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].

@ __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* = ∫ [d*N*(*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 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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