Quantization of First-Class Constrained Systems  

In General > s.a. quantization of second-class systems; dirac procedure.
* Methods: There are various methods; The Dirac and reduced-phase-space formalisms are not equivalent, but this is not obvious from some of the simplest examples (like QED without sources); When they differ, the Dirac procedure seems to be the correct one if the constrained degrees of freedom are in principle excitable; They are equivalent for cotangent bundle phase spaces with canonical symplectic structure [@ Puta LMP(84)]; When the constraints are power of a linear function (irregular, type II), the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent.

Reduced Phase Space
* Idea: Use the space of orbits of the constraint vector fields on the constraint surface Γ' as phase space.
* Example: Consider the gauge vector field v on Γ', with gab vavb = λ2 (or λ?); Then, go to the space of orbits of v, on which there is a metric hab; Wave functions are densities of weight 1/2 on this reduced phase space, and the Hamiltonian is H = −\(\hbar\)2 hab Pa Pb + potential; When defining the inner product, the measure should be λ1/2 dvh, not just dvh.
@ References: Blyth & Isham PRD(75) [applications]; Pons et al JPA(99)mp/98 [theory for gauge theory]; Chingangbam & Sharan qp/99 [examples]; Muslih NCB(02)mp/01; Thiemann CQG(06)gq/04 [and partial observables]; Anastopoulos gq/04 [geometric procedure].

Batalin-Vilkovisky, Batalin-Fradkin-Vilkovisky, BRST Methods > s.a. BRST quantization; lagrangian dynamics; renormalization; symplectic structures.
* Idea: The BV method is a powerful Lagrangian method, generalizing the BRST approach, to analyze functional integrals with (infinite-dimensional) gauge symmetries, invented to fix gauges associated with symmetries that do not close off-shell; The BFV method is Hamiltonian.
* Fradkin-Vilkovisky theorem: The Batalin-Fradkin-Vilkovisky path integral is complete independent of the gauge fixing 'fermion', even within a non-perturbative context.
@ General references: Hasiewicz et al JMP(91) [and Gupta-Bleuler]; Govaerts & Troost CQG(91) [BFV and Faddeev]; Khudaverdian & Nersessian MPLA(93) [geometrical]; Batalin & Tyutin IJMPA(96)ht/95 [perturbative equivalence]; Govaerts & Scholtz JPA(04)ht [Fradkin-Vilkovisky theorem]; Bashkirov et al ht/05 [field theories, necessary and sufficient conditions]; Bashkirov & Sardanashvily ht/06 [and Ward identities]; Fredenhagen & Rejzner CMP(12)-a1101 [on generic globally hyperbolic spacetimes]; Bonechi et al a1907 [equivariant extension]; Rejzner a2004-proc [intro and motivation].
@ BV-BFV formalism, intros: in Cattaneo & Schiavina LMP(16)-a1607; Mnev a1707-in [and topological quantum field theory]; Cattaneo & Moshayedi a1905-ln.
@ Batalin-Vilkovisky method: Albert et al JMP(10)-a0812 [in finite dimensions]; Anselmi PRD(14)-a1311 [and background-field method]; Getzler JHEP(16)-a1511 [for the spinning particle]; Clavier & Nguyen a1609-proc [as integration for polyvectors]; Iseppi RVMP(19)-a1610 [application to a matrix model]; Ikeda & Strobl a2007 [from BFV, and spacetime covariance].
@ And geometric quantization: Duval et al CMP(90); Figueroa-O'Farrill & Kimura CMP(91).
@ Specific types of theories: McMullan JMP(87) [BFV and Yang-Mills theories]; Browning & McMullan JMP(87) [BFV for other theories]; Hüffel APS(02)ht-in [2-point non-commutative Yang-Mills model]; Dayi IJMPA(04)ht/03 [generalized fields]; Bashkirov ht/04 [BV, quadratic \(\cal L\)]; Fredenhagen &Rejzner CMP(13)-a1110 [perturbative algebraic quantum field theory]; Rejzner PhD-a1111 [locally covariant field theory]; > s.a. 3D quantum gravity; BF theories; modified QED [scalar]; perturbative quantum gravity; quantization of gauge theories.

Path-Integral Quantization > s.a. Faddeev-Popov; Ghost Fields; path integrals.
* First-class: Choose gauge-fixing conditions χi(q, p) = f i, for fixed f i, with {χi, χj} = 0 and det|{C, χ}| ≠ 0; Then

Z = \(\cal D\)p \(\cal D\)q δ(χif i) δ(Cj) det|{C, χ}| exp{ i dt \((\dot pq-H)\)} .

@ General references: Garczyński PLB(87); Abrikosov PLA(93); Ferraro et al PLB(94); Muslih & Güler NCB(97); Klauder & Shabanov qp/98; Muslih mp/00; Rabei NCB(00); Klauder LNP(01)ht/00 [rev]; Ohnuki JPA(04) [particle on the D-sphere]; LaChapelle a1212.
@ Coherent states: Klauder AP(97)qp/96, qp/96, qp/96; Klauder & Shabanov PLB(97)ht/96 [including Yang-Mills]; Junker & Klauder EPJC(98)qp/97, in(99)ht/98 [with fermions].

