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 ' as phase space.
* Example: Consider the gauge vector field v on ', with gab vavb = 2 (or ?); 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 = –2 hab Pa Pb + potential; When defining the inner product, the measure should be 1/2 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; symplectic structures.
* Idea: The BV method is Lagrangian, the BFV method Hamiltonian.
* Fradkin–Vilkovisky theorem: The Batalin–Fradkin–Vilkovisky path integral is complete independent of the gauge fixing 'fermion', even within a nonperturbative context.
@ General references: McMullan JMP(87) [BFV and Yang-Mills theories]; Browning & McMullan JMP(87) [BFV for other theories]; 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]; Hüffel ht/02-in [2-point non-commutative Yang-Mills model]; Dayi IJMPA(04)ht/03 [generalized fields]; Bashkirov ht/04 [BV, quadratic ]; Govaerts & Scholtz JPA(04)ht [Fradkin-Vilkovisky theorem]; Bashkirov et al ht/05 [field theories, nsc's]; Bashkirov & Sardanashvily ht/06 [and Ward identities].
@ And geometric quantization: Duval et al CMP(90); Figueroa-O'Farrill & Kimura CMP(91).

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

Z = Dp Dq (i–f i) (Cj) det|{C,}| exp{ i dt (p· q – H)} .

@ General references: Garczynski 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 D-sphere].
@ 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, ht/98-in [with fermions].

Other Methods > s.a. deformation quantization [Fedosov, Moyal].
* Expectation values: Define as physical states those for which |C| = 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 .
* 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.
@ 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].
@ Faddeev-Jackiw: Faddeev & Jackiw PRL(88); Barcelos-Neto & Wotzasek MPLA(92), IJMPA(92); Jackiw ht/93; Barcelos-Neto & Silva IJMPA(95) [reducible theory]; Müller-Kirsten & Zhang PLA(95), PLA(95); García & Pons IJMPA(98); > s.a. dirac approach [comparison].
@ 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-in [-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 a0804 [effective constraints].

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 and reviews: Gitman & Tyutin 90; Klauder LNP(01)ht/00; Grundling RPMP(06) [Grundling, Hurst approach].
@ General: in Ashtekar & Horowitz PRD(82); Ashtekar & Stillerman JMP(86); Kuchar PRD(87) [factor ordering], in(88) [covariant]; Dresse et al PLB(90); Hájícek in(94); Klauder qp/96-in; Kaplan et al PRA(97)qp/98; Klauder NPB(99)ht; Corichi CQG(08)-a0801 [geometrical].
@ Related topics: Goldberg et al JMP(91); Rovelli PRL(98) [gauge transformations in quantum mechanics]; Lavrov et al MPLA(99) [osp(1,2) susy]; Facchi et al JOB(04)qp/03 [and Zeno dynamics]; Konopka & Markopoulou gq/06 [states, from noiseless subsystems]; > s.a. regularization, superselection, 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]; Miskovic & Zanelli JMP(03), Klauder & Little CQG(06)gq [irregular].
@ 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.
@ Generally covariant: Montesinos GRG(01)gq/00 [relational evolution]; Sforza gq/00-PhD; > 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.

Main page – Abbreviations – Journals – Comments – Other sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified 20 jun 2008