viewed at a smaller scale, with different parameters describing the components of the system. The components, or fundamental variables, may relate to atoms, elementary particles, atomic spins, etc. The parameters of the theory typically describe the interactions of the components. These may be variable "[[coupling constant|couplings]]" which measure the strength of various forces, or mass parameters themselves. The components themselves may appear to be composed of more of the self-same components as one goes to shorter distances.
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| For example, in [[quantum electrodynamics]] (QED), an electron appears to be composed of electrons, positrons (anti-electrons) and photons, as one views it at higher resolution, at very short distances. The electron at such short distances has a slightly different electric charge than does the "[[dressed particle|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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| ==History==
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| The idea of scale transformations and scale invariance is old in physics. Scaling arguments were commonplace for the [[Pythagoreanism|Pythagorean school]], [[Euclid]] and up to [[Galileo]].<ref>[http://www.av8n.com/physics/scaling.htm Pedagogical site on Scaling]</ref> 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.
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| The renormalization group was initially devised in particle physics, but nowadays its applications extend to [[solid-state physics]], [[fluid mechanics]], [[cosmology]] and even [[nanotechnology]]. An early article<ref>[[Ernst Stueckelberg|Stueckelberg, E.C.G.]] and Petermann, A. (1953). "La renormalisation des constants dans la theorie de quanta", ''Helv. Phys. Acta'' '''26''', 499.</ref> by [[Ernst Stueckelberg]] and Andre Petermann in 1953 anticipates the idea in [[quantum field theory]]. Stueckelberg and Petermann opened the field conceptually. They noted that [[renormalization]] exhibits a group of transformations which transfer
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| quantities from the bare terms to the counterterms. They introduced a function ''h''(''e'') in QED, which is now called the [[Beta-function#Quantum_electrodynamics|beta function]] (see below).
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| [[Murray Gell-Mann]] and [[Francis E. Low]] in 1954 restricted the idea to scale transformations in QED,<ref>{{cite journal |last=Gell-Mann |first=M. |authorlink=Murray Gell-Mann | coauthor=[[Francis E. Low|Low, F.E.]] |year=1954 |title=Quantum Electrodynamics at Small Distances | journal=Physical Review |volume=95|issue=5|pages=1300–1312 | doi=10.1103/PhysRev.95.1300| bibcode=1954PhRv...95.1300G}}</ref> 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
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| {{Equation box 1
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| |indent =:
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| |equation = {{math|''g''(''μ'') {{=}} ''G''<sup>−1</sup>( (''μ''/''M'')<sup>d</sup> ''G''(''g''(''M'')) ) }},
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| |cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |bgcolor=#F9FFF7}}
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| for some function ''G'' (unspecified—nowadays called [[Franz Wegner|Wegner]]'s scaling function) and a constant ''d'', in terms of
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| the coupling ''g(M)'' at a reference scale ''M''.
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| Gell-Mann 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'':
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| {{Equation box 1
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| |indent =:
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| |equation = {{math|''g''(κ) {{=}} ''G''<sup>−1</sup>( (κ/''μ'')<sup>d</sup> ''G''(''g''(''μ'')) ) {{=}} ''G''<sup>−1</sup>( (κ/''M'')<sup>d</sup> ''G''(''g''(''M'')) )}} .
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| |cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |bgcolor=#F9FFF7}}
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| The gist of the RG is this group property: as the scale μ varies, the theory presents a self-similar replica of itself, and ''any scale can be accessed similarly from any other scale'', by group action, a formal conjugacy of couplings<ref>{{cite journal |last=Curtright |first=T.L. |authorlink=Thomas Curtright |coauthor=Zachos, C.K. |date=March 2011 |title=Renormalization Group Functional Equations | journal=Physical Review D|volume=83|issue= 6|pages=065019| doi=10.1103/PhysRevD.83.065019|bibcode = 2011PhRvD..83f5019C |arxiv = 1010.5174 }}</ref> in the mathematical sense ([[Schröder's equation]]).
