BRST Transformations and Quantization |
Background
* Idea: In the quantization
of gauge theories using the Faddeev-Popov procedure, Becchi, Rouet, Stora
and Tyutin realized that the effective action, including ghost and gauge-fixing
terms, although not gauge-invariant in the regular sense, was invariant under
a group of gauge-like transformations; This is now considered to be the
simplest quantization method for non-Abelian theories.
Classical Structure
* BRST extended phase space:
Add one pair (η, π) of Grassmann-valued (ghost)
variables for each constraint C of the theory.
* BRST charge: The expression
Q:= Ca
ηa
+ Cabc
ηb
ηa
πc
+ Cabcde
ηc
ηb
ηa
πe
πd
+ ..., where the Cas are the constraints,
and the other Cs the (first-order, second-order, ...) are structure functions.
* BRST transformations:
The canonical transformations on the extended phase space generated by Q;
For any function F of (q, p, η, π),
δF = {Q, F} .
* Observables: An observable is a function F with ghost number zero which is (strongly) invariant under BRST transformations; If we consider the BRST δ as a coboundary operator, which gives a grading to the Fs by ghost number, then observables are the 0-th cohomology group of δ; Every observable in the usual sense as a function of (q, p) admits a BRST-invariant extension.
References
> s.a. non-commutative geometry; symplectic
structures [covariant]; types of cohomology.
@ General: Becchi et al CMP(75),
AP(76);
Tyutin pr(75);
Bonora & Tonin PLB(81);
Kostant & Sternberg AP(87);
Cheng & Tsai PRD(89);
Henneaux CMP(91) [spacetime locality];
Rivelles CQG(02) [new transformations];
Becchi Pra(12)-a1107-talk [history];
Krishna et al MPLA(11)-a1109 [dual-BRST symmetry];
Ahmad CTP(13)-a1309 [general formalism].
@ Intros / reviews: Henneaux PRP(85);
Nemeschansky et al AP(88);
Niemi PRP(89);
Becchi ht/96-ln;
Bes & Civitarese AJP(02)may [toy models].
@ Geometric: Thierry-Mieg JMP(80);
Quirós et al JMP(81);
Bonora & Cotta-Ramusino CMP(83);
Kastler & Stora JGP(86),
JGP(86);
Loll CMP(88);
Castellani CQG(90);
Bonora & Malik JPA(10) [geometrical interpretation of the Curci-Ferrari conditions];
Dragon & Brandt fs(12)-a1205 [and cohomology].
@ Related topics:
Moshin & Reshetnyak IJMPA(14)-a1405 [in the generalized Hamiltonian formalism].
And Physics, Other Concepts
> s.a. lovelock gravity; noether's theorem.
@ Classical mechanics: Gozzi PLB(88),
et al PRD(89) [path-integral formulation];
Henneaux & Teitelboim CMP(88);
McMullan CMP(92);
Marnelius MPLA(00)ht.
@ BRST extensions of a Lie group:
Henneaux NPB(88).
@ Representations of BRST algebra:
Horuzhy & Voronin CMP(89);
Voronin & Khoruzhii TMP(92).
@ Inner products: Düchting et al NPB(99)ht/98 [and the Gribov problem].
@ Anti-BRST transformations:
Curci & Ferrari NCA(76);
Grigore a2011 [causal approach].
@ Supersymmetric models: Brandt fs(12)-a1201;
Buchbinder & Koutrolikos JHEP(15)-a1510 [higher-spin fields].
@ Other systems:
Barducci et al IJMPA(89) [spacetime symmetries];
Okumura JMP(00) [spontaneously broken gauge theory];
Gracia-Bondía in(10)-a0808-ln [massive boson fields].
@ Related topics: Dubois-Violette et al CMP(86);
Azizov & Khoruzhii TMP(89) [ghost number];
Abud et al AP(90) [gauged];
Lusanna JMP(90);
Rybkin NPB(91);
Batakis & Kehagias MPLA(93) [topological obstructions];
Brandt CMP(97)ht/96 [and covariance];
Picariello & Torrente-Lujan CPC(04)ht/03 [Mathematica package for symbolic computations];
Lavrov et al JHEP(11)-a1108 [soft breaking of BRST symmetry is inconsistent].
BRST Quantization > s.a. quantization of 1st-class
and 2nd-class systems; dirac method;
lagrangian dynamics; particles; QED;
bosonic strings.
* Motivation: BRST invariance
effectively implements gauge invariance in the path-integral quantization
of gauge theories without having to fix the gauge, and having to confront the
related tricky issues.
@ General references:
Fradkin & Vilkovisky PLB(75);
Batalin & Vilkovisky PLB(77);
Batalin & Fradkin AIHP(88);
Batalin et al NPB(89);
Hwang & Marnelius NPB(89);
Duval et al AP(91);
Loll pr(91);
van Holten LNP(05)ht/02;
Fuster et al IJGMP(05)ht;
Constantinescu & Ionescu AIP(09)-a1112.
