Renormalization Group  

In General > s.a. [renormalization]; for applications, see specific types of theories.
* History: The technique was developed in 1973 by Ken Wilson.
* Idea: A group of transformations on the (renormalized) parameters of a theory (mass, wave function, coupling constants) corresponding to changes of the renormalization conditions (subtraction point), under which the physics is required to be invariant.
* Applications: The invariance requirement provides non-trivial constraints on the asymptotic behavior of the theory; Renormalization group ideas are largely responsible for the considerable success achieved in developing a quantitative theory of phase transitions.
> Applications: see Disordered Systems; phase transitions; random walk.

Renormalization Group Equation > s.a. chaos [period-doubling bifurcation].
* Idea: It expresses the connection between scale transformations and renormalizability of a theory; Like a mathematical microscope, it allows us to look at very-small-scale physics from the behavior at larger scales.
* Callan-Symanzik equation: An analytic form of the renormalization group invariance; For 4 theory, it is

* Beta function: (or Gell-Mann-Low function).
@ General references: Callan PRD(70); Symanzik CMP(70); Dolan IJMPA(95), IJMPA(95), IJMPA(97) [geometrical view].
@ Functional renormalization group: Polonyi CEJP(03)ht/01-ln; Pawlowski AP(07)ht/05; Weyrauch JPA(06) [and tunneling].
@ Related topics: Simionato IJMPA(00)ht/98, IJMPA(00)ht/98, IJMPA(00) [and gauge symmetry]; Gosselin et al PLA(99)qp/00, Gosselin & Mohrbach JPA(00)qp/98 [1-particle quantum mechanics, finite T]; Litim & Pawlowski PRD(02) [perturbative expansion].

References
@ Textbooks, reviews: Coleman in(71); Wilson & Kogut PRP(74); Wallace & Zia RPP(78); in Cheng & Li 84; Shirkov IJMPA(88), IJMPB(98)ht/97; Binney et al 92; Goldenfeld 93; Benfatto & Gallavotti 95; Intriligator hp/98-in; Mironov & Morozov PLB(00); O'Connor & Stephens ed-PRP(01); Rivero ht/02-ln; Amit & Martín-Mayor 05; Mitter mp/05-in [mathematical]; Sonoda ht/06-ln [and perturbation theory]; Hollowood a0909-ln [and quantum field theory and supersymmetry].
@ Simple: Maris & Kadanoff AJP(78)jun; Wilson SA(79)aug; Hans AJP(83)aug.
@ History: Stückelberg & Peterman HPA(53) [discovery]; Shirkov & Kovalev PRP(01)mp/00-in.
@ Critical phenomena: Wilson RMP(75), RMP(83); Barber PRP(77) [intro]; Schmidhuber AJP(97)nov-ht; Bhattacharjee cm/00-ln; Pelissetto & Vicari PRP(02); Requardt mp/02 [many-body systems]; > s.a. phase transitions.
@ Non-perturbative: Phillips et al AP(98); Aoki et al PTP(02)qp; Berges et al PRP(02) [and statistical mechanics]; Blaizot et al PLB(06)ht/05 [solution], PRE(06)ht/05, PRE(05) [and p-dependence of n-point functions]; Delamotte cm/07 [intro]; Canet & Chaté JPA(07) [Model A, critical dynamics]; Pinson CMP(08); Dupuis & Sengupta a0807 [for lattice models]; > s.a. N-point functions.
@ Holographic: Balasubramanian & Kraus PRL(99) [and AdS]; Álvarez & Gómez PLB(00)ht; de Boer FdP(01)ht-in; Erdmenger PRD(01)ht; Bianchi et al NPB(02)ht/01; Skenderis CQG(02)ht-ln; Fukuma et al PTP(03)ht/02 [rev].
@ Without fixed points: Glazek & Wilson PRL(02)ht [limit cycles and chaos]; Rosten JHEP(09)-a0808 [from exact renormalization group].
@ Related topics: Wilson AiM(75); Lässig NPB(90); Minic & Nair IJMPA(96) [wave functionals and eigenvalues]; Jona-Lasinio PRP(01) [and probability theory]; Requardt cm/01 [scaling limit]; Duetsch & Fredenhagen ht/05-in [in terms of algebraic quantum field theory]; RG2005 JPA(06); Chishtie et al IJMPE(07)ht/06 [improvement]; Sonoda JPA(07)ht/06 [ordinary vs exact renormalization group]; Alexandre a0711 [misleading, free fixed point]; Streets JGP(09) [singularity formation]; Yin a0911 [spectral properties at infinite temperature]; Faris & Yin a0911 [cluster-expansion approach].

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