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| On the basis of this (finite) group equation and its scaling property, Gell-Mann and Low could then focus on infinitesimal transformations, and invented a computational method based on a mathematical flow function {{math|''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:'''''
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| {{Equation box 1
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| |indent =:
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| |equation = {{math| ∂''g''/∂ln(''μ'') {{=}} ''ψ''(''g'') {{=}} ''β''(''g'') }}.
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| |cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |bgcolor=#F9FFF7}}
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| The modern name is also indicated, the [[Beta function (physics)|beta function]], introduced by [[Curtis Callan|C. Callan]] and [[Kurt Symanzik|K. Symanzik]] 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''
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| in this perturbative approximation. The renormalization group prediction (cf Stueckelberg-Petermann and Gell-Mann-Low 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 low-energy physics value of 1/137. (Early applications to [[quantum electrodynamics]] are discussed in the influential book of [[Nikolay Bogolyubov]] and [[Dmitry Shirkov]] in 1959.<ref>[[Nikolay Bogolyubov|N.N. Bogoliubov]], [[Dmitry Shirkov|D.V. Shirkov]] (1959): ''The Theory of Quantized Fields''. New York, Interscience.</ref>)
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| The renormalization group emerges from the [[renormalization]] of the quantum field variables, which normally has to address the problem of ''infinities'' in a quantum field theory (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 Feynman]], [[Julian Schwinger]] and [[Sin-Itiro Tomonaga]], who received the 1965 Nobel prize for these contributions. They effectively devised the theory of mass and charge renormalization, in which the infinity in the momentum scale is [[cutoff|cut-off]] by an ultra-large [[Regularization (physics)|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 longer-distance scales at which the physical quantities are measured, and, as a result, all observable quantities end up being ''finite'', instead, even for an ''infinite'' Λ. Gell-Mann and Low thus realized in these results that, while, infinitesimally, a tiny change in '' g'' is provided by the above RG equation given ψ(''g''), the self-similarity is expressed by the fact that ψ(''g'') depends explicitly ''only upon the parameter(s) of the theory, and not upon the scale'' μ. Consequently, the above renormalization group equation may be solved for (''G'' and thus) ''g''(μ).
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| A deeper understanding of the physical meaning and generalization of the
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| 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 "block-spin" renormalization group.<ref>[[Leo Kadanoff|L.P. Kadanoff]] (1966): "Scaling laws for Ising models near <math>T_c</math>", Physics (Long Island City, N.Y.) '''2''', 263.</ref> The ''blocking idea'' is a way to define the components of the theory at large distances as aggregates of components at shorter distances.
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| This approach covered the conceptual point and was given full computational substance<ref>[[Kenneth G. Wilson|K.G. Wilson]](1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. '''47''', 4, 773.</ref> in the extensive important contributions of [[Kenneth G. Wilson|Kenneth Wilson]]. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the [[Kondo effect|Kondo problem]], in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and [[critical phenomena]] in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.
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| Meanwhile, the RG in particle physics had been reformulated in more practical terms by C. G. Callan and K. Symanzik in 1970.<ref>{{Cite journal
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| | last1 = Callan | first1 = C. | title = Broken Scale Invariance in Scalar Field Theory
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| | doi = 10.1103/PhysRevD.2.1541 | journal = Physical Review D | volume = 2 | issue = 8 | pages = 1541 | year = 1970
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| | pmid = | pmc = |bibcode = 1970PhRvD...2.1541C }}; {{cite doi| 10.1007/BF01649434}}</ref> The above beta function, which describes the "running of the coupling" parameter with scale, was also found to amount to the "canonical trace anomaly", which represents the quantum-mechanical breaking of scale (dilation) symmetry in a field theory. (Remarkably, quantum mechanics itself can induce mass through the trace anomaly and the running
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| coupling.) Applications of the RG to particle physics exploded in number in the 1970s with the establishment of the [[Standard Model]].
