Renormalization of Quantum Gravity |

**In General**
> s.a. renormalization group [including holographic renormalization].

@ __General references__: Christiansen et al PRD(16)-a1403 [global phase diagram];
Falls a1702 [and asymptotic safety];
de Alwis a1707 [exact RG flow];
Doboszewski & Linnemann a1712 [how to formulate];
Dhumuntarao a1807
[renormalization group flow and stress-energy tensor as source for heat equation];
Wiesendanger a1905 [renormalizability].

@ __Ultraviolet behavior__: Christiansen et al PRD(15)-a1506 [functional renormalisation group approach];
> s.a. asymptotic safety.

@ __Effective G and cosmological constant__: Parker gq/98/PRL [and cosmology];
Reuter & Weyer PRD(04)gq/03 [improved action];
Shapiro et al JCAP(05).

@

@

>

**Perturbative Renormalization** > s.a. renormalization
/ gravitation; quantum gravity.

* __The non-renormalizability issue__: The theory
appears non-renormalizable (around *g*_{ab}
= *η*_{ab}) from power counting; This would
not imply non-existence of quantum gravity, but by itself if quantum gravity was an ordinary theory
it would be close to a kiss of death.

* __Argument for non-renormalizability__:
The coupling constant *G* has dimensions of squared length.

* __Proposed approaches__:
Study methods for handling non-renormalizable theories; Choose appropriate matter
terms (> see supergravity); Euclideanize,
study instantons and sum over manifolds; Include higher-derivative terms.

@ __General references__:
Deser RMP(57);
't Hooft & Veltman AIHP(74);
Deser in(75);
Deser et al in(75);
**·**Weinberg
in(79);
Martellini PRL(83) [with cosmological constant];
Goroff & Sagnotti NPB(86) [non-renormalizable ultraviolet divergences];
Crane & Smolin NPB(86) [virtual black holes and fermions];
Deser et al PLB(89);
van de Ven NPB(92) [non-renormalizable infinity in the two-loop effective action];
Shomer a0709 [pedagogical];
Marin a1002 [reformulation of the theory];
Morris JHEP-a1802 [conformal sector];
Morris a1805-GRF,
Das et al IJMPD(18)-a1805-GRF [perturbatively renormalizable theory];
Lavrov & Shapiro a1902 [gauge-invariant];
> s.a. minisuperspace [and wormholes].

@ __With matter__: Deser & van Nieuwenhuizen PRD(74) [Einstein-Maxwell theory],
PRD(74) [Einstein-Dirac theory],
et al PRD(74) [Einstein-Yang-Mills theory].

**Non-Perturbative Renormalization** > s.a. 3D quantum gravity;
approaches to quantum gravity; canonical
and covariant quantum gravity.

* __Idea__: The usual non-renormalizability
arguments are not reliable for a theory without a background metric; One needs to use
different ones, and some arguments, both classical and quantum, point to the possibility
that the theory, despite being non-renormalizable with the usual perturbative methods,
is non-perturbatively renormalizable (although Witten claimed it is not, because the
4D action is cubic),

- __Classical analog__: The total mass
of a charged particle in general relativity should satisfy

*mc*^{2}
= *m*_{0}*c*^{2}
+ *e*^{2}/*ε* −
*Gm*^{2}/*ε* ,

i.e., *m* = 2*e* *G*^{−1/2}
for *ε* → 0, independent of *m*_{0}
(nice for particles from geometry); This comes from the diffeomorphism invariance of
the theory and the fact that therefore energy is expressed as a surface integral
at infinity.

- __Examples from other theories__:
The 3D Gross-Neveu model; Gravity is exactly soluble in 3D, where can be written as
a theory with quadratic action, where one perturbs around the zero triad
[@ Witten NPB(88)].

@ __General references__:
Arnowitt et al PRL(60);
in Ashtekar; Weinberg in(79);
Smolin NPB(82);
Crane & Smolin NPB(86);
Kawai & Ninomiya NPB(90);
Abe & Nakanishi MPLA(95);
Maeda & Sakamoto PRD(96)ht [strong-coupling expansion];
Hamada PTP(02)ht [higher-order renormalization];
Kreimer AP(08)-a0705,
in(09)-a0805 [from structure of Dyson-Schwinger equations];
Percacci PoS-a0910;
Hamber a1002-MG12;
Knorr a1710 [new approximation scheme].

**Other Theories of Gravity **
> s.a. 2D quantum gravity; higher-order
gravity; modified theories [hořava gravity].

@ __General references__:
Haba ht/02 [Brans-Dicke theory];
Nikolić a0708 [in linearized gravity];
Freidel et al PRD(09)-a0905 [3D Boulatov model, group-field-theory approach];
Narain & Percacci CQG(10)-a0911,
Narain & Rahmede CQG(10)-a0911 [scalar-tensor theories];
Pagani & Percacci CQG(15)-a1506 [with torsion and non-metricity];
> s.a. Topologically Massive Gravity.

@ __Gravity and coupled matter__: Granda EPL(98)ht/05 [general relativity + *N* scalars, *G* and Λ],
NCB(99)ht/05 [general relativity + *N*-spinor];
Ibiapina Bevilaqua et al CQG(16)-a1506
[Einstein gravity coupled to scalar electrodynamics, effective field theory].

@ __Discrete models__:
Requardt gq/02;
Oeckl NPB(03)gq/02 [without background];
> s.a. dynamical triangulations; lattice gravity;
regge calculus; spin-foam models.

@ __Functional renormalization group__: Benedetti & Caravelli JHEP(12)-a1204 [and local potential approximation];
Donà & Percacci PRD(13)-a1209 [with fermions and tetrads];
Nagy et al PRD(13)-a1307 [critical exponents];
Reuter & Schollmeyer AP(16)-a1509 [on a theory space];
Alkofer a1809-MS [with one extra compact dimension];
Reuter & Saueressig 19.

@ __Related topics__: Reuter & Saueressig PRD(02)ht/01,
PRD(02)ht [non-local truncation of general relativity];
Dienes & Mafi PRL(02) [compactification geometry];
Bonanno & Reuter JHEP(05)ht/04 [comparison between renormalization group equations];
Anselmi JHEP(07)ht/06 [semiclassical];
Canfora PRD(06)ht [UV behavior at large *N*];
Neugebohrn PhD(07)-a0704 [and effective actions];
Narain & Percacci APPB-a0910-proc [beta function, scheme dependence];
Donkin & Pawlowski a1203 [phase diagram of quantum gravity from diffeomorphism-invariant RG-flows].

> __Related topics__: see cosmological constant [running];
general-relativistic cosmology [with varying *G* and Λ];
quantum gravity phenomenology.

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