View source for Renormalization group analysis

The theoretical physicist aims to elaborate theories at the microscopic scale, from which observed phenomena can be explained. Renormalization group analysis allows one to determine effective theories at each length scale, from microscopic to macroscopic, by averaging over  "degrees of freedom"  of the previous scale. For instance, given a system defined on a [[Lattice|lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r1100801.png" /> of spacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r1100802.png" /> (e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r1100803.png" /> is a lattice of spacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r1100804.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r1100805.png" />) and a [[Lagrangian|Lagrangian]], or a Hamiltonian, involving an interaction between sites, a new Lagrangian is obtained on a lattice of spacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r1100806.png" /> by summing over possible values of the initial variables at each site of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r1100807.png" />, for given values of variables on the new lattice, etc. In some theories, e.g. few-body Newtonian mechanics,  "degrees of freedom"  at very different scales  "decouple" ; as a consequence, particles (or planets in the solar system) can be approximated by point-like objects in large-scale analysis. This is no longer true in [[Quantum field theory|quantum field theory]] in particle physics, nor in the related study of phase transitions and critical phenomena in classical statistical physics, domains in which renormalization group analysis has been mainly developed. The renormalization group, which is actually a semi-group, is the set of transformations, in an infinite-dimensional space of Lagrangians, that give the effective Lagrangian of each scale from those of shorter scales. In cases studied, effective Lagrangians tend to fixed points; moreover, there will exist  "universality classes"  such that the effective Lagrangians will all tend to a common fixed point or to a set of fixed points of finite dimension. The dominant long-distance physics will then depend only on a finite number of parameters.

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r1100808.png" />-models.==
In field theory, a model is defined by a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r1100809.png" /> on the space of fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008010.png" />. Fields are functions or distributions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008011.png" />, or on a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008013.png" />. They have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008014.png" /> real-valued components, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008015.png" />. Correlation functions are the moments of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008016.png" />, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008017.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008018.png" />. In applications, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008019.png" /> is either the usual space or Euclidean space-time. (In the latter case, real-time physics follows by analytic continuation in time variables: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008020.png" />.)

Given a Lagrangian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008022.png" /> has the heuristic ill-defined form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008023.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008024.png" />. In the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008025.png" />-model on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008026.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008027.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008028.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008029.png" /></td> </tr></table>

with integrals replaced by discrete sums (and derivatives by differences) on a lattice. At <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008031.png" />, it describes a free theory of mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008032.png" />. To define the model on a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008033.png" /> one may first restrict <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008034.png" /> to a finite volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008036.png" /> and then consider the limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008037.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008038.png" /> one may initially consider in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008039.png" /> lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008040.png" /> with spacings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008041.png" /> and parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008044.png" />, and consider the continuous limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008045.png" />. As explained below, this will allow one, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008047.png" />, to define the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008048.png" />-model and will yield an effective Lagrangian on a lattice of spacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008049.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008050.png" />. The limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008051.png" /> is treated in turn.

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008052.png" />-models in a finite volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008054.png" />.==
For any given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008055.png" />, renormalization group analysis yields Lagrangians <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008057.png" />, on lattices with spacings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008058.png" />, including terms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008061.png" /> with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008064.png" />, plus (already at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008065.png" />) infinitely many other terms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008066.png" />. However, the latter will be  "irrelevant" . Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008069.png" />, it is then expected at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008070.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008071.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008072.png" /> will be well defined in the limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008073.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008076.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008079.png" /> are suitably chosen; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008081.png" /> remain close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008083.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008084.png" />. The results are confirmed by a rigorous analysis in which a truncated version of effective Lagrangians (involving first terms only) is used.

The theory is asymptotically free at short distances (the weight of the interaction, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008085.png" />, tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008086.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008087.png" />).

There is no result at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008088.png" />, the physical dimension in particle physics; a conjecture is that only more refined  "gauge"  theories can then exist.

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008089.png" />-models on a lattice.==
(The limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008090.png" />.) Effective theories are obtained on lattices with spacings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008092.png" />. Two-point correlations decrease exponentially as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008093.png" /> in  "massive"  cases (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008094.png" />) and as inverse powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008095.png" /> in  "critical"  mass-less cases (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008096.png" />).

i) There exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008097.png" /> such that the theory is critical. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r11008099.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080100.png" />: asymptotic freedom at large distances follows with a trivial (Gaussian) fixed point and a dominant long-distance behaviour in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080101.png" /> as in the free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080102.png" /> theory.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080104.png" /> large, a non-trivial fixed point is obtained as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080105.png" />, by means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080106.png" /> expansions: dominant behaviour, at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080107.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080108.png" />, with a critical exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080109.png" /> independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080110.png" /> (the  "anomalous dimensionanomalous dimension" ). These results can be proved under suitable conditions.

On the other hand, a heuristic analysis for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080111.png" /> can be made using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080113.png" />-expansions. One considers a space dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080114.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080115.png" /> small, a non-trivial fixed point close to the Gaussian fixed point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080116.png" /> is obtained. Results are then extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080117.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080118.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080119.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080120.png" />). The critical indices obtained at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080121.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080122.png" /> are close to the experimental ones in related situations, or to those obtained from numerical calculations, or also, at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080123.png" /> to exact results on the related Ising model.

ii) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080124.png" /> one obtains a massive Gaussian fixed point.

iii) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080125.png" /> there is a discrete symmetry breaking at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080126.png" />: in dependence on, e.g., boundary conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080127.png" />, different non-trivial fixed points, which are mixtures of two pure  "phases"  are obtained in the limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080128.png" /> (symmetry then relates different solutions).

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080129.png" />, there is for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080130.png" /> a continuous symmetry breaking: an infinite number of non-trivial fixed points linked by a  "Goldstone bosonGoldstone boson" .

==Physical theories.==
Applications in statistical physics (cf. also [[Statistical physics, mathematical problems in|Statistical physics, mathematical problems in]]) related to the previous models include phase transitions: liquid-vapour (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080131.png" />), superfluid helium (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080132.png" />), ferromagnetic systems (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080133.png" />), and statistical properties of long polymers ( "N= 0" ). Among other applications of renormalization group analysis is the BCS theory of superconductivity. In particle physics, the theory of strong interactions (QCD) makes sense as an asymptotically free theory at short distances; the situation is different for theories of strong and electroweak interactions, which might be viewed as effective theories arising from a more fundamental theory at  "ultrashort"  distances.

Applications of renormalization group ideas have been made in many other domains; of particular mathematical interest are the recent studies of partial differential equations using this method.

For general references on renormalization group analysis, see [[#References|[a1]]], [[#References|[a2]]]; see [[#References|[a3]]] for rigorous results.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.G. Wilson,   J. Kogut,   "The renormalization group and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110080/r110080134.png" />-expansion"  ''Phys. Rep.'' , '''12'''  (1974)  pp. 75–200</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Zinn-Justin,   "Quantum field theory and critical phenomena" , Oxford Univ. Press  (1996)  (Edition: Third)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Magnen,   D. Iagolnitzer (ed.) , ''XI-th Internat. Congress Math. Physics'' , Internat. Press , Boston  (1995)  pp. 121–141</TD></TR></table>