Renormalization
Theory |

**In General** > s.a. regularization;
renormalization group.

* __Idea__: A procedure
for calculating the variation of effective quantities in quantum field
theory such as coupling constants and wavefunctions, as the length scales
or energy scales change; Can be seen as a way to give a unified
description of fundamental and composite/effective degrees of freedom, and
understand how they are related.

* __History__: Used by
Feynman, Schwinger, Tomonaga in QED, by 't Hooft and Veltman for
non-abelian gauge theory; Improved by Ken Wilson's work; Dyson's formalism
has been superseded by dimensional regularization, especially for gauge
theories.

* __Motivation__: One
usually sees the need to renormalize a quantum field theory when one meets
the infinities coming up in the calculation of amplitudes for some
diagrams, but the concept of renormalization is much more general, and is
needed for physical reasons in any interacting theory; For example, the
mass of an electron in a crystal is renormalized, as is the mass of a
He-filled balloon rising in the air {Carl's class demonstration}.

* __Conditions__:
Renormalizability can be obtained only for *s* = 0, 1/2,
1.

__Specific aspects, applications__:
see critical phenomena; heat
[kernel]; mass [classical]; particle
models; specific
theories.

**Approaches, Techniques** > s.a. quantum
field theory approaches [including avoiding renormalization] and techniques.

* __Remark__: When one
reads that some results in quantum field theory are independent of the
regularization method, this really means that the (reasonable) methods
that have been tried all give the same result.

* __Perturbative
renormalization__: (i) Expand the quantities of interest in a
perturbative series; (ii) Regularize the divergent terms, in order to be
able to manipulate them; (iii) Manipulate the free parameters to absorb
the regularization parameters in their definitions; If this succeeds, the
theory is called (perturbatively) renormalizable, or superrenormalizable,
which doesn't necessarily mean that the series we started with will now
converge.

* __1-loop__: Can be
formulated in terms of the heat-kernel expansion.

* __Non-perturbative
renormalization__: Done, e.g., in the lattice
approach.

* __BPH prescription__:
(i) Start with a lagrangian density \(\cal L\) written in terms of
renormalized quantities (mass, wave function, coupling constant); (ii)
Isolate the divergent parts of the 1PI diagrams by Taylor expansions
(around arbitrary points); (iii) Construct a set of counterterms Δ\(\cal
L\)^{(1)} to cancel the 1-loop divergences;
(iv) Use the new \(\cal L\) + Δ\(\cal L\)^{(1)}
to generate 2-loop diagrams and construct Δ\(\cal L\)^{(2)}
to cancel their divergences; (v) Repeat the above steps until the next one
does not produce divergences; The resulting lagrangian density is of the
form

\(\cal L\)^{∞} = \(\cal L\)
+ Δ\(\cal L\) = \(\cal L\) + ∑_{i}
Δ\(\cal L\)^{(i)} .

* __Epstein-Glaser
approach__: A causal perturbation theory framework in which
renormalization is equivalent to the extension of time-ordered
distributions to coincident points by a modified Taylor subtraction on the
corresponding test functions.

* __Renormalon__: A
pattern of divergence of perturbative quantum field theory expansions,
related to their small and large momentum behavior.

@ __Algebraic__: Grassi et al NPB(01)hp;
Brennecke & Duetsch a0801-conf
[quantum action principle in causal perturbation theory].

@ __Epstein-Glaser approach__: Epstein & Glaser AIHP(73);
Prange JPA(99);
Keller JMP(09)-a0902
[euclidean]; Ceyhan a1002
[and Connes-Marcolli renormalization group]; > s.a.
renormalization of specific theories [gauge theory, scalar field]; regularization.

