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 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].

> __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 topics__: Boulware & Deser AP(75)
[and classical general relativity];
Tsamis & Woodard AP(92) [Green functions];
Hamada PTP(00)ht/99 [2-loop renormalizable];
Bell et al gq/00-proc ["versatile"];
Modesto GRG(05)ht/03 [bosonic tensor fields];
Nojiri & Odintsov PLB(10) [renormalizable].

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