Dirac Quantization of First-Class Constrained Systems  

In General > s.a. quantum particle models.
* Idea: Impose the constraints as operators on functions Ψ: Γ → \(\mathbb C\) (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) ψ = –i\(\hbar\) \(\cal L\)v ψ(q) = 0, and the Hamiltonian is different,

H = –\(\hbar\)2 gabab + potential = –\(\hbar\)2 gab Da Db – \(\hbar\)2 λ–1(Dbλ) 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.
@ General references: Dirac CJM(50); Bergmann HPA(56); Dirac PRS(58), PR(59), 64; Matschull qp/96 [review]; Deriglazov PLB(05) [without primary constraints]; Lantsman a1110 [Dirac variables and gauge-invariant and Poincaré-covariant states]; Kiriushcheva et al a1112 [field-parametrization dependence].
@ Geometric version: Tulczyjev SM(74); Lichnerowicz CRAS(75); Barvinsky gq/96; Gozzi NPPS(97)dg; Lian et al a1703 [geometric potential].
@ Objections, problems: Shanmugadhasan JMP(73); Kundt.
@ Applications, examples: DeWitt PR(67); Kuchař PRD(89); Hájíček CQG(90) [quadratic constraint]; Hájíček & Kuchař PRD(90), JMP(90) [quadratic + linear constraints, operator ordering and transversal affine connection]; Montesinos et al PRD(99)gq [2 non-commuting Hamiltonian constraints]; Mišković & Zanelli JMP(03) [irregular systems]; Rosenbaum et al JPA(07)ht/06 [spacetime non-commutative theories]; > s.a. modified electromagnetism; parametrized theories.

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 \(\cal H\)kin; Given a kinematical \(\cal H\)kin, instead of imposing constraints on the kinematical states to get physical ones, rig \(\cal H\)kin to obtain Ω ⊂ \(\cal H\)kin⊂ Ω*, and choose a suitable \(\cal H\)phy ⊂ Ω*, with inner product (η(ψ1), η(ψ2)):= η(ψ2)[ψ1], where η is the rigging map; One also defines an action of physical observables; The rigging map can be defined when the constraints form a Lie algebra (not an algebra with non-trivial structure functions).
@ General references: Marolf gq/95; Embacher gq/97-MG8, HJ(98)gq/97; Giulini & Marolf CQG(99)gq/98, CQG(99)gq; Giulini NPPS(00)gq [rev]; Louko & Martínez-Pascual JMP(11)-a1107 [constraint rescaling]; Martínez-Pascual JMP(13)-a1305 [with non-constant gauge-invariant structure functions].
@ Specific theories: Ashtekar & Tate JMP(94)gq [examples]; Louko & Rovelli JMP(00)gq/99 [SL(2, \(\mathbb R\)) gauge theory]; Shvedov ht/01 [systems with structure functions]; Louko & Molgado CQG(05)gq [Ashtekar-Horowitz-Boulware model]; Rumpf gq/97 [relativistic particle in curved spacetime]; Gambini & Olmedo CQG(14)-a1304 [totally constrained model, and other quantization methods].

Group Averaging
* Try: Do the case C = px + α py on T2, with α 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 η: Ω → Ω* by (|ψ\(\rangle\) ∈ Ω) \(\mapsto\) (ψ|:= V–1 ∫ dμG(g) \(\langle\)ψ| U(g), with U a representation of G.
@ References: Giulini NPPS(00)gq [rev]; Marolf gq/00-MG9; Marolf PRD(09)-a0902 [perturbation theory for rigging map].
@ Non-compact groups: Gomberoff ht/00-MG9; Louko JPCS(06)gq/05.
@ Examples: Gomberoff & Marolf IJMPD(99)gq [SO(n,1)]; Louko & Molgado JMP(04)gq/03 [(p, q)-oscillator representation of SL(2, \(\mathbb R\))], IJMPD(05)gq/04 [subgroup of SL(2, \(\mathbb R\))]; Kamiński et al CQG(09)-a0907 [lqc-related examples]; > s.a. quantum klein-gordon theory.

Other Approaches and Comparisons > s.a. canonical quantum mechanics [group quantization].
* Master constraint approach: Replace the individual constraints by a weighted sum of absolute squares of the constraints.
@ Master constraint approach: Dittrich & Thiemann CQG(06)gq/04 [framework], CQG(06)gq/04 [finite-dimensional], CQG(06)gq/04 [SL(2, \(\mathbb R\)) models], CQG(06)gq/04 [free field theories], CQG(06)gq/04 [interacting field theories]; Han & Thiemann JMP(10)-a0911 [and refined algebraic quantization]; > s.a. loop quantum gravity.
@ 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-conf; 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); Hájíček JMP(86) [path-integral version of the 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]; Han & Thiemann CQG(10)-a0911 [and master constraint and reduced phase space].
@ And Faddeev-Jackiw approach: García & Pons IJMPA-ht/96, IJMPA(98)ht; Liao & Huang AP(07); Manavella IJMPA(14) [composite particles, with Grassmann dynamical variables]; > s.a. Stückelberg Model.
@ 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 FizB(09)ht/06 [and Faddeev-Popov].
@ Related topics: Tuynman JMP(90) [modified]; Barvinsky CQG(93) [operator ordering]; Tate PhD(92)gq/93 [algebraic approach]; 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]; Brody et al JPA(09)-a0903 [variant, metric approach].

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