@ __Other approaches__: Egoryan & Manvelyan TMP(86)
[stochastic]; Müller & Rau PLB(96)ht/95
[Fock space projectors]; Schnetz JMP(97)ht/96
[differential]; Ni qp/98;
Yang ht/98,
ht/98-conf;
Pernici NPB(00)
[dimensional]; 't Hooft IJMPA(05)ht/04-in
[without divergences]; Ng & van Dam IJMPA(06)ht/05
[neutrix calculus]; Alexandre ht/05-conf;
Gracey ht/06-conf
[practicalities]; Costello a0706
[and
Batalin-Vilkovisky formalism, on compact manifolds]; Prokhorenko a0707
[non-equilibrium];
Bostelmann et al CMP(09)-a0711
[scaling
algebras]; Ebrahimi-Fard & Patras AHP(10)-a1003
[exponential]; Pivovarov a1312
[non-perturbative]; Berghoff CNTP(15)-a1411
[De Concini-Procesi wonderful models]; Todorov NPB-a1611 [causal position space renormalization]; > s.a. types
of theories [asymptotic safety].

**References** > s.a. history
of quantum physics; Scaling; quantum
fields in curved spacetime.

@ __Books__: Fröhlich 83;
Collins 84; Piguet & Sorella 95
[algebraic]; Salmhofer 98
[mathematical introduction]; McComb 03.

@ __Reviews, intros__: Coleman in(73);
Velo & Wightman ed-76; Delamotte AJP(04)feb-ht/02
[intro]; Stefanovich ht/05
[mass and charge]; Collins ht/06-en;
Sonoda a0710-ln;
Mastropietro 08
[non-perturbative]; Trinchero a1004
[simple example]; Rivasseau a1102
[and interacting Fermi liquids]; Gurău et al a1401
[advanced overview].

@ __Renormalizability and finiteness__: Gegelia & Kiknadze ht/95;
Castagnino
IJTP(01)qp/00;
Bergbauer et al a0908
[resolution of singularities]; Tillman a1009.

@ __Coupling-constant renormalization__: Elias & Mckeon IJMPA(03)
[vanishing
of bare in
4D]; Toms PRL(08)
[fine-structure
constant,
cosmological constant contribution]; > s.a. gauge-theory
renormalization; QCD; quantum-gravity
renormalization [for *G*].

@ __Green-function renormalization__: Hepp CMP(66)
[Bogoliubov-Parasiuk
subtraction].

@ __Wave-function renormalization__: Esposito et al PRD(98)
[fermions
at
finite *T*].

@ __Hopf algebra approach__:
Connes & Kreimer
JHEP(99)ht,
CMP(00)ht/99
[and
Riemann-Hilbert
problem]; Ionescu & Marsalli ht/03
[Hopf
algebra deformation]; Ebrahimi-Fard et al PLB(06)ht/05
[combinatorics];
Ebrahimi-Fard & Guo ht/05;
Ratsimbarison mp/05
[rev];
Ebrahimi-Fard et al CMP(07)
[and
Lie algebra];
Panzer a1407-proc
[intro]; > s.a. theories
[electromagnetism, gauge
theory].

@ __Mathematical setting__: Joglekar
JPA(01)ht/00;
Vladimirov ht/02
[proofs]; Anselmi PRD(16)-a1511
[cohomology, and the background-field method].

@ __Coarse-graining in field theory__: Yang ht/00;
Calzetta et al PRP(01)ht-proc
[semiclassical
gravity]; Zapata JPCS(05),
Manrique JPCS(05)
[without background
metric].

@ __Conceptual__: Teller PhSc(89)jun,
Huggett
& Weingard PhSc(96)sep
[and
philosophy]; Kadanoff SHPMP(13);
Bény & Osborne NJP(15)-a1402
[information in renormalization, and equivalence classes of theories];
Butterfield JPhil-a1406
[reduction and emergence]; Butterfield & Bouatta a1406-in
[for philosophers].

@ __Related topics__: Slavnov TMP(95)
[ambiguities],
PLB(01)
[gauge-invariant];
Kreimer JKTR(97)qa/96-Hab
[and
knot theory]; Yang ht/00
[constraints
on schemes]; Gainutdinov ht/01
[ultraviolet
finite non-local effective theory]; Beneke PRP(99)
[renormalons];
Polonyi & Sailer
PRD(01)
[of
composite operators]; Thorn ht/03-conf
[in
string language]; Agarwala LMP(10)-a0909
[geometric formulation]; Caginalp & Esenturk JPA(14)
[for higher-order differential equations]; > s.a. Multiscale
Physics.

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