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, gab = ηab + G1/2hab (hab = "gravitational potential"), and treats this as an interacting spin-2 field theory (G1/2 here really is lP, 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[gc, 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 the 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 \(\langle h, t \mid h', t' \rangle\).

Issues and Techniques > s.a. approaches [perturbative]; graviton [production, propagator, scattering]; 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/r3 corrections to the Newtonian gravitational potential.
* With cosmological constant: The theory with a massive graviton has discontinuities at m2 → 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 R2 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 PRD(18)-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 CSF(99)-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 PRD(18)-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 [mg → 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-MS [simplified Feynman rules]; Abreu et al PRL(20)-a2002 [two-loop four-graviton scattering amplitudes].
> Other: see 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]; Jakobsen a2010-MSc [classical general relativity is derived from quantum field theory].
@ 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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