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.

__Based on modified gravity__: see higher-order gravity; modified
approaches [including linearized, deformed, pilot-wave, simplified models].

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send feedback and suggestions to bombelli at olemiss.edu – modified 18
jan 2018