Canonical Quantum Gravity: Approaches and Variables  

In General > s.a. canonical quantum gravity; canonical quantum mechanics; gravity theories; modified quantum gravity [pilot-wave theory].
* Factor ordering: Factor ordering ambiguities arise even in the simplest minisuperspace models; Solutions have been proposed based on requirements of hermiticity of constraint operators, isomorphism (at least to leading order in \(\hbar\)) between the classical and quantum algebras, invariance under redefnition of variables, equivalence of kinetic part of scalar constraint to Laplacian in superspace; In order to be meaningful, such conditions require that a Hilbert space of states be well defined; > s.a. FLRW quantum cosmology.
@ Using the space of solutions: in Bergmann pr(69); Geroch AP(71); Esposito IJGMP(16)-a1511 [parametrix approach].
@ With matter: Pavšič PLB(11)-a1106 [point particle, and time coordinate]; Husain & Pawłowski PRL(12)-a1108 [dust-based time, physical Hamiltonian]; Bojowald et al QM-a1302 [general properties and description]; Schuller& Witte PRD(14)-a1402 [canonical gravitational theory from quantizable matter dynamics]; Purohit a1704 [scalar field]; Anselmi a2006 [massive quantum fields of arbitrary spin]; > s.a. models in canonical quantum gravity.
@ With material reference systems: Kuchař & Torre PRD(91) [metric variables]; Giesel & Thiemann a1206 [lqg].
@ Gauge fixing: Montani NPB(02)gq ["kinematical action"], IJMPD(03)gq; Mercuri & Montani IJMPD(04)gq/03; Battisti & Montani IJMPA(08)-a0801-proc.
@ Measure, inner product: Marolf in(95)gq/94 [spectral analysis, for minisuperspace]; Menotti NPPS(98)hl/97 [finite-dimensional space of geometries]; > s.a. geometrodynamics; path-integral quantum gravity; regge calculus.
@ Linearized: Khrustalev & Tchitchikina gq/01, gq/01 [around arbitrary solution]; > s.a. perturbations in general relativity; quantum gravity.
@ Related topics: Isham & Kakas CQG(84), CQG(84) [group quantization]; Crane PLB(91); González-Díaz G&C(97)gq; Montesinos GRG(01)gq/00 [relational evolution]; Gambini & Pullin PRL(03) [discrete]; Wang CQG(05) [non-linear quantization]; Giesel & Thiemann CQG(07)gq/06, CQG(07)gq/06, CQG(07)gq/06 [algebraic]; Kober IJMPA(12)-a1109; > s.a. diffeomorphisms; geometrical operators; lattice field theory; theta sectors.

Metric Variables > s.a. 2D quantum gravity; 3D quantum gravity; ADM form of general relativity; geometrodynamics; quantum cosmology.
* Elementary variables: The \(q_{ab}\) and \(p_{ab}\) of the classical ADM canonical formulation.
* States: In the Schrödinger representation, they are given by diffeomorphism-invariant functionals Ψ(q), satisfying the Wheeler-DeWitt equation.
* Problem: No operator ordering of the constraints found such that the quantum constraint algebra reproduces the classical one.
@ General references: DeWitt PR(67); Kuchař in(73); Ashtekar & Geroch RPP(74); Kuchař in(81); Christodoulakis in(86); Christodoulakis & Zanelli CQG(87); Gerhardt ATMP(18)-a1501.
@ Solutions: Kowalski-Glikman & Meissner PLB(96)ht; Błaut & Kowalski-Glikman PLB(97)gq [+ scalar], gq/97 [pure gravity].

Other Variables and Representations > s.a. connection and loop representation; holonomy.
@ Similar to ADM: Bousso & Hawking PRD(99)ht/98 [Kab, including examples]; Gerhardt ATMP(13)-a1205.
@ Vielbein variables: Paternoga & Graham PRD(00)gq [triad, Chern-Simons state]; Cianfrani & Montani CQG(07) [tetrad, no gauge fixing]; Kanatchikov JPCS(13)-a1302 [de Donder-Weyl Hamiltonian formulation and precanonical quantization]; Soo & Yu ChJP-a1601 [dreibein and SU(3) generators]; Donoghue PRD(17)-a1609 [the spin connection as confined, and metricity].
@ Affine quantization: Klauder JMP(99)gq; Watson & Klauder JMP(00)qp, CQG(02)gq/01; Klauder JMP(01)gq, CQG(02)gq/01, IJMPD(03)gq; Klauder gq/04-proc, gq/04-proc, PAN(05)ht/04-in, IJGMP(06)gq/05, JPCS(07)gq/06, a0711-proc, a1003-proc; Klauder JMP(12)-a1203, a1206-proc [rev]; Klauder a1811, a1903, a2004, a2006; > s.a. quantum black holes.
@ Complex SL(2,C) Chern-Simons theory: Haggard et al PLB(16)-a1509, a1512 [with a cosmological constant].
@ Covariant lqg: Alexandrov PRD(02)gq/01, & Vassilevich PRD(01)gq [area spectrum]; Salisbury FP(01)gq [physical operators]; Alexandrov & Livine PRD(03)gq/02 [spacetime connection]; Alexandrov PRD(02)gq [Hilbert space]; Alexandrov & Kadar CQG(05)gq [timelike surfaces]; Livine in(09)gq/06; Cianfrani & Montani a0904-proc; Geiller et al Sigma(11)-a1103, PRD(11)-a1105; Rovelli & Wilson-Ewing PRD(12)-a1205 [discrete symmetries and improved Holst action]; Han PRD(14)-a1308 [low-energy perturbation theory]; Cianfrani a2103 [without Immirzi parameter]; > s.a. spin-foam models.
@ Other covariant approaches: Larsson ht/05, in(06)-a0709 [and diffeomorphism anomalies]; Cremaschini & Tessarotto EPJC(17)-a1609, EPJC(17)-a1609; > s.a. covariant quantum gravity.
@ Tomographic: Man'ko et al GRG(05), GRG(05); Stornaiolo a2007 [quantum to classical cosmology]; > s.a. minisuperspace.
@ Other types and topics: Rayner CQG(90) [metric-based loop variables]; Matschull CQG(95)gq/94 [combination of Ashtekar and Wheeler-DeWitt]; Crane gq/97; Bobieński et al gq/01-MG9 [2-surfaces]; Okołów & Lewandowski CQG(03)gq, CQG(05)gq/04 [holonomy-flux star-algebra]; Varadarajan CQG(08)-a0709 [uniqueness results and new representations]; Aastrup & Grimstrup CQG(16)-a1602 + news conf(16)may [quantum holonomy theory]; Gomes CQG(17)-a1706 [timeless configuration space]; > s.a. connection representation [Chang-Soo variables].

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