Canonical Quantum Gravity

In General > s.a. approaches and variables; models [spherical symmetry, other reductions]; supergravity.
* Advantages: It provides a convenient analysis of the structure of the theory in terms of identifying different types of degrees of freedom; Done with Hamiltonian methods, using a Hilbert space of states and an algebra of observables, which emphases 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 Σ × \(\mathbb R\) 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: Kuchař gq/93; Baez gq/99-in [higher-dimensional algebra]; Thiemann gq/01/LRR [hard]; Pullin IJTP(99), AIP(03)gq/02 [simple]; Giulini & Kiefer LNP(07)gq/06 [and geometrodynamics]; Montani gq/07-MGXI [critical view]; Cianfrani et al a0805; Ashtekar GRG(09)-a0904 [diffeomorphisms, background independence]; Cianfrani et al 14 [pedagogical]; Sharatchandra a1806; Thiemann a2003 [constructive QFT and renormalisation].
@ 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 models.
@ 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.
@ Approximations: Christodoulou & Francaviglia GRG(77).
@ Related topics: Gambini & Pullin PRL(00) [expansion around Λ → ∞]; Bojowald et al PRD(14)-a1402 [discreteness corrections and higher spatial derivatives]; Patrascu a1406 [quantum gravity and topology change]; Małkiewicz a1512-MG14 [internal clock and physical Hilbert space]; Lin CQG(16)-a1508 [quantum Cauchy surfaces].

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

\(P = \int {\cal D}N(x)\, \exp\{-{\rm i}\,N(x)\, H(x)\}\;, \)

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 in(07)gq/05 [simplification]; Thébault Symm(11)-a1108 [interpretation in quantum gravity].
@ 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; Anderson CQG(10)-a0905 [conformal, in quantum cosmology]; Maitra APPS(09)-a0910 [for FLRW models, and causal dynamical triangulation approach]; > s.a. geometrodynamics.
@ Variations: Kheyfets & Miller PRD(95)gq/94 [as expectation values]; Tibrewala a1207-proc [modified constraint algebra and spacetime interpretation]; Gryb & Thébault FP(14)-a1303 [with spatial conformal diffeomorphisms as gauge group]; Tibrewala CQG(14)-a1311 [deformation from inhomogeneities and lqg corrections].

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