@ Hamiltonian and Lagrangian approaches:
Grigoryan et al NPB(92);
Nirov & Razumov JMP(93);
Barnich & Grigoriev CMP(05) [for gauge theories];
Gao LMP(06) [new construction];
Constantinescu & Ionescu IJMPA(06) [equivalence];
Kaparulin et al JGP(13)-a1207
[general mechanical systems, link between path-integral quantization and deformation quantization];
Moshin & Reshetnyak PLB(14)-a1406 [Lagrangian].
@ Batalin-Vilkovisky and deformations:
Stasheff qa/97-proc.
@ On inner product spaces:
Batalin & Marnelius NPB(95)ht [gauge theories];
Marnelius & Sandström IJMPA(00)ht/98.
@ For superfields:
Aoyama et al PLB(89) [4D superparticle];
Batalin et al NPB(98)ht/97.
@ With curved phase space:
Batalin & Fradkin NPB(89);
Batalin et al NPB(90).
@ Related topics:
Slavnov PLB(89) [unitarity];
Thomi JMP(88);
Dayi MPLA(89),
IJMPA(96) [odd-time BV];
Bizdadea & Saliu NPB(95) [second-class constraints];
Scholtz & Shabanov AP(98) [Gribov problem];
Lyakhovich & Sharapov JHEP(05)ht/04 [without Hamiltonian and Lagrangian];
Fulp FP(07)m.DG/06 [and geometric quantization].
Specific Systems > s.a. quantum gauge theories
and path integrals for gauge theories.
@ Particle: Thomi JMP(89);
in Nemeschansky et al AP(88);
Batlle et al PRD(89);
Marnelius NPB(94).
@ Gauge theory:
in Nemeschansky et al AP(88) [Maxwell];
Hull et al NPB(91) [Yang-Mills];
Marnelius NPB(93);
Bizdadea JPA(96),
& Saliu PLB(98)ht/99,
EPL(98)ht/99 [p-forms];
Ferraro & Sforza PRD(97)ht/96,
PRD(01)gq [generally covariant];
Moss & Silva PRD(97)gq/96 [boundary conditions];
Federbush ht/99 [Yang-Mills theory];
Bizdadea et al CQG(98)ht/99 [spin-5/2];
Barnich et al PRP(00) [local BRST cohomology];
Rivelles PLB(03) [higher-derivative, and quantum gravity];
Rai & Mandal MPLA(11)-a1003 [abelian 2-form theory];
Bratchikov a1203 [reducible gauge theories];
Wrochna & Zahn RVMP(17)-a1407
[on curved spacetime, classical phase space and Hadamard states];
Shestakova a1410-conf [and gravity];
Batalin & Lavrov PLB(15)-a1507 [intrinsic "BRST" operator];
Acharyya et al PRD(16)-a1604 [in manifolds with spatial boundary];
Öttinger PRD(18)-a1803 [Hamiltonian approach on Fock space];
> s.a. renormalization.
@ Gauge theory, field-dependent transformations:
Lavrov & Lechtenfeld PLB(13) [Yang-Mills theory];
Upadhyay PhD-a1308.
@ Gravity / cosmology: Hájíček JMP(86);
Louko CQG(87);
Ashtekar et al PRD(87);
Labastida & Pernici PLB(88) [topological gravity];
Barnich et al NPB(95)ht [Einstein-Yang-Mills theory];
Blaga et al PRD(95);
Federbush ht/99 [quantum gravity model];
Hamada ht/00;
Bashkirov & Sardanashvily ht/04 [for spacetime transformations];
Castellana & Montani CQG(08)-a0710,
IJMPA(08) [and physical states, vs lqg];
Upadhyay AP(14) [perturbative quantum gravity, and gaugeon formalism];
Upadhyay & Paul EPJC(16)-a1506 [minisuperspace models];
Upadhyay et al BJP(17)-a1510 [unimodular gravity];
> s.a. 3D quantum gravity; cosmological perturbations;
Metric-Affine Gravity; non-commutative gravity.
@ Other field theories:
Folacci PRD(92)-a0911 [massless scalar field, minimally-coupled, in de Sitter space];
Noltingk JMP(02) [histories electromagnetism];
Soroush PRD(03)ht [non-commutative gauge theory];
Buchbinder et al MPLA(09) [bosonic antisymmetric tensor fields];
Hasiewicz & Walczyk RPMP(11) [high-spin systems with anomalies];
Anastasiou et al PRL(18)-a1807 [gravity/two-form/dilaton system];
> s.a. high-spin fields; sigma-models;
topological field theories.
@ Other quantum systems:
van Nieuwenhuizen ht/04-proc [quantum-mechanical model];
López & Rogers a1208 [systems with secondary constraints].
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