Covariant Quantum Gravity |

**In General** > s.a. effective
quantum field theory; perturbations in general relativity.

* __Idea__: A perturbative
approach to quantum gravity, similar to those of other field theories, which
focuses on scattering processes involving gravitons; The name refers to
Poincaré covariance with respect to the Minkowski background.

* __Procedure__: One chooses a
background metric *η*, usually Minkowski, divides the physical metric
into 2 terms, *g*_{ab}
= *η*_{ab}
+ *G*^{1/2}*h*_{ab}
(*h*_{ab} = "gravitational
potential"), and treats this as an interacting spin-2 field theory
(*G*^{1/2} here really is *l*_{P},
that's where it appears); To calculate amplitudes, use the Lorentzian path-integral approach (with the Faddeev-Popov
trick) and the stationary-phase (saddle-point) approximation; For general relativity, the partition function is

*Z*[*g*_{c}, *j*]
= ∫ \(\cal D\)*h* \(\cal D\)*φ*
exp**{**(i/\(\hbar\))
∫ *G*^{−1}
|*g*|^{1/2} [*R*(*g*)−Λ]
+ \(\cal L\)(*φ*, *g*, *j*)**}** ;

To generate 1-loop diagrams, expand to second order; The external lines are
on shell iff *η* satisfies the Einstein equation.

* __Motivation__: It is a
pragmatic approach in which one knows how to do certain things without the
need for a new framework, and it should work if one stays
well above Planck length; It is unitary [@ DeWitt].

* __Drawbacks__: (1) Not
a deep approach, misses many features that distinguish quantum gravity from
other field theories; (2) Linearized approach, which uses a fixed background;
(3) The resulting quantum field theory appears to be non-renormalizable [@
't Hooft & Veltman AIHP(74), ...], although some ways of overcoming
this problem have been proposed; (4) The semiclassical ground state is unstable [@ Horowitz, Hartle];
(5) It cannot address questions related to the regime near the initial singularity;
(6) One does not know what the path integral measure is, nor how to give a covariant
meaning to \(\langle\)*h*, *t* | *h*', *t*'\(\rangle\).

**Issues and Techniques** > s.a. approaches [perturbative];
renormalization; semiclassical general relativity.

* __Stability of Minkowski__:
Flat spacetime cannot decay, because of the positive-energy theorem, but
it can have large fluctuations.

* __Graviton propagator__:
One-loop corrections to it induce 1/*r*^{3}
corrections to the Newtonian gravitational potential.

* __With cosmological constant__: The theory
with a massive graviton has discontinuities at *m*^{2}
→ 0 (5 \(\mapsto\) 2 degrees of freedom) and (2/3) Λ (5 \(\mapsto\) 4 degrees of freedom).

@ __Around Minkowski__: Brout et al PRL(79),
NPB(80) [zero point energy and cosmological constant];
Modanese NPB(00) [dipole fluctuations];
de Berredo-Peixoto et al MPLA(00)gq/01 [1-loop calculation].

@ __Stability of Minkowski__: Hartle & Horowitz PRD(81);
Horowitz in(81);
Gunzig et al PLB(90);
Mazzitelli & Rodrigues PLB(90) [with *R*^{2} term];
Simon PRD(91);
Garattini IJMPA(99)gq/98 [foamy];
Modanese PLB(99)gq.

@ __Other spacetimes__: Gross et al PRD(82) [Schwarzschild, *T* ≠ 0];
Tsamis & Woodard CQG(90),
CMP(94);
Forgacs & Niedermaier ht/02,
Niedermaier JHEP(02)ht [2-Killing-vector-field reduction, renormalization];
Christiansen et al a1711 [constant-curvature backgrounds].

@ __With coupled matter__: Mackay & Toms PLB(10)-a0910 [scalar,
Vilkovisky-DeWitt effective action]; > s.a. matter phenomenology.

@ __Excitations in general__:
Chapline ht/98 [branes and conformal gravity].

@ __Infrared behavior__:
Antoniadis et al PLB(94) [scale invariance];
Tsamis & Woodard AP(95) [strong infrared effects];
Ware et al JHEP(13)-a1308 [asymptotic S matrix];
Wetterich a1802;
> s.a. 3D quantum gravity; matter phenomenology.

@ __Ultraviolet behavior__: Korepin a0905 [one-loop cancellation of UV divergences];
Christiansen et al PRD(15)-a1506 [functional renormalisation group approach];
Deser GRG(16)-a1609 [in *D* > 4 not all 1-loop divergences can be removed];
Anselmi & Piva JHEP(18)-a1803.

@ __Related topics__: Donoghue & Torma PRD(96) [loop diagrams];
Grigore CQG(00)ht/99 [and ghosts];
Bern et al PRL(00)ht/99 [strings and graviton-matter coupling];
Dilkes et al PRL(01)ht,
Duff et al PLB(01)ht
[*m*_{g} → 0 and (2/3) Λ discontinuities];
Datta et al PLB(04)hp/03 [angular momentum selection rules];
Bjerrum-Bohr et al JHEP(10)-a1005 [Kawai-Lewellen-Tye relations to gauge-theory amplitudes];
Ohta et al JHEP(16)-a1605 [off-shell one-loop divergences, and unimodular gravity];
Rafie-Zinedine a1808 [simplified Feynman rules].

> __Other__: see graviton
[production, scattering, etc]; higher-order and other modified
theories [linearized, and propagator]; tests of general relativity.

**References** > s.a. quantum gravity
/ canonical [relationship]; quantum cosmology.

@ __General__: DeWitt PR(67),
PR(67);
DeWitt in(72);
Faddeev & Popov SPU(73);
Duff in(75);
Ward a0810-conf [status and update];
Hodges a1108 [tree-level gravitational scattering amplitudes].

@ __Boundary conditions__: Avramidi & Esposito CQG(98)ht/97,
ht/97-GRF;
Esposito IJMPA(00)gq [boundary operators].

@ __Corrections to classical theory__: Iliopoulos et al NPB(98) [on spatially flat FLRW models];
Gibbons CQG(99)ht;
Khriplovich & Kirilin JETP(04)gq/04.

@ __Relationship with gauge theory__: Bern et al NPB(99)gq/98;
Bern LRR(02)gq;
Bjerrum-Bohr et al JHEP(10)-a1007 [and Yang-Mills amplitudes, tree level];
Bern et al PRD(10)-a1004 [as the "square of gauge theory"].

@ __Other relationships__: Baryshev Grav(96)gq/99 [vs geometrodynamics];
Bern ht/01-conf [and string theory];
Mattei et al NPB(06)gq/05 [and path integrals/spin-foams].

@ __Causal perturbation theory__: Grillo ht/99,
ht/99,
ht/99;
Grillo AP(01)ht/99 [and scalar matter];
Wellmann PhD(01)ht [spin-2 quantum gauge theory];
Grigore CQG(10)-a1002 [second-order, conditions on interactions with matter].

@ Related theories: Hamada PTP(00)ht/99 [2-loop renormalizable];
Bell et al gq/00-proc ["versatile"];
Nojiri & Odintsov PLB(10) [renormalizable];
Tessarotto & Cremaschini Ent(18)-a1807 [Generalized Lagrangian Path approach].

@ __Related topics__: Boulware & Deser AP(75) [and classical general relativity];
Tsamis & Woodard AP(92) [Green functions];
Modesto GRG(05)ht/03 [bosonic tensor fields].

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