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; Now considered the simplest quantization method for non-Abelian theories.

Classical Structure
* BRST extended phase space: Add one pair (,) of Grassmann-valued (ghost) variables for each constraint C of the theory.
* BRST charge: The expression Q:= Caa + Cabc b a c + Cabcde c b a ed + ..., where the Ca's are the constraints, and the other C's the (first-order, second-order, ...) are structure functions.
* BRST transformations: The canonical transformations on the extended phase space generated by the BRST charge; For any function F of (q, p, , ),

F = {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 as a coboundary operator, which gives a grading to the F's by ghost number, then observables are the 0-th cohomology group of ; 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].
@ 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].
@ 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).

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 Gribov problem].
@ Anti-BRST transformations: Curci & Ferrari NCA(76).
@ Other systems: Barducci et al IJMPA(89) [spacetime symmetries]; Okumura JMP(00) [spontaneously broken gauge theory]; Gracia-Bondía 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]; Federbush ht/99 [and Yang-Mills], ht/99 [and gravity]; Picariello & Torrente-Lujan CPC(04)ht/03 [Mathematica package].

BRST Quantization > s.a. 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 ht/02-ln; Fuster et al IJGMP(05)ht.
@ Hamiltonian and Lagrangian: 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].
@ Batalin-Vilkovisky and deformations: Stasheff qa/97-in.
@ 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. 3D quantum gravity; Metric-Affine Gravity; 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]; 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]; > s.a. renormalization.
@ Gravitation / cosmology: Hájícek 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]; 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, compared to lqg].
@ 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]; > s.a. sigma model; topological field theories.
@ Other quantum systems: van Nieuwenhuizen ht/04-in [quantum-mechanical model].

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