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| In 1973, it was discovered that a theory of interacting colored quarks, called [[quantum chromodynamics]] had a '''''negative beta function'''''. This means that an initial high-energy value of the coupling will eventuate a special value of μ at which the coupling blows up (diverges). This special value is the scale
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| of the strong interactions, [[Coupling constant#QCD scale|μ = Λ<sub>QCD</sub>]] and occurs at about 200 MeV. Conversely, the coupling becomes weak at very high energies ([[asymptotic freedom]]), and the quarks become observable as point-like particles, in [[deep inelastic scattering]], as anticipated by Feynman-Bjorken scaling. QCD was thereby established as the quantum field theory controlling the strong interactions of particles.
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| 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, [[Calculus of variations|variational]] approaches are a better alternative. During the 1980s some real-space RG techniques were developed in this sense, the most successful being the density-matrix RG (DMRG), developed by S. R. White and R. M. Noack in 1992.
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| The conformal symmetry is associated with the vanishing of the
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| beta function. This can occur naturally
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| if a coupling constant is attracted, by running, toward a
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| '''fixed point''' at which β(''g'') = 0. In QCD, the fixed point occurs at short distances where ''g'' → 0 and is called a (trivial)
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| [[ultraviolet fixed point]]. For heavy quarks, such
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| as the [[top quark]], it is calculated that the coupling to the
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| mass-giving [[Higgs boson]] runs toward a fixed non-zero (non-trivial) [[infrared fixed point]].
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| In [[string theory]] conformal invariance of the string
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| world-sheet is a fundamental symmetry: β=0 is a requirement. Here, β is a function of the geometry of the space-time in which the string moves. This determines the space-time dimensionality of the string theory and enforces Einstein's equations of [[general relativity]] on the geometry.
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| The RG is of fundamental importance to string theory and
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| theories of [[grand unification]].
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| It is also the modern key idea underlying [[critical phenomena]] in condensed matter physics.<ref>
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| {{Cite journal
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| | last1 = Shankar | first1 = R.
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| | doi = 10.1103/RevModPhys.66.129
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| | title = Renormalization-group approach to interacting fermions
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| | journal = Reviews of Modern Physics
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| | volume = 66
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| | pages = 129
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| | year = 1994|arxiv = cond-mat/9307009 |bibcode = 1994RvMP...66..129S }} (For nonsubscribers see [http://arxiv.org/abs/cond-mat/9307009, arXiv:cond-mat/9307009].)</ref> Indeed, the RG has become one of the most important tools of modern physics. It is often used<ref name="CallawayPetronzio1984">{{cite journal|last1=Callaway|first1=David J.E.|last2=Petronzio|first2=Roberto|title=Determination of critical points and flow diagrams by Monte Carlo renormalization group methods|journal=Physics Letters B|volume=139|issue=3|year=1984|pages=189–194|issn=03702693|doi=10.1016/0370-2693(84)91242-5|bibcode = 1984PhLB..139..189C }}</ref> in combination with the [[Monte Carlo method]].
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| ==Block spin==
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| This section introduces pedagogically a picture of RG which may be
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| easiest to grasp: the block spin RG. It was devised by [[Leo P. Kadanoff]] in 1966.
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| Let us consider a 2D solid, a set of atoms in a perfect square array,
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| as depicted in the figure. Let us assume that atoms interact among
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| themselves only with their nearest neighbours, and that the system is
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| at a given temperature <math>T</math>. The strength of their
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| interaction is measured by a certain [[coupling constant]] <math>J</math>. The
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| physics of the system will be described by a certain formula, say
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| <math>H(T,J)</math>.
