Approaches to Quantum Gravity  

In General > s.a. observables; phenomenology; quantum cosmology; semiclassical general relativity; statistical mechanics.
* Main types: One may distinguish between (a) "conventional" approaches, which can be based on the quantization of a classical gravity theory (quantum geometrodynamics, loop quantum gravity, spin-foam models, Regge calculus and causal dynamic triangulations, group field theory) or on an extension of quantum field theory (string theory), and (b) other approaches, often discrete, sometimes combinatorial or based on condensed-matter physics ideas (causal sets, quantum graphity).
* "Conventional" approaches: The spacetime manifold remains, although one sometimes considers contributions from different ones; 1988, Most results have come just from quantum field theory in curved spacetime calculations; The next step is to include the back-reaction, necessary to achieve above goals and to understand things like dissipation of anisotropies.
@ General references: Mielczarek & Trześniewski a1708 [links between approaches]; > s.a. quantum gravity [overviews].

Perturbative Approaches > s.a. covariant quantum gravity; string theory.
* Idea: The fuzziness in the metric and causal relations is obtained by quantizing deviations from a background, reference metric, usually taken to be flat or (anti-)de Sitter, on a fixed manifold; Initially based on the Lorentz-covariant spin-2 field theory in 4D Minkowski (possibly including higher-derivative theories or supergravity, and/or using path integrals), and thus called "covariant quantum gravity"; It is the approach that has been most used in combination with other interactions, and later merged in part with superstrings/M-theory.
@ References: Mandelstam AP(62), PR(68); Floreanini & Percacci PRD(92) [mean-field approach]; Parentani NPB(97)gq/96 [radiative processes in quantum cosmology]; Ichinose & Ikeda IJMPA(99)ht/98; Deser AdP(99)gq/99 [infinities, including supergravity]; Bern LRR(02)gq [relationship with gauge theory]; Tsuneyama ht/04/PRD; Freidel & Starodubtsev ht/05 [topological field theory approach]; Brunetti & Fredenhagen gq/06-proc [background-independent]; Anselmi & Benini JHEP(07)-a0704 [higher-derivative corrections]; Bern et al PRD(08)-a0707 [unexpected cancellations]; Ward IJMPD(08) [resummed]; Benedetti & Speziale JHEP(11)-a1105 [1st-order tetrad formalism and Immirzi parameter]; Vereshkov & Marochnik JModP(13)-a1108 [Heisenberg representation]; Akhoury et al PRD(11) [collinear and soft divergences]; Hossenfelder PLB(13)-a1208 [as an effective theory]; Brunetti et al CMP(16)-a1306 [as a locally covariant quantum field theory]; Upadhyay PLB(13)-a1305 [Batalin-Vilkovisky formalism], EPJC(14)-a1312 [gaugeon formalism]; Huang a1312 [in imaginary time]; Bellazzini et al PRD(16)-a1509 [constraints from unitarity and analyticity]; > s.a. covariant quantum gravity [including infrared behavior]; friedmann equation [quantum corrections].

Non-Perturbative Approaches > s.a. canonical quantum gravity [including affine, lqg]; path-integral quantization.
* Idea: (i) Canonical approach, including spin networks; (ii) Asymptotic quantization; (iii) Path integral quantization (Lorentzian or Euclidean), including spin foams; (iv) Numerical/discrete approaches auch as lattice gravity, Regge calculus, causal sets or dynamical triangulations; (v) Stochastic quantization.
@ References: Ambjørn et al PRP(12)-a1203 [CDT, asymptotic safety, lattice gravity]; Tavernelli a1801 [based on a geometrization of quantum mechanics].
@ And topos theory: Flori PhD(09)-a0911, Dahlen a1111 [loop quantum gravity]; Döring a1306 [and neo-realist interpretation].

Discrete Approaches > s.a. causal sets; discrete geometries and spacetime; lattice gravity and regge calculus.
@ References: Schmelzer gq/98 [condensed-matter type]; Holfter & Paschke JGP(03)ht/02 [and Dirac operator]; Gambini & Pullin in(09)gq/05; Hamma & Markopoulou NJP(11)-a1011 [condensed-matter-type spin models]; Gudder a1109-wd; Rivasseau AIP(12)-a1112, a1209-conf, FdP(14)-a1311, FdP(14)-a1406-proc, a1604-proc [tensor-track approach, including a sum over all topologies]; Oriti a1710-fs [spacetime as a quantum many-body system]; > s.a. quantum spacetime.
@ Information-based: Mandrin a1408, a1409, a1411 [from minimum information]; Chen a1412 [informationally-complete unification].

Asymptotic Quantization
* At null infinity: It lies somewhere in between the canonical approach and the covariant one; One avoids the 3+1 decomposition of the former and the linearization and use of a fixed background of the latter, and quantizes the radiation degrees of freedom (idea taken from QED), using the asymptotic properties of the gravitational field.
@ Null surfaces formulation: Frittelli et al PRD(97)gq/96; Domínguez & Tiglio PRD(99)gq [large effects].
@ At null infinity: Ashtekar PRL(81), JMP(81), in(81), 87.
@ At spatial infinity: Alexander & Bergmann FP(84) [electromagnetism], FP(86); Bergmann GRG(87), GRG(89).

Other Approaches > s.a. effective theories; types of approaches.
@ Other dimensions: Bjerrum-Bohr NPB(04)ht/03 [D → ∞]; Nieto a0704 [8D, background-independent Kaluza-Klein à la lqg]; Nieto RMF-a1003 [8D and (2+2) loop quantum gravity], IJGMP(12)-a1107 [in 2+2 dimensions]; Vaugon a1512 [pure geometry, of signature (–, –, +, ..., +)]; Deser GRG(16)-a1609 [1-loop divergences cannot all be removed in D > 4]; > s.a. 2D and 3D quantum gravity; regge calculus.
@ Using condensed matter ideas: Olmo & Rubiera-García IJMPD(15)-a1507 [point defects and dislocations described by non-metricity and torsion].
@ Analog models: Kiefer PLA(89) [analogy with brownian motion]; Baker a0810 [elastic-solid model]; Krein et al PRL(10)-a1006 [phonons in random fluids].
@ Categorical: Crane ht/93, gq/00; Isham ATMP(03)gq, ATMP(03)gq, ATMP(04)gq/03; Isham FP(05)qp/04-in; Baez qp/04; Raptis IJTP(06)gq/04-conf, IJTP(07) [and abstract differential geometry]; Crane gq/06-ch; > s.a. categories in physics; quantum spacetime models; Topos Theory.
blue bullet Based on modified gravity: see higher-order gravity; modified approaches [including linearized, deformed, pilot-wave, simplified models].

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