quantum gravity from the qft perspective · classical gravity – newton’s law, ~f 12 = − gm1...

Quantum Gravity from the QFT Perspective

Ilya L. Shapiro

Universidade Federal de Juiz de Fora, MG, Brazil

Friedrich-Schiller-Universit at, Jena, 08/01/2015

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - Jan uary, 2015

Contents of the mini-course

• GR and its limits of applicability. Why should we quantizegravity. Semiclassical approach and higher derivatives.

• Gauge-invariant renormalization. Power counting. Quantu mGR vs Higher derivative QG. How we can (if so) live with ghosts ?More HD’s and more ghosts. Superrenormalizable QG.

• Renormalization Group (RG). For which versions of QG canRG be used? Decoupling: what remains in the IR?

• Alternative approaches: Effective approach to QG andInduced Gravity paradigm.

BibliographyI.L. Buchbinder, S.D. Odintsov, I.Sh., Effective Action in QuantumGravity. (1992 - IOPP).

I.Sh., Class. Q. Grav. 25 (2008) 103001 (Topical review); 0801.0216.


Lecture 1.

GR and singularities. Limits of applicability, Planck scal e.

Dimensional approach and Planck scale.

Quantum & gravity or Quantum Gravity (QG)?

Quantum Field Theory in curved space: hints for a complete QG .

QG: how we can do it? Linearized parametrization vscovariance. Locality.


Classical Gravity – Newton’s Law,

~F12 = − G M1 M2

r212

r12 or U(r) = − G M1 M2

r.

Newton’s law work well from laboratory up to the galaxy scale .

The “Pioneer anomaly” which put shadow to the Newton’s lawfor the last decades, has been resolved recently by elementa ry(albeit not simple!) approach.

S. Turyshev et al., .... Support for the thermal origin of the Pioneeranomaly. PRL 108 (2012); arXiv:1204.2507 [gr-qc].

For galaxies and their clusters one needs, presumably, tointroduce a Dark Matter, which consists from particles ofunknown origin. Or modify gravity in some way.

Definitely, Newton’s laws do not “ask” for being quantized.


General Relativity and Quantum Theory

General Relativity (GR) is a complete theory of classicalgravitational phenomena. It proved valid in the wide range o fenergies and distances.

The basis of the theory are the principles of equivalence andgeneral covariance.There are covariant equations for the matter (fields and part icles,fluids etc) and Einstein equations for the gravitational fiel d gµν

Gµν = Rµν − 12

Rgµν = 8πG Tµν − Λ gµν .

We have introduced Λ, cosmological constant (CC) forcompleteness.

The most important solutions of GR have specific symmetries.

1) Spherically-symmetric solution. Planets, Stars, Black holes.2) Isotropic and homogeneous metric. Universe.


I. Spherically-symmetric solution of Schwarzschild.

For a point-like mass in the origin of the spherical coordina tesystem, the solution is (Schwarzschild, 1916)

ds2 =(

1 − rg

r

)

dt2 − dr2

1 − rg/r− r2dΩ , rg = 2GM

r = 0 singularity is physical and indicates a serious problem.

An object with horizon may be formed as a consequence of thegravitational collapse, leading to the formation of physic alsingularity at r = 0.

Assuming GR is valid at all scales, we arrive at the situationwhen the r = 0 singularity becomes real.

Then, the matter has infinitely high density of energy, andcurvature invariants are also infinite. Our physical intuit iontells us that this is not a realistic situation.


II. Standard cosmological model

Another important solution of GR is the one for thehomogeneous and isotropic metric (FLRW solution).

ds2 = dt2 − a2(t) ·(

dr2

1 − kr2 + r2dΩ)

,

Radiation-dominated epoch is characterized by the EOSp = ρ/3 and T µ

µ = 0. The Friedmann equation is

a2 =8πG

3ρ0a4

0

a4 =⇒

The solutions becomes singular at t → 0.

a(t) =( 4

3· 8πGρ0a4

0

)1/4×

√t , H = a/a =

12t

The situation is qualitatively similar to the black hole sin gularity.