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| [[File:Rgkadanoff.png|180px]]
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| Now we proceed to divide the solid into '''blocks''' of <math>2\times
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| 2</math> squares; we attempt to describe the system in terms of
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| '''block variables''', i.e.: some variables which describe the
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| average behavior of the block. Also, let us assume that, due to a
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| lucky coincidence, the physics of block variables is described by a
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| formula of the same kind, but with '''different''' values for
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| <math>T</math> and <math>J</math>: <math>H(T',J')</math>. (This isn't exactly true, of course, but it is often approximately true in practice, and that is good enough, to a first approximation.)
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| Perhaps the initial problem was too hard to solve, since there were
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| too many atoms. Now, in the '''renormalized''' problem we have only
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| one fourth of them. But why should we stop now? Another iteration of
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| the same kind leads to <math>H(T'',J'')</math>, and only one sixteenth
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| of the atoms. We are increasing the '''observation scale''' with each
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| RG step.
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| 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 <math>(T,J)\to (T',J')</math> and <math>(T',J')\to (T'',J'')</math>. Usually, when iterated many times, this RG transformation leads to a certain number of '''fixed points'''.
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| Let us be more concrete and consider a [[magnetic]] system (e.g.: the
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| [[Ising model]]), in which the ''J'' coupling constant denotes the
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| trend of neighbour [[spin (physics)|spin]]s to be parallel. The configuration of the system is the result of
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| the tradeoff between the ordering ''J'' term and the disordering
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| effect of temperature. For many models of this kind there are three
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| fixed points:
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| # <math>T=0</math> and <math>J\to\infty</math>. 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.
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| # <math>T\to\infty</math> and <math>J\to 0</math>. Exactly the opposite, temperature dominates, and the system is disordered at large scales.
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| # A nontrivial point between them, <math>T=T_c</math> and <math>J=J_c</math>. In this point, changing the scale does not change the physics, because the system is in a [[fractal]] state. It corresponds to the [[Curie point|Curie]] [[phase transition]], and is also called a [[critical point (thermodynamics)|critical point]].
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| So, if we are given a certain material with given values of ''T''
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| and ''J'', all we have to do in order to find out the large scale
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| behaviour of the system is to iterate the pair until we find the
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| corresponding fixed point.
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| ==Elementary theory==
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| In more technical terms, let us assume that we have a theory described
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| by a certain function <math>Z</math> of the state variables
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| <math>\{s_i\}</math> and a certain set of coupling constants
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| <math>\{J_k\}</math>. This function may be a [[partition function (quantum field theory)|partition function]],
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| an [[Action (physics)|action]], a [[Hamiltonian (quantum mechanics)|Hamiltonian]], etc. It must contain the
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| whole description of the physics of the system.
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| Now we consider a certain blocking transformation of the state
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| variables <math>\{s_i\}\to \{\tilde s_i\}</math>,
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| the number of <math>\tilde s_i</math> must be lower than the number of
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| <math>s_i</math>. Now let us try to rewrite the <math>Z</math>
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| function ''only'' in terms of the <math>\tilde s_i</math>. If this is achievable by a
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| certain change in the parameters, <math>\{J_k\}\to
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| \{\tilde J_k\}</math>, then the theory is said to be
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| '''renormalizable'''.
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| For some reason, most fundamental theories of physics such as [[quantum electrodynamics]], [[quantum chromodynamics]] and [[Electroweak force|electro-weak]] interaction, but not gravity, are exactly
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| renormalizable. Also, most theories in condensed matter physics are
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| approximately renormalizable, from [[superconductivity]] to fluid
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| turbulence.
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| The change in the parameters is implemented by a certain
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| beta function: <math>\{\tilde
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| J_k\}=\beta(\{ J_k \})</math>, which is said to induce a
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| '''renormalization flow''' (or RG flow) on the
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| <math>J</math>-space. The values of <math>J</math> under the flow are
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| called '''running couplings'''.
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| As was stated in the previous section, the most important
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| information in the RG flow are its '''fixed points'''. The possible
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| macroscopic states of the system, at a large scale, are given by this
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| set of fixed points.
