Quantum Geometrodynamics: Dynamics in Superspace

The Program in General > s.a. canonical general relativity and ADM formulation [classical version].
* Idea: Geometrodynamics is the program for canonical quantum gravity based on superspace and the dynamics given there by the Wheeler-DeWitt equation; A 4-geometry is represented by a classical set of points C of superspace, contained in it as a hypersurface; The set of 3D geometries contains the time information; Initial data consists of a neighborhood in C, from which the Einstein equation gives C.
* Problems: Mainly interpretational, such as What is the meaning of (q,)? We need an inner product for 's before we say that ||² is a probability.

Kinematics: Superspace > s.a. canonical quantum gravity; regge calculus [simplicial superspace]; riemannian geometry; topology change.
* Idea: Superspace is a reduced configuration space for a Hamiltonian formulation of gravity, in which a point is a diffeo equivalence class of Riemannian metrics on a spacelike hypersurface : S():= Riem()/Diff().
* Constraints: This choice of configuration space eliminates the diffeomorphism constraint, but the scalar one (coming from the gauge arbitrariness in choosing a slicing of spacetime) remains, and becomes the dynamical equation.
* Topology: S() is not a manifold, but is made up out of a finite number of manifolds (strata).
* Extended: Glue together copies of the same space [@ DeWitt (70)].
* Grand: Need singular geometries.
* Minisuperspace: Impose enough symmetries to get finite number of degrees of freedom.
@ Structure of superspace: & Peres < 79; Fischer in(70); in Brill in(72); Reula pr(GR12); Christodoulakis & Zanelli pr(89); Giulini HPA(95)gq/93, PRD(95)gq/93; Fischer & Moncrief GRG(96) [including conformal superspace]; Giulini GRG(09)-a0902.
@ Supermetric: Pekonen JGP(87); Szydlowski JMP(99); Schmidt gq/01-in.
@ Minisuperspace: Misner in(72); Kuchar & Ryan in(86); Kiefer AP(91); Kerbrat et al RPMP(92); Saremi gq/01; > s.a. gowdy spacetimes.
@ Related topics: Jacobson in(88) [self-dual representation]; Rainer gq/96-in [locally homogeneous].

States: Wave Functionals > s.a. time in quantum gravity.
* Idea: States are functionals [q_ab,] of metrics (and matter fields) belonging to an appropriate space; Notice that these states will end up without a time dependence, since they must satisfy the constraints; Instead, time can be taken as one of the components of q_ab.
@ Inner product: DeWitt PR(67); Tsamis & Woodard PRD(87); Christodoulakis & Zanelli CQG(87); Vilenkin PRD(89).

Dynamics: Wheeler-DeWitt Equation > s.a. 3D quantum gravity; cosmological constant; FRW models [solutions].
* Idea: The "zero-energy" Schrödinger-like equation one gets as the operator version of the scalar constraint, by replacing q and p by their coordinate representation operators (multiplication operator and –i(/q), respectively), and acting with it on a wave functional [q_ab,].
* Expression: For the dynamics of gravity coupled to a field , it is

up to linear derivative terms depending on the choice of factor ordering, where the supermetric, with "signature" (–,+,+,+,+,+) –the minus sign coming from the conformal mode– and thought of as a 6 × 6 matrix, is defined by

G_abcd = q^–1/2 (q_ac q_bd + q_ad q_bc – q_ab q_cd) .

@ General references: Woodard CQG(93); Jackiw gq/95 [modified]; Blaut & Kowalski-Glikman gq/96 [solutions, and quantum potential]; Norbury EJP(98)phy [II, from Newtonian physics]; Soo CQG(07)gq [in terms of gauge-invariant 3-geometry elements]; Sawayama a0904 [small universe].
@ As spinor equation: Dereli et al PLB(94) [including degenerate metrics and signature change].
@ And stability of cosmological models: Gurzadyan & Kocharyan JETP(87); Kocharyan CMP(91).

References > s.a. canonical quantum gravity; quantum cosmology [measure, third quantization]; supergravity; supersymmetric field theories.
@ General: Wheeler 62; Baierlein et al PR(62); Fletcher in(62); Marzke & Wheeler in(64); Wheeler in(64); DeWitt PR(67); Wheeler in(68); Wheeler 68; Brill & Gowdy RPP(70); DeWitt in(70); Fischer in(70); Misner in(72); Christodoulou & Francaviglia AAST(76); Shestakova a0801-in [problems and approaches to their solution]; Kiefer GRG(09)-a0812 [overview].
@ Operator ordering: Hawking & Page NPB(86); Kontoleon & Wiltshire PRD(99)gq/98; Hall gq/04-in [from "exact uncertainty"].
@ And time evolution: Kheyfets & Miller gq/94, IJMPA(00); Cosgrove CQG(96)gq/95 [time slicings and consistency]; Gil-Medrano JMP(96); Gentle et al IJMPA(04)gq/03, George et al gq/03-in [constraints and true degrees of freedom].
@ Relationships: Kubota et al PLB(04) [Wheeler-DeWitt and AdS-cft].
@ WKB approximation: Gerlach PR(69); Horiguchi NCB(96).
@ Related topics: Parentani NPB(97)gq/96 [and Green functions for matter fields]; Carroll gq/05 [uncertainties and statistical geometrodynamics]; Nelson & Sakellariadou PLB(08)-a0709 [quantum corrections and effective matter]; Pedram & Jalalzadeh PRD(08) [signature change with fluids]].
@ Generalizations: Shestakova G&C(99)gq/00, G&C(05) [extended phase space], G&C(00)gq [and cosmological constant], a0810-in; Belinchón IJMPD(02)gq/01 [varying constants]; > s.a. phenomenology [with non-linear correction].

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