Renormalization Group  

In General > s.a. renormalization / for applications, see specific types of theories and quantum gravity.
* 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; statistics.

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 λφ4 theory, it is

(μ ∂/∂μ + β ∂/∂λ) ΓR(n)(pi; λ, μ) = –i μ2 α Γφ^2 R(n)(0, pi; λ, μ) .

* Beta function: (or Gell-Mann-Low function).
@ General references: Callan PRD(70); Symanzik CMP(70); Curtright & Zachos PRD(11)-a1010 [global structure of group trajectories].
@ Geometric view: Dolan IJMPA(95), IJMPA(95), IJMPA(97); Jackson et al a1312 [for holographic theories].
@ Functional renormalization group: Polonyi CEJP(03)ht/01-ln; Pawlowski AP(07)ht/05; Weyrauch JPA(06) [and tunneling]; Benedetti et al JHEP(11)-a1012; Vacca & Zambelli PRD(11)-a1103 [regularization and coarse-graining in phase space]; Metzner et al RMP(12) [and correlated fermion systems]; Nagy AP(14)-a1211-ln [intro, and asymptotic safety]; Codello et al PRD(14)-a1310 [scheme dependence and universality]; Mati PRD(15)-a1501 [Vanishing Beta Function curves]; Codello et al PRD(15)-a1502 [and local renormalization group], a1505 [and effective action].
@ 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]; Baume et al JHEP(14)-a1401 [Callan-Symanzik equation]; Harst & Reuter AP(15)-a1410 [new, simpler to use functional flow equation].

References > s.a. Emergence; higgs mechanism; types of metrics [information geometry].
@ 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-proc; Mironov & Morozov PLB(00); O'Connor & Stephens ed-PRP(01); Rivero ht/02-ln; Amit & Martín-Mayor 05; Mitter mp/05-en [mathematical]; Sonoda ht/06-ln [and perturbation theory]; Hollowood a0909-ln [and quantum field theory and supersymmetry]; Meurice et al PTRS(11)-a1102 [new applications]; Dimock RVMP(13)-a1108, JMP(13)-a1212, AHP(14)-a1304 [Balaban's approach]; Sfondrini a1210-ln [and universality]; Hollowood 13.
@ 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-proc; Peskin JSP(14)-a1405 [Ken Wilson and strong interactions].
@ Critical phenomena: Fisher RMP(74); 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]; Singh a1402 [and mean-field theories phase transitions, pedagogical]; > s.a. critical phenomena; 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 EPJB(08)-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]; Heemskerk & Polchinski JHEP(11)-a1010 [and Wilsonian RG]; Park & Mann JHEP(12) [asymptotically flat gravity].
@ UV fixed points: Gies & Janssen PRD(10)-a1006 [3D Thirring model]; Eichhorn et al EPJC(16)-a1510 [in multi-field models]; > s.a. asymptotic safety in quantum gravity.
@ IR fixed points: Ryttov & Shrock PRD(12)-a1206 [scheme transformations in the vicinity of an infrared fixed point]; > s.a. quantum-gravity renormalization.
@ Fixed points, other: Alexandre a0711 [misleading, free fixed point]; Berges & Wallisch a1607 [non-thermal fixed points].
@ Without fixed points: Glazek & Wilson PRL(02)ht [limit cycles and chaos]; Rosten JHEP(09)-a0808 [from exact renormalization group]; Bulycheva & Gorsky a1402-fs [limit cycles].
@ And information theory: Apenko PhyA(12) [information loss and irreversibility of RG flow]; Bény & Osborne PRA-a1206 [information-geometry approach]; Li a1604 [irreversibility of the renormalization group flow and entropy]; DeBrota a1609.
@ And conformal symmetry: Komargodski & Schwimmer JHEP(11)-a1107 [4D renormalization group flows and spontaneously broken conformal symmetry]; Komargodski JHEP(12)-a1112.
@ 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]; Dütsch & Fredenhagen ht/05-proc [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]; Streets JGP(09) [singularity formation]; Yin CMP(11)-a0911 [spectral properties at infinite temperature]; Yin JMP(11)-a0911 [cluster-expansion approach]; Rosten PRP(12)-a1003 [exact renormalization group, fundamentals]; Gaberdiel & Hohenegger JHEP(10) [supersymmetric flows]; Dütsch CM(12)-a1012-in [connection between the Stückelberg-Petermann and Wilson renormalization groups]; Bény & Osborne NJP(15)-a1402 [and effectively indistinguishable microscopic theories]; Dias et al PLB(14)-a1407 [renormalization-group improved effective potential]; Altaisky a1604 [and wavelets]; Bal et al a1703 [using tensor networks].


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