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| Since the RG transformations in such systems are '''lossy''' (i.e.: the number of
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| variables decreases - see as an example in a different context, [[Lossy data compression]]), there need not be an inverse for a given RG
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| transformation. Thus, in such lossy systems, the renormalization group is, in fact, a
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| [[semigroup]].
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| ==Relevant and irrelevant operators, universality classes==<!-- This section is linked from [[Power law]] -->
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| Consider a certain observable {{mvar|A}} of a physical
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| system undergoing an RG transformation. The magnitude of the observable
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| 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
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| observable is said to be a '''relevant''' observable; in the second, '''irrelevant''' and in the third, '''marginal'''.
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| A relevant operator is needed to describe the macroscopic behaviour of the system; an irrelevant observable is not. Marginal observables
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| may or may not need be taken into account. A remarkable broad fact is that ''most observables are irrelevant'', i.e., ''the macroscopic physics is dominated by only a few observables in most systems''. As an example, in microscopic physics, to describe a system consisting of a [[Mole (unit)|mole]] of carbon-12 atoms we need of the order of 10<sup>23</sup> ([[Avogadro constant|Avogadro's number]]) variables, while to describe it as a macroscopic system (12 grams of carbon-12) we only need a few.
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| Before the RG, there was an astonishing empirical fact to explain: the coincidence of the [[critical exponents]] (i.e.: the behaviour near a
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| [[second order phase transition]]) in very disparate phenomena, such as magnetic systems, superfluid transition ([[Lambda transition]]), alloy physics, etc. This was called [[universality (dynamical systems)|universality]] and is now successfully explained by RG, just by showing that the differences between all those phenomena are, in fact, related to such '''irrelevant observables'''.
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| Thus, many macroscopic phenomena may be grouped into a small set of '''universality classes''', specified by the set of relevant observables.
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| {{See also|Dangerously irrelevant operator}}
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| ==Momentum space==
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| Renormalization groups, in practice, come in two main flavours. The Kadanoff picture
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| explained above refers mainly to the so-called '''real-space
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| RG'''. '''Momentum-space RG''' on the other hand, has a longer history
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| despite its relative subtlety.{{Citation needed|date=February 2007}} It can be used for systems where the degrees of freedom can be cast in terms of the
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| [[Fourier modes]] of a given field. The RG transformation proceeds
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| by ''integrating out'' a certain set of high momentum (large wavenumber) modes. Since large wavenumbers are related to short length scales, the momentum-space RG results in an essentially similar coarse-graining effect as with real-space RG.
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| Momentum-space RG is usually performed on a [[perturbation theory|perturbation]] expansion. The validity of such an expansion is predicated upon the true physics of our system being close to that of
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| a [[free field]] system. In this case, we may calculate observables by summing the leading terms in the expansion.
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| 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.
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| As an example of the physical meaning of RG in particle physics we will
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| give a short description of charge renormalization in quantum electrodynamics
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| (QED). Let us suppose we have a point positive charge of a certain true
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| (or '''bare''') magnitude. The electromagnetic field around it has a certain
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| 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
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| by the charge, and the positron will be repelled. Since this happens continuously,
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| 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
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| may enter. We have a dependence of a certain coupling constant (the electric
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| charge) with distance.
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| Momentum and length scales are related inversely according to the
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| [[de Broglie relation]]: the higher the energy or momentum scale we may reach, the lower the length scale we may probe and resolve. Therefore, the momentum-space RG practitioners sometimes declaim to ''integrate out'' high momenta or high energy from their theories.
