Dirac Quantization of First-Class Constrained Systems  

In General > s.a. quantum particle models.
* Idea: Impose the constraints as operators on functions : C (this requires a choice of operator ordering and regularization); Their kernel defines physical states, and only on these we define an inner product such that the observables are selfadjoint.
* Example: Wave functions are densities of weight 1/2 on phase space, which have to satisfy (va Pa) = –i v (q) = 0, and the Hamiltonian is different,

H = –2 gab ab + potential = –2 gab Da Db 2 –1(Db) Db + potential .

* Remark: An anomaly in the commutators would mean, e.g., that the wave function on a given surface depends on gauge equivalent paths used to get there!
* Criticism: Might lead to non-normalizable states if the gauge orbits are non-compact.

Comparisons, Relationships > s.a. canonical quantum mechanics [group quantization].
@ And reduced phase space: Buchbinder & Lyakhovich TMP(89) [including inner product]; Mladenov IJTP(89); Romano & Tate CQG(89); Loll PRD(90); Schleich CQG(90); Kunstatter CQG(92); Ordóñez & Pons PRD(92), JMP(95)ht/93; Plyushchay & Razumov IJMPA(96)ht/93, ht/94-in; Epp PRD(94); Vathsan JMP(96)ht/95 [for simple gauge theory]; Shimizu PTP(97)gq/96; > s.a. dirac quantum field theory.
@ And path integral: Faddeev TMP(69); Maskawa & Nakajima PTP(76); Hájícek JMP(86) [path integral version of projector]; Blau AP(91); Cabo et al PLB(91); Halliwell & Hartle PRD(91); Govaerts JPA(97)ht/96 [phase space coherent states]; Barvinsky NPB(98)ht/97, PLB(98)ht/97 [solution].
@ And Faddeev-Jackiw: García & Pons IJMPA(98)ht; Liao & Huang AP(07).
@ And other approaches: in Faddeev & Slavnov 80 [BRST]; Barvinsky & Krykhtin CQG(93) [BFV, 1-loop]; Ogawa et al PTP(96)ht/97 [Schwinger]; Louis-Martinez PLA(00) [and Moyal]; Shvedov AP(02)ht/01 [BRST-BFV]; Lantsman ht/06 [and Faddeev-Popov].

References
@ General: Dirac CJM(50); Bergmann HPA(56); Dirac PRS(58), PR(59), 64; Matschull qp/96 [review]; Deriglazov PLB(05) [without primary constraints].
@ Geometric version: Tulczyjev SM(74); Lichnerowicz CRAS(75); Barvinsky gq/96; Gozzi NPPS(97)dg.
@ Objections, problems: Shanmugadhasan JMP(73); Kundt.
@ Applications, examples: DeWitt PR(67); Kuchar PRD(89); Hájícek CQG(90) [quadratic constraint]; Hájícek & Kuchar PRD(90), JMP(90) [quadratic + linear constraints, operator ordering and transversal affine connection]; Montesinos et al PRD(99)gq [2 non-commuting Hamiltonian constraints]; Miskovic & Zanelli JMP(03) [irregular systems]; Golovnev IJGMP(06)qp/05 [particle motion on a surface]; Rosenbaum et al ht/06 [spacetime non-commutative theories]; > s.a. parametrized theories.
@ Master constraint approach: Dittrich & Thiemann CQG(06)gq/04 [framework], CQG(06)gq/04 [finite-dimensional], CQG(06)gq/04 [SL(2, R) models], CQG(06)gq/04 [free field theories], CQG(06)gq/04 [interacting field theories]; > s.a. lqg.
@ Related topics: Tuynman JMP(90) [modified]; Barvinsky CQG(93) [operator ordering]; Tate gq/93-PhD [algebraic]; Seiler & Tucker JPA(95) [from the pde point of view]; Rovelli gq/97 [space of solutions]; Kempf & Klauder JPA(01)qp/00 [0 continuous spectrum and projection].

Refined Algebraic Quantization
* Idea: A variation of the Dirac prescription, motivated by the fact that in many cases, physical states of interest are not in kin; Given a kinematical kin, instead of imposing constraints on the kinematical states to get physical ones, rig kin to obtain kin *, and choose a suitable phy *, with inner product ((1), (2)):= (2)[1], where is the rigging map; One also defines an action of physical observables.
@ General references: Marolf gq/95; Embacher gq/97, HJ(98)gq/97; Giulini & Marolf CQG(99)gq/98, CQG(99)gq; Giulini NPPS(00)gq [rev].
@ Specific theories: Ashtekar & Tate JMP(94)gq [examples]; Louko & Rovelli JMP(00)gq/99 [SL(2, R) gauge theory]; Shvedov ht/01 [systems with structure functions]; Louko & Molgado CQG(05)gq [Ashtekar-Horowitz-Boulware model]; Rumpf gq/97 [relativistic particle in cst].

Group Averaging
* Try: Do the case C = px + py on T2, with irrational.
* Idea: An implementation of the refined algebraic quantization method, which one can use (modulo expression being well-defined) when the constraint algebra closes, but has been formally used even in a more general case.
* Prescription: Define the rigging map : * by (| ) (|:= V–1 dG(g) | U(g), U a representation of G.
@ References: Gomberoff & Marolf IJMPD(99)gq [SO(n,1)]; Giulini NPPS(00)gq [rev]; Gomberoff ht/00-MG9 [non-compact G]; Marolf gq/00-MG9; Louko & Molgado JMP(04)gq/03 [(p, q)-oscillator representation of SL(2, R)], IJMPD(05)gq/04 [subgroup of SL(2, R)]; Louko gq/05-in [non-compact groups].


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