Renormalization Theory

In General > s.a. mass [classical]; history of physics; quantum field theory techniques and quantum fields in curved spacetime; regularization.
* 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in QED, by 't Hooft and Veltman for non-abelian gauge theory; Improved by K G Wilson's work; Dyson's formalism has been superseded by dimensional regularization, especially for gauge theories.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.

Approaches, Techniques > s.a. heat [kernel]; quantum field theory approaches [avoiding renormalization]; renormalization group.
* 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 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 ⁽¹⁾ to cancel the 1-loop divergences; (iv) Use the new + ⁽¹⁾ to generate 2-loop diagrams and construct ⁽²⁾ to cancel their divergences; (v) Repeat the above steps until the next one does not produce divergences; The resulting lagrangian is of the form

^infty = + = + _i ⁽ⁱ⁾ .

* 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-in [proof of quantum action principle in causal perturbation theory].
@ 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-in; Pernici NPB(00) [dimensional]; 't Hooft ht/04-in [without divergences]; Ng & van Dam IJMPA(06)ht/05 [neutrix calculus]; Alexandre ht/05-in; Gracey ht/06-in [practicalities]; Costello a0706 [and Batalin-Vilkovisky formalism, on compact manifolds]; Prokhorenko a0707 [non-equilibrium].

References > s.a. critical phenomena; particle models; Scaling; specific theories.
@ 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)ht/02 [intro]; Stefanovich ht/05 [m and charge]; Collins ht/06-in; Sonoda a0710-in; Mastropietro 08 [non-perturbative].
@ Renormalizability and finiteness: Gegelia & Kiknadze ht/95; Castagnino IJTP(01)qp/00.
@ Coupling constant: Elias & Mckeon IJMPA(03) [vanishing of bare in 4D]; Robinson & Wilczek PRL(06) [gauge couplings, gravitational contribution]; > s.a. QCD, renormalization of quantum gravity [for G].
@ Green function: Hepp CMP(66) [Bogoliubov-Parasiuk subtraction].
@ Wave function: 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]; > s.a. theories [em, gauge theory].
@ Mathematical setting: Joglekar JPA(01)ht/00; Vladimirov ht/02 [proofs].
@ Coarse-graining in field theory: Yang ht/00; Calzetta et al PRP(01)ht-in [semiclassical gravity].
@ Related topics: Teller PhSc(89) [and philosophy]; 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 [uv finite non-local effective theory]; Beneke PRP(99) [renormalons]; Polonyi & Sailer PRD(01) [of composite operators]; Thorn ht/03-in [in string language].

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