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| ==Exact Renormalization Group Equations==
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| An '''exact renormalization group equation''' ('''ERGE''') is one
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| that takes [[irrelevant]] couplings into account. There
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| are several formulations. | |
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| The '''Wilson ERGE''' is the simplest conceptually,
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| but is practically impossible to implement. [[Fourier transform]] into [[momentum space]] after [[Wick rotation|Wick rotating]] into [[Euclidean space]]. Insist upon a hard momentum [[cutoff]], <math>p^2 \leq \Lambda^2</math> so that the only degrees of freedom are those with momenta less than Λ. The [[partition function (quantum field theory)|partition function]] is
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| :<math>Z=\int_{p^2\leq \Lambda^2} \mathcal{D}\phi \exp\left[-S_\Lambda[\phi]\right].</math>
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| For any positive Λ′ less than Λ, define S<sub>Λ′</sub> (a functional over field configurations φ whose Fourier transform has momentum support within <math>p^2 \leq \Lambda'^2</math>) as
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| :<math>\exp\left(-S_{\Lambda'}[\phi]\right)\ \stackrel{\mathrm{def}}{=}\ \int_{\Lambda' \leq p \leq \Lambda} \mathcal{D}\phi \exp\left[-S_\Lambda[\phi]\right].</math>
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| Obviously,
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| :<math>Z=\int_{p^2\leq \Lambda'^2}\mathcal{D}\phi \exp\left[-S_{\Lambda'}[\phi]\right].</math>
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| In fact, this transformation is [[transitive relation|transitive]]. If you compute S<sub>Λ′</sub> from S<sub>Λ</sub> and then compute S<sub>Λ″</sub> from S<sub>Λ′</sub>, this gives you the same Wilsonian action as computing S<sub>Λ″</sub> directly from S<sub>Λ</sub>.
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| The '''Polchinski ERGE''' involves a [[smooth function|smooth]] UV [[regularization (physics)|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 function (quantum field theory)|partition functional]]s have to match exactly.
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| In other words, (for a real scalar field; generalizations to other fields are obvious)
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| :<math>Z_\Lambda[J]=\int \mathcal{D}\phi \exp\left(-S_\Lambda[\phi]+J\cdot \phi\right)=\int \mathcal{D}\phi \exp\left(-\frac{1}{2}\phi\cdot R_\Lambda \cdot \phi-S_{\text{int}\,\Lambda}[\phi]+J\cdot\phi\right)</math>
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| and Z<sub>Λ</sub> is really independent of Λ! We have used the condensed [[deWitt notation]] here. We have also split the bare action S<sub>Λ</sub> into a quadratic kinetic part and an interacting part S<sub>int Λ</sub>. 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<sub>Λ</sub> is a function of the momentum p and the second term in the exponent is
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| :<math>\frac{1}{2}\int \frac{d^dp}{(2\pi)^d}\tilde{\phi}^*(p)R_\Lambda(p)\tilde{\phi}(p)</math>
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| when expanded. When <math>p \ll \Lambda</math>, R<sub>Λ</sub>(p)/p^2 is essentially 1. When <math>p \gg \Lambda</math>, R<sub>Λ</sub>(p)/p^2 becomes very very huge and approaches infinity. R<sub>Λ</sub>(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.
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| The condition that
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| :<math>\frac{d}{d\Lambda}Z_\Lambda=0</math>
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| can be satisfied by (but not only by)
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| :<math>\frac{d}{d\Lambda}S_{\text{int}\,\Lambda}=\frac{1}{2}\frac{\delta S_{\text{int}\,\Lambda}}{\delta \phi}\cdot \left(\frac{d}{d\Lambda}R_\Lambda^{-1}\right)\cdot \frac{\delta S_{\text{int}\,\Lambda}}{\delta \phi}-\frac{1}{2}\operatorname{Tr}\left[\frac{\delta^2 S_{\text{int}\,\Lambda}}{\delta \phi\, \delta \phi}\cdot R_\Lambda^{-1}\right].</math>
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| [[Jacques Distler]] claimed [http://golem.ph.utexas.edu/~distler/blog/archives/000648.html] without proof that this ERGE isn't correct [[nonperturbative]]ly.
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| The '''Effective average action ERGE''' involves a smooth IR regulator cutoff.
