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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