Path-Integral Approach to Quantum Gravity |

**In General** > s.a. histories formulations
and path integrals in quantum mechanics.

* __Advantages__: It allows
to ask more meaningful questions about the evolution of spacetime than canonical
quantum gravity (& Sorkin); Time, and timelike diffeomorphisms, are treated
on an equal footing as others.

* __Disadvantages__:
(i) May be too rooted in the classical notion of history (& Isham).

* __Problems__: (i) The
sum over all physically distinct manifolds is well-defined only in 2D – it
may or may not be in 3D, and surely it is not in 4D or higher (a possible solution
is to enlarge the set of manifolds); (ii) There is a divergence due to the conformal
modes of the metric; (iii) Perturbation theory *D* > 2 requires higher
derivatives in the free action, which seem to lead to ghosts.

**Lorentzian** > s.a. quantum cosmology;
quantum regge calculus.

* __Regularization__:
It can be done by dynamical triangulation methods (> see
dynamical triangulations).

@ __General references__:
Teitelboim PRD(82) [closed spaces],
PRD(83) [asymptotically flat spaces];
Cline PLB(89);
Farhi PLB(89);
Ambjørn et al PRL(00)ht,
PRD(01)ht/00,
Loll LNP(03)ht/02 [non-perturbative];
Chishtie & McKeon CQG(12)-a1207 [first-order form of the Einstein-Hilbert action].

@ __Measure__: Leutwyler PR(64);
DeWitt in(72);
Fradkin & Vilkovisky PRD(73);
Faddeev & Popov SPU(74);
Kaku & Senjanović PRD(77);
Teitelboim PRD(83) [proper time gauge];
Botelho PRD(88).

@ __FLRW with scalar__: Bernido PRD(96);
Simeone PLA(03)gq [ambiguities, and canonical].

@ __Issues__: Teitelboim PRL(83) [gauge invariance];
Dasgupta & Loll NPB(01)ht,
GRG(11)-a0801 [conformal problem, fix].

**Euclidean** > s.a. 3D quantum gravity;
semiclassical quantum gravity; Wick Rotation.

* __Idea__: It generalizes the idea
of Feynman path integrals, using euclideanized (positive-definite) metrics; An amplitude
is a sum of exp(–*S*_{E}[*g*])
over all manifolds *M*, differentiable structures and geometries interpolating
between two 3-manifolds.

* __Motivation__: (i) Conceptually,
it develops a perturbative scheme not based on the coupling constant; (ii) It allows
to sum over all spacetime manifolds, thus including the effects of topology change.

* __Drawbacks__:

- Interpretational problems,
like relating the calculations to the Lorentzian case (easier in flat spacetime),
and causality;

- Difficulty of defining the
measure, the usual problem in path-integral methods;

- It is usually impossible to
represent (*M*,* g*) as a "Lorentzian" section
of a complex manifold with a "Euclidean" section;

- Even if the previous problem
was not present (static spacetimes), there is no guarantee of analyticity;

- The Euclidean action is not
positive definite in general – the "conformal factor problem
– (but see references below);

- One cannot classify all distinct
4-manifolds, so one cannot construct the space of histories, let alone inequivalent
ones.

* __Applications__: It has
become important in quantum cosmology.

@ __References__: in Deser, Duff & Isham PLB(80) [meaning];
Hayward PRD(96)gq/95 [complex lapse];
Hawking & Hertog PRD(02)ht/01 [without ghosts].

**Specific Approaches and Models** > s.a. 2D quantum gravity;
quantum cosmology; spin-foam models.

@ __Connection variables__: Kshirsagar CQG(93);
Alexandrov & Vassilevich PRD(98)gq [Ashtekar];
Ita HJ-a0804 [finiteness, generalized Kodama states];
Han CQG(10)-a0911 [for master constraint of loop quantum gravity];
Engle et al CQG(10)-a0911,
Han CQG(10)-a0911 [for Holst and Plebański gravity].

@ __Approaches__: Muslih GRG(02)mp/00 [Hamilton-Jacobi];
Krishnan et al JHEP(16)-a1609 [semi-classical, with Neumann boundary conditions];
Sharatchandra a1806
[extraction of the Hilbert space and constraints from the formal functional integral].

@ __Models__: Halliwell & Louko PRD(89) [de Sitter, steepest-descent contour and boundary conditions],
PRD(90)
[general homogeneous models, steepest-descent approximation];
Giribet & Simeone IJMPA(02)gq/01 [Taub universe];
> s.a. bianchi-I quantum cosmology; FLRW models;
minisuperspace.

**References** > s.a. canonical quantum
gravity [relationship];
quantum cosmology; quantum
regge calculus.

@ __Books, reviews__: Hawking in(79); Esposito
01; Simeone 02 [quantum cosmology]; Hamber a0704/RMP [discrete
and continuum].

@ __General__: Misner RMP(57); Clarke CMP(77);
Hawking PRD(78);
Taylor PRD(79);
Fujikawa & Yasuda NPB(84);
Arisue et al PRD(87);
Gibbons, Hawking & Stewart
NPB(87);
DeWitt in(88); Hartle PRD(88),
PRD(88),
pr(88); Bern, Blau & Mottola PRD(91)
[covariance]; Mottola JMP(95)ht.

@ __Measure__: Mazur PLB(91)ht/97;
Anselmi PRD(92);
Hamamoto & Nakamura PTP(00)ht [higher-order];
Aros et al CQG(03)gq;
Dasgupta a1106 [Euclidean].

@ __Positivity of action__: Schoen & Yau PRL(79).

@ __Conformal factor problem and stability__: Mazur & Mottola NPB(90); Dasgupta GRG(11) [and the trace of the diffeomorphisms].

@ __Related topics__: Turok PLB(99)gq [stability of Minkowski space];
Pfeiffer gq/04 [and manifold invariants];
Marlow IJTP(06)gq [histories algebra and Bayesian probabilities];
Káninský a1712-dipl [probabilistic spacetime];
> s.a. gravitational instanton.

**Variations on the Theory**

@ __General references__: Gamboa & Mendez NPB(01)ht/00 [*t* = spacetime volume];
Mandrin a1602 [non-equilibrium extension].

@ __Modified theories__: Borzou a1805 [Lorentz gauge theory of gravity];
> s.a. unimodular gravity.

@ __Generalized manifolds__: Schleich & Witt NPB(93)gq,
NPB(93)gq [singular],
CQG(99)gq [exotic]; > s.a. differentiable manifolds.

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