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__: 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].

@ __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]; Pfeiffer gq/04 [and
manifold
invariants]; Marlow IJTP(06)gq
[histories
algebra and Bayesian probabilities]; > s.a. gravitational
instanton.

**Variations on the Theory**

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

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

main page – abbreviations – journals – comments – other
sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 12
mar 2017