Canonical Quantum Gravity  

In General > s.a. models in canonical quantum gravity; supergravity.
* Advantages: Done with Hamiltonian methods, using a Hilbert space of states and an algebra of observables, emphasizing the geometrical character of quantum gravity; Therefore, as compared to the covariant approach, it is broader and deeper; If non-perturbative, it is applicable to strong gravity and can ensure unitarity.
* Drawbacks: More difficult to handle than the covariant approach, mainly because of the constraints (kinematical difficulty); Only R topologies are allowed, and only spatial diffeomorphisms implemented as symmetries, while timelike ones are mixed with dynamics; It is difficult to ask spacetime questions, since wave functions are t-independent.
* Observables: They have to be non-local; In the spatially closed case, we don't know a single one (> see observables).
@ General references: Kuchar gq/93; Baez gq/99 [higher-dimensional algebra]; Thiemann gq/01/LRR [hard]; Pullin IJTP(99), gq/02-in [simple]; Giulini & Kiefer gq/06-in [and geometrodynamics]; Montani gq/07-in [critical view]; Cianfrani et al a0805.

Constraints > s.a. time in quantum gravity.
* Idea: Conditions on the quantum states that correspond to the classical constraints of general relativity, and represent the gauge (difeomorphism) invariance of the theory; Usually implemented following the Dirac prescription by defining operator constraints and imposing that physical states belong to their kernel; In the refined algebraic quantization / group averaging variant, they result from the action of a projection operator.
* Constraint projection operator: If H is the Hamiltonian constraint operator, the projection operator onto its kernel is

P = [dN(x)] exp{–i N(x) H(x)} ,

and it is also expected to provide a link between the canonical and the path integral formulations of the theory.
@ General references: Moncrief PRD(72); Christodoulou & Francaviglia AAST(76); Jackiw gq/95 [commutator anomalies]; Salisbury FP(01)gq [projector]; Shojai & Shojai Pra(02)gq/01, CQG(04) [algebra, de Broglie-Bohm]; Gentle et al IJMPA(04)gq/03 [geometrodynamics]; Christodoulakis & Papadopoulos gq/04 [and covariance]; Soo gq/05-in [simplification].
@ Factor ordering: Anderson PR(59), in(63); Schwinger PR(63), PR(63); DeWitt PR(67); Komar PRD(79); Christodoulakis & Zanelli NCB(86), CQG(87) [and field redefinitions]; Tsamis & Woodard PRD(87); Friedman & Jack PRD(88), in(91); McMullan & Paterson PLB(88); Carlip PRD(93) [in 3D]; Rosales PRD(96)gq [and time]; Ferraro & Sforza NPPS(00)gq-in; > s.a. geometrodynamics.
@ Variations: Kheyfets & Miller PRD(95)gq/94 [as expectation values].

Issues and Approaches > s.a. approaches to quantum gravity [pilot wave]; canonical quantum mechanics; gravity theories; lattice field 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 ) 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.
@ Group quantization: Isham & Kakas CQG(84), CQG(84).
@ Using the space of solutions: in Bergmann pr(69); Geroch AP(71).
@ With boundary: Baez et al PRD(95)gq; Smolin JMP(95)gq; Pervushin et al PLB(96); Major CQG(00)gq/99 [spin nets].
@ And covariant: Landsman CQG(95)gq; Barvinsky & Kiefer NPB(98)gq/97 [semiclassical]; Kanatchikov IJTP(01)gq/00; > s.a. spin foam.
@ And path integrals: Halliwell PRD(88); Guven & Ryan PRD(92); Sorkin & Sudarsky CQG(99)gq [black hole horizon fluctuations]; Muslih GRG(02); Savvidou CQG(04)gq/03, CQG(04)gq/03; > s.a. path integral quantum gravity.
@ Gauge fixing: Montani NPB(02)gq ["kinematical action"], IJMPD(03)gq; Mercuri & Montani IJMPD(04)gq/03; Battisti & Montani a0801-in.
@ Measure, inner product: Marolf in(95)gq/94 [spectral analysis, for minisuperspace]; Menotti NPPS(98)hl/97-in [finite-dimensional space of geometries]; > s.a. geometrodynamics, path integrals, regge calculus.
@ Linearized: Khrustalev & Tchitchikina gq/01, gq/01 [around arbitrary solution]; > s.a. perturbations in general relativity, quantum gravity.
@ Approximations: Christodoulou & Francaviglia GRG(77).
@ Related topics: Crane PLB(91); González-Díaz G&C(97)gq; Montesinos GRG(01)gq/00 [relational evolution]; Gambini & Pullin PRL(00) [expansion around → ], PRL(03) [discrete]; Wang CQG(05) [non-linear quantization]; Giesel & Thiemann CQG(07)gq/06, CQG(07)gq/06, CQG(07)gq/06 [algebraic]; > s.a. diffeomorphisms; geometrical operators; theta sectors.

Metric Variables > s.a. 2D quantum gravity; 3D quantum gravity; ADM form of general relativity; geometrodynamics; quantum cosmology.
* Elementary variables: The qab and pab of the classical ADM canonical formulation.
* States: In the Schrödinger representation, they are given by diffeomorphism-invariant functionals (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); Kuchar in(73); Ashtekar & Geroch RPP(74); Kuchar in(81); Christodoulakis in(86); Christodoulakis & Zanelli CQG(87); Kuchar & Torre PRD(91) [and reference fluid].
@ Solutions: Kowalski-Glikman & Meissner PLB(96)ht; Blaut & Kowalski-Glikman PLB(97)gq [+ scalar], gq/97 [pure gravity].

Other Variables and Representations > s.a. connection and loop representation.
@ Similar to ADM: Bousso & Hawking PRD(99)ht/98 [Kab, including examples].
@ Vielbein variables: Paternoga & Graham PRD(00)gq [triad, Chern-Simons state]; Cianfrani & Montani CQG(07) [tetrad, no gauge fixing].
@ Affine variables: Klauder JMP(99)gq; Watson & Klauder JMP(00)qp; Klauder JMP(01)gq, CQG(02)gq/01, ht/04-in, IJGMP(06)gq/05 [rev].
@ 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 gq/06-in.
@ Tomographic: Man'ko et al GRG(05), GRG(05); > s.a. minisuperspace.
@ Related topics: Rayner CQG(90) [metric-based loop variables]; Matschull CQG(95)gq/94 [combination of Ashtekar and Wheeler-DeWitt]; Crane gq/97; Bobienski et al gq/01-MG9 [2-surfaces]; Sahlmann gq/02 [comments on representations]; Okolów & Lewandowski CQG(03)gq, CQG(05)gq/04 [holonomy-flux star-algebra]; Larsson ht/05, in(06)-a0709 [covariant, and diffeomorphism anomalies]; Varadarajan a0709 [uniqueness results and new representations].

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