Path Integral Approach to Quantum Gravity  

In General
* 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; regge calculus.
* Regularization: Can be done by dynamical triangulation methods (> see regge calculus).
@ 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].
@ Measure: Leutwyler PR(64); DeWitt in(72); Fradkin & Vilkovisky PRD(73); Faddeev & Popov SPU(74); Kaku & Senjanovic PRD(77); Teitelboim PRD(83) [proper time gauge]; Botelho PRD(88).
@ FRW with scalar: Bernido PRD(96); Simeone PLA(03)gq [ambiguities, and canonical].
@ Issues: Teitelboim PRL(83) [gauuge invariance]; Dasgupta & Loll NPB(01)ht [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(–SE[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;
- 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);
- Cannot classify all distinct 4-manifolds, so 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.
@ Connection variables: Kshirsagar CQG(93); Alexandrov & Vassilevich PRD(98)gq [Ashtekar]; Ita a0804 [finiteness, generalized Kodama states].
@ Approaches: Muslih GRG(02)mp/00 [Hamilton-Jacobi].
@ Models: Halliwell & Louko PRD(89) [de Sitter, steepest descent contour and boundary conditions], PRD(90) [general homogeneous models, steepest descent]; Giribet & Simeone IJMPA(02)gq/01 [Taub universe]; > s.a. FRW models, minisuperspace.

References > s.a. canonical quantum gravity [relationship]; path integrals in quantum mechanics; quantum cosmology; quantum regge calculus.
@ Books, reviews: Hawking in(79); Esposito 01; Simeone 02 [quantum cosmology]; Hamber a0704-RMP [discrete and continuum].
@ General: 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.
@ Positivity of action: Schoen & Yau PRL(79).
@ Conformal factor problem and stability: Mazur & Mottola NPB(90).
@ Related topics: Turok PLB(99)gq [stability of Minkowski]; Pfeiffer gq/04 [and manifold invariants]; Marlow IJTP(06)gq [histories algebra and Bayesian prob's]; > s.a. gravitational instanton.

Variations on the Theory
@ General references: Gamboa & Mendez NPB(01)ht/00 [t = spacetime volume].
@ Generalized manifolds: Schleich & Witt NPB(93)gq, NPB(93)gq [singular], CQG(99)gq [exotic]; > s.a. differentiable manifolds.

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 27 jun 2008