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(−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;
- 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];
Baulieu a2012 [ADM variables, unimodular gauge];
> 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
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