BRST Transformations and Quantization  

* 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:= Caηa + Cabc ηb ηa πc + Cabcde ηc ηb ηa πeπd + ..., 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 a1205-fs [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).
@ Supersymmetric models: Brandt a1201-fs; 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 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]; > 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 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]; > 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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