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| The idea is to take all fluctuations right up to an 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]].
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| For a real [[scalar field]], we add an IR cutoff
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| :<math>\frac{1}{2}\int \frac{d^dp}{(2\pi)^d} \tilde{\phi}^*(p)R_k(p)\tilde{\phi}(p)</math>
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| to the [[action (physics)|action]] S where R<sub>k</sub> is a function of both k and p such that for
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| <math>p \gg k</math>, R<sub>k</sub>(p) is very tiny and approaches 0 and for <math>p \ll k</math>, <math>R_k(p)\gtrsim k^2</math>. R<sub>k</sub> 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]]
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| :<math>\frac{1}{2} \phi\cdot R_k \cdot \phi</math>
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| for this IR regulator.
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| So,
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| :<math>\exp\left(W_k[J]\right)=Z_k[J]=\int \mathcal{D}\phi \exp\left(-S[\phi]-\frac{1}{2}\phi \cdot R_k \cdot \phi +J\cdot\phi\right)</math>
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| where J is the [[source field]]. The [[Legendre transform]] of W<sub>k</sub> ordinarily gives the [[effective action]]. However, the action that we started off with is really S[φ]+1/2 φ⋅R<sub>k</sub>⋅φ and so, to get the effective average action, we subtract off 1/2 φ⋅R<sub>k</sub>⋅φ. In other words,
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| :<math>\phi[J;k]=\frac{\delta W_k}{\delta J}[J]</math>
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| can be inverted to give J<sub>k</sub>[φ] and we define the effective average action Γ<sub>k</sub> as
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| :<math>\Gamma_k[\phi]\ \stackrel{\mathrm{def}}{=}\ \left(-W\left[J_k[\phi]\right]+J_k[\phi]\cdot\phi\right)-\frac{1}{2}\phi\cdot R_k\cdot \phi.</math>
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| Hence,
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| :<math>\frac{d}{dk}\Gamma_k[\phi]=-\frac{d}{dk}W_k[J_k[\phi]]-\frac{\delta W_k}{\delta J}\cdot\frac{d}{dk}J_k[\phi]+\frac{d}{dk}J_k[\phi]\cdot \phi-\frac{1}{2}\phi\cdot \frac{d}{dk}R_k \cdot \phi</math>
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| ::::<math>=-\frac{d}{dk}W_k[J_k[\phi]]-\frac{1}{2}\phi\cdot \frac{d}{dk}R_k \cdot \phi=\frac{1}{2}\left\langle\phi \cdot \frac{d}{dk}R_k \cdot \phi\right\rangle_{J_k[\phi];k}-\frac{1}{2}\phi\cdot \frac{d}{dk}R_k \cdot \phi</math>
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| ::::<math>=\frac{1}{2}\operatorname{Tr}\left[\left(\frac{\delta J_k}{\delta \phi}\right)^{-1}\cdot\frac{d}{dk}R_k\right]=\frac{1}{2}\operatorname{Tr}\left[\left(\frac{\delta^2 \Gamma_k}{\delta \phi \delta \phi}+R_k\right)^{-1}\cdot\frac{d}{dk}R_k\right]</math>
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| thus
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| :<math>\frac{d}{dk}\Gamma_k=\frac{1}{2}\operatorname{Tr}\left[\left(\frac{\delta^2 \Gamma_k}{\delta \phi \delta \phi}+R_k\right)^{-1}\cdot\frac{d}{dk}R_k\right]</math>
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| is the ERGE which is also known as the [[Christof Wetterich|Wetterich]] equation.
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| As there are infinitely many choices of ''R''<sub>''k''</sub>, there are also infinitely many different interpolating ERGEs.
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| Generalization to other fields like spinorial fields is straightforward.
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| 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.
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| ==See also==
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| {{Div col}}
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| * [[Renormalization]] with reference to perturbation theory, associated to momentum-space RG.