Other Methods > s.a. deformation quantization [Fedosov, Moyal]; Faddeev-Jackiw Method.
* Expectation values: Define as physical states those for which \(\langle\)ψ |C | ψ\(\rangle\) = 0; One drawback is that it is not a linear condition on the states, so it is not preserved by linear combinations and the solutions don't obey the superposition principle–they don't form a subspace of \(\cal H\).
* Triplectic: The Sp(2)-covariant version of the field-antifield quantization in the Lagrangian formalism.
@ Expectation values: Marinov FPL(89); Kheyfets & Miller PRD(95)gq/94; Kheyfets et al IJMPA(96) [for gravity].
@ Triplectic: Batalin & Marnelius NPB(96)ht/95; Geyer et al MPLA(99)ht/98; Grigoriev PLB(99) [Lie group structure]; Geyer & Lavrov IJMPA(04) [general coordinates]; [> s.a. BRST formalism].
@ Hamilton-Jacobi approach: Baleanu & Güler NCB(99), NCB(00); Güler NCB(05).
@ Rieffel induction: Landsman dg/96; Wren JGP(98).
@ Related topics: Dayi PLB(89) [gauge fixing]; Klauder qp/98-fs [infinite-dimensional]; Savvidou & Anastopoulos CQG(00)gq/99 [histories quantization]; Rabei et al PRA(02) [WKB approximation, semiclassical]; Little & Klauder PRD(05)gq [second-class on quantization, model]; Bojowald et al RVMP(09)-a0804 [effective constraints]; Wachsmuth & Teufel PRA(10)-a1005 [configuration-space constraints in terms of confining potential].

References > s.a. coherent states; geometric quantization; quantum states [semiclassical].
* Remark: Grundling has proposed a method for obtaining an algebra on reduced phase space, which works even for classically ergodic systems, where other methods like group averaging fail (from Ray).
@ Texts, revs: Gitman & Tyutin 90; Klauder LNP(01)ht/00; Grundling RPMP(06) [Grundling, Hurst approach]; Rothe & Rothe 10; Prokhorov & Shabanov 11.
@ General: in Ashtekar & Horowitz PRD(82); Ashtekar & Stillerman JMP(86); Kuchař PRD(87) [factor ordering], in(88) [covariant]; Dresse et al PLB(90); Hájíček in(94); Klauder qp/96-proc; Kaplan et al PRA(97)qp/98; Klauder NPB(99)ht; Corichi CQG(08)-a0801 [geometrical]; Brody et al JPA(08), JPA(09); Gustavsson JPCS(09)-a0903 [symplectic vs metric formulation]; Fairbairn & Meusburger a1204-in; Bojowald & Tsobanjan a1906 [symplectic reduction].
@ Semiclassical aspects: Kirwin MZ-a0810 [reduction and quantization, semiclassical]; Bojowald & Tsobanjan RVMP(09), PRD(09)-a0906 [effective constraint methods]; Wachsmuth & Teufel MAMS-a0907 [effective Hamiltonian]; Tsobanjan AIP(09)-a0911 [leading-order corrections to dynamics].
@ Related topics: Goldberg et al JMP(91); Rovelli PRL(98) [gauge transformations in quantum mechanics]; Lavrov et al MPLA(99) [osp(1,2) supersymmetry]; Facchi et al JOB(04)qp/03 [and Zeno dynamics]; Konopka & Markopoulou gq/06 [states, from noiseless subsystems]; > s.a. regularization; superselection rules; theta sectors.

Specific Types of Systems > s.a. quantum gauge theories; canonical quantum gravity; Proca Theory; supergravity.
* Parametrized field theories: Torre and Varadarajan showed that for generic foliations emanating from a flat initial slice in D > 2 spacetimes, scalar field evolution along arbitrary foliations is not unitarily implemented on the Fock space, which implies an obstruction to Dirac quantization; The no-go result can be overcome however using lqg techniques.
@ General references: Banerjee & Chakraborty AP(96) [Chern-Simons]; Arik & Ünel ht/96 [quadratic C]; Grundling & Hurst JMP(98) [not preserved by dynamics]; Montesinos et al PRD(99)gq [general relativity toy model]; Mišković & Zanelli JMP(03), Klauder & Little CQG(06)gq [irregular]; Serhan et al IJTP(09) [holonomic]; Belhadi et al AP(14)-a1406 [classically soluble constrained systems]; Bojowald & Brahma JPA(16)-a1407 [fluctuations and structure functions].
@ Special configuration spaces: Kleinert & Shabanov PLA(97) [on Sd]; Maraner ht/98 [on a line]; Ikemori et al MPLA(99) [on S2]; Scardicchio PLA(02) [on S1]; > s.a. quantum systems.
@ Reparametrization-invariant: Klauder JPA(01)qp/00 [ultralocal fields]; Varadarajan PRD(07)gq/06 [parametrized field theory]; > s.a. parametrized theories.
@ Totally constrained: Kodama PTP(95)gq, PTP(95)gq [classical and quantum theory]; Doldán et al IJTP(96)ht/94; Olmedo IJMPD(16)-a1604 [Schrödinger vs Heisenberg pictures].
@ Generally covariant: Montesinos GRG(01)gq/00 [relational evolution]; Sforza PhD(00)gq; > s.a. types of quantum field theories.
@ Time-dependent constraints: Muslih CzJP(02)mp/01 [canonical path integral].
@ Discrete spacetime lattice: Di Bartolo et al CQG(02)gq.

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