Quantum Geometrodynamics: Dynamics in Superspace |

**The Program in General** > s.a. canonical general relativity
and ADM formulation [classical version]; canonical quantum gravity.

* __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 Ψ(*q*, *φ*)? We need an inner
product for Ψs before we say that |Ψ|^{2}
is a probability.

**Kinematics: Superspace**
> s.a. canonical quantum gravity; regge
calculus [simplicial superspace]; topology change.

* __Idea__: Superspace is a reduced
configuration space for a Hamiltonian formulation of gravity, in which a point
is a diffeomorphism equivalence class of Riemannian metrics on a spacelike
hypersurface Σ: *S*(Σ):= Riem(Σ)/Diff(Σ).

* __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*(Σ)
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;
Gomes JMP(11);
Liu a1509
[Gromov's *ε*-approximation topology].

@ __Supermetric__: Pekonen JGP(87);
Szydłowski JMP(99);
Schmidt gq/01-proc;
Hehl & Kiefer GRG(18)-a1711 [comparison with the 4th-rank constitutive tensors in electrodynamics and elasticity theory].

@ __Minisuperspace__: Misner in(72);
Kuchař & 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-conf [locally homogeneous];
Barbour & Ó Murchadha a1009 [conformal superspace];
> s.a. Configuration Space; riemannian geometry.

**States: Wave Functionals**
> s.a. time in quantum gravity.

* __Idea__: States are functionals
Ψ[*q*_{ab},*φ*]
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; FLRW quantum cosmology;
lattice gravity [discretized].

* __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\(\hbar\)(δ/δ*q*), respectively),
and acting with it on a wave functional
Ψ[*q*_{ab}, *φ*].

* __Expression__: For the dynamics of gravity
coupled to a field *φ*, it is

{−*G*_{abcd}
(δ^{2}/δ*q*_{ab}
δ*q*_{cd})
+ *q*^{1/2}[–^{3}*R*
+ 2Λ + (16π*G*)^{−1}
*T*_{00}(iδ/δ*φ*,
*φ*)]} Ψ(*q*, *φ*) = 0 ,

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}
= \(1\over2\)*q*^{−1/2}
(*q*_{ac} *q*_{bd}
+ *q*_{ad} *q*_{bc}
− *q*_{ab}
*q*_{cd}) .

@ __General references__:
Woodard CQG(93);
Jackiw gq/95 [modified];
Błaut & 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];
Cherkas & Kalashnikov DANB-a1406 [near small scale factors, discrete vs continuum spectrum];
Rovelli CQG(15)-a1506 [rev].

@ __Related topics__: Gurzadyan & Kocharyan JETP(87),
Kocharyan CMP(91) [stability of cosmological models];
Dereli et al PLB(94) [as spinor equation, including degenerate metrics and signature change];
Mehta a1912 [in null-foliated spacetimes].

**References** > s.a. quantum cosmology [measure, third quantization];
semiclassical quantum gravity; 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-proc,
MG12(12)-a0911 [problems and approaches to their solution];
Kiefer GRG(09)-a0812 [overview];
Shestakova G&C(19)-a1912 [and gauge invariance].

@ __Operator ordering__:
Hawking & Page NPB(86);
Kontoleon & Wiltshire PRD(99)gq/98;
Hall GRG(05)gq/04-in [from "exact uncertainty"];
Huang et al EPJC(16)-a1505 [and the existence of a classical universe].

@ __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-proc [constraints and true degrees of freedom];
Ita et al PTEP(15)-a1501 [with intrinsic time].

@ __Relationships__: Kubota et al PLB(04) [Wheeler-DeWitt and AdS-cft];
Gielen Sigma(11)-a1111 [and connection dynamics].

@ __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],
in(10)-a0810;
Belinchón IJMPD(02)gq/01 [varying constants];
Brännlund et al CQG(14)-a1312 [topology and volume effects];
Faizal IJMPA(14)-a1406,
IJMPA(15)-a1503 [deformed canonical commutation relations];
Anderson a1412 [space of spaces].

__Based on modified gravity__:
see phenomenology [with non-linear correction]; lattice gravity
and regge calculus [discrete versions]; modified approaches
[scalar-tensor].

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