Renormalization of Quantum Gravity  

Perturbative Non-Renormalizability > s.a. [renormalization]; gravitation; quantum gravity.
* The issue: The theory appears non-renormalizable (around gab = 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.
@ References: Deser RMP(57); Deser & van Nieuwenhuizen PRD(74) [Einstein-Maxwell], PRD(74) [Einstein-Dirac], et al PRD(74) [Einstein-Yang-Mills]; '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]; > s.a. minisuperspace [and wormholes].

Non-Perturbative Renormalizability > 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

mc2 = m0c2 + e2/ – Gm2/ ,

i.e., m = 2e G–1/2 for → 0, independent of m0 (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, a0805 [from structure of Dyson-Schwinger equations]; Percacci a0910-ln.
@ UV fixed point: Souma PTP(99)ht, gq/00; Lauscher & Reuter PRD(02)ht/01, CQG(02)ht/01, IJMPA(02)ht/01; Percacci & Perini PRD(03)ht [with scalar field]; Litim PRL(04)ht/03 [Euclidean, arbitrary dimension]; Percacci PRD(06)ht/05; Niedermaier & Reuter LRR(06) [asymptotic safety]; Emoto gq/06-in; Niedermaier CQG(07)gq/06 [rev]; Percacci a0709-in [asymptotic safety]; Reuter & Weyer PRD(09)-a0804 [and diffeomorphism invariance]; Ward MPLA(08)-a0808 [predictions for G and , resummation techniques]; Litim PoS-a0810 [rev]; Codello et al AP(09) [Wilsonian renormalization group equation]; Reuter & Weyer GRG(09)-a0903-in [role of background independence]; Niedermaier PRL(09) [from perturbation theory]; > s.a. covariant quantum gravity.
@ UV fixed point, other theories: Fischer & Litim PLB(06)ht, Litim ht/06-in [D > 4]; Codello & Percacci PRL(06)ht, Codello et al IJMPA(07)-a0705 [higher-order, f(R) gravity]; Reuter & Weyer PRD(09)-a0801 [conformally reduced gravity]; Benedetti et al MPLA(09)-a0901 [higher-derivative]; > s.a. Horava Gravity [candidate UV completion].
@ UV fixed point, phenomenology: Hewett & Rizzo JHEP(07)-a0707, Litim & Plehn PRL(08)-a0707 [collider signals].

Topics > s.a. cosmological constant [running]; effective theories; general-relativistic cosmology [with varying G and ].
@ Running of G: Dou & Percacci CQG(98)ht/97; Percacci & Perini CQG(04)ht [fixed point]; Hamber & Williams PRD(05)ht [vacuum polarization, effective field equations], PRD(07)ht/06 [static isotropic]; López Nacir & Mazzitelli PRD(07)ht/06 [and non-integer powers of ]; > s.a. Newton's constant.
@ Scale dependence of G: Reuter ht/00; Reuter & Weyer JCAP(04), EAS(06)ap/05-in [astrophysical distances].
@ Effective G and cosmological constant: Parker gq/98 [and cosmology]; Reuter & Weyer PRD(04)gq/03 [improved action]; Shapiro et al JCAP(05).
@ Various theories: Granda EPL(98)ht/05 [general relativity + N scalars, G and ], NCB(99)ht/05 [general relativity + N-spinor]; Haba ht/02 [Brans-Dicke theory]; Nikolic a0708 [in linearized gravity]; Freidel et al PRD(09)-a0905 [3D Boulatov model, group-field-theory approach]; Narain & Percacci a0911, Narain & Rahmede a0911 [scalar-tensor theories]; > s.a. modified theories [including higher-order]; Topologically Massive Gravity.
@ Discrete: Requardt gq/02; Oeckl NPB(03)gq/02 [without background]; > s.a. discrete spacetime, lattice gravity, regge calculus, spin-foam models.
@ Other topics: Smolin NPB(79) [and asymptotically scale-invariant geometry]; Mazzitelli PRD(92) [and higher-order terms in the action]; 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 a0704-PhD [and effective actions]; Narain & Percacci a0910 [beta function, scheme dependence].

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