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| * [[Scale invariance]]
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| * [[Schröder's equation]]
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| * [[Regularization (physics)]]
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| * [[Density matrix renormalization group]]
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| * [[Functional renormalization group]]
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| * [[Critical phenomena]]
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| {{Div col end}}
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| ==References==
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| {{Reflist}}
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| ===Pedagogical and Historical reviews===
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| * S.R. White (1992): Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. '''69''', 2863. The most successful variational RG method.
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| * N. Goldenfeld (1993): Lectures on phase transitions and the renormalization group. Addison-Wesley.
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| * [[Dmitry Shirkov|D.V. Shirkov]] (1999): Evolution of the Bogoliubov Renormalization Group. [http://arxiv.org/abs/hep-th/9909024 arXiv.org:hep-th/9909024]. A mathematical introduction and historical overview with a stress on group theory and the application in high-energy physics.
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| * B. Delamotte (2004): A hint of renormalization. [http://scitation.aip.org/journals/doc/AJPIAS-ft/vol_72/iss_2/170_1.html American Journal of Physics, Vol. 72, No. 2, pp. 170\u2013184, February 2004]. A pedestrian introduction to renormalization and the renormalization group. For nonsubscribers see [http://arxiv.org/abs/hep-th/0212049 arXiv.org:hep-th/0212049]
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| * H.J. Maris, L.P. Kadanoff (1978): Teaching the renormalization group. [http://dx.doi.org/10.1119/1.11224 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.
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| * K. Huang (2013): A Critical History of Renormalization. [http://arXiv.org/pdf/1310.5533 arXiv:1310.5533]
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| *{{cite web
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| | last=Shirkov | first=D. V. | date=2001-08-31
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| | url=http://cerncourier.com/cws/article/cern/28487
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| | title=Fifty years of the renormalization group
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| | publisher=CERN Courier | accessdate=2008-11-12 }}
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| ===Books===
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| *[[Tsung-Dao Lee|T. D. Lee]]; ''Particle physics and introduction to field theory'', Harwood academic publishers, 1981, [ISBN 3-7186-0033-1]. Contains a Concise, simple, and trenchant summary of the group structure, in whose discovery he was also involved, as acknowledged in Gell-Mann and Low's paper.
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| *L.Ts.Adzhemyan, N.V.Antonov and A.N.Vasiliev; ''The Field Theoretic Renormalization Group in Fully Developed Turbulence''; Gordon and Breach, 1999. [ISBN 90-5699-145-0].
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| *Vasil'ev, A.N.; ''The field theoretic renormalization group in critical behavior theory and stochastic dynamics''; Chapman & Hall/CRC, 2004. [ISBN 9780415310024] (Self-contained treatment of renormalization group applications with complete computations);
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| *Zinn-Justin, Jean ; ''Quantum field theory and critical phenomena'', Oxford, Clarendon Press (2002), ISBN 0-19-850923-5 (a very thorough presentation of both topics);
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| *The same author: ''Renormalization and renormalization group: From the discovery of UV divergences to the concept of effective field theories'', in: de Witt-Morette 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, 375-388 (1999) [ISBN ]. Full text available in [http://www-spht.cea.fr/articles/t98/118/ ''PostScript''].
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| * [[Hagen Kleinert|Kleinert, H.]] and Schulte Frohlinde, V; ''Critical Properties of φ<sup>4</sup>-Theories'', [http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore, 2001)]; Paperback ISBN 981-02-4658-7''. Full text available in [http://www.physik.fu-berlin.de/~kleinert/b8 PDF].
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| {{DEFAULTSORT:Renormalization Group}}
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| [[Category:Quantum field theory]]
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| [[Category:Statistical mechanics]]
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| [[Category:Renormalization group]]
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| [[Category:Scaling symmetries]]
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| [[Category:Fixed points (mathematics)]]
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| [[Category:Mathematical physics]]
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