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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send feedback and suggestions to bombelli at olemiss.edu – modified 3
nov 2009