Applicability of GR

The singularities are significant, because they emerge in th emost important solutions, in the main areas of application o f GR.

Extrapolating backward in time we find that the use of GR leadsto a problem, while at the late Universe GR provides a consist entbasis for cosmology and astrophysics. The most naturalresolution of the problem of singularities is to assume that

• GR is not valid at all scales .

At the very short distances and/or when the curvature become svery large, the gravitational phenomena must be described b ysome other theory, more general than the GR.

But, due to success of GR, we expect that this unknown theorycoincides with GR at the large distance & weak field limit.

The most probable origin of the deviation from the GR arequantum effects.


Dimensional arguments.

The expected scale of the quantum gravity effects is associa tedto the Planck units of length, time and mass. The idea of Planckunits is based on the existence of the 3 fundamental constant s:

c = 3 · 1010 cm/s ,

~ = 1.054 · 10−27 erg · sec ;

G = 6.67 · 10−8 cm3/sec2 g .

One can use them uniquely to construct the dimensions of

length lP = G1/2~

1/2 c−3/2 ≈ 1.4 · 10−33 cm;

time tP = G1/2~

1/2 c−5/2 ≈ 0.7 · 10−43 sec;

mass MP = G−1/2~

1/2 c1/2 ≈ 0.2 · 10−5g ≈ 1019 GeV .


Three choice for Quantum Gravity (QG)

One may suppose that the fundamental units indicate to thepresence of fundamental physics at the Planck scale.

We can classify the possible approaches to three distinctgroups, namely we can:

• Quantize both gravity and matter fields. This is, definitely,the most fundamental possible approach.

• Quantize only matter fields on classical curved background(semiclassical approach).

• Quantize something else. E.g., in case of (super)stringtheory both matter and gravity are induced.

Which approach is “better”?


QFT and Curved space-time are well-established notions,which passed many experimental/observational tests.

Therefore, our first step should be to consider QFT of matterfields on classical curved background.

Different from quantum theory of gravity, QFT of matter field s incurved space is renormalizable and free of conceptual probl ems.

Moreover, understanding renormalization of semiclassica ltheory can be very helpful in order to impose some constrainsfor a complete QG.

Metric enters the generating functional of the Green functi ons asan external parameter,

Z (J , gµν) =

∫

dΦ expiS(Φ, gµν) + iΦJ .

We arrive at the Effective Action of the mean field(s), Γ[Φ, gµν ].Ilya Shapiro, Lectures on QG from QFT perspective. Jena - Jan uary, 2015

The main specific feature of QFT in curved space is that theEffective Action depends on the background metric, Γ[Φ, gµν ]

In terms of Feynman diagrams, one has to consider graphs withinternal lines of matter fields & external lines of both matter andmetric. In practice, one can consider gµν = ηµν + hµν .

→ + +

+ + + + ...


An important observation is thatall those “new” diagrams with hµν legs have superficial degreeof divergence equal or lower that the “old” flat-space diagra ms.

Consider the case of scalar field which shows why thenonminimal term ξRϕ2 is necessary for a scalar field:

→ + +

+ + + ... .


It is easy to see that the theory can be renormalizable only if weinclude higher derivative terms in the vacuum sector.

Renormalization involves fields, couplings, masses,non-minimal parameter ξ and vacuum action parameters.

Introduction: I.L. Buchbinder, S.D. Odintsov & I.Sh. (IOPP, 1992).

Relevant diagrams for the vacuum sector

+ + + + ... .

All possible covariant counterterms have the same structur e as

Svac = SEH + SHD


The renormalizable QFT in curved space requires introducin g ageneralized form of the gravity (external field), “vacuum ac tion”.

Svac = SEH + SHD

where SEH = − 116πG

∫

d4x√−g R + 2Λ .

is the Einstein-Hilbert action with the cosmological const ant.

SHD includes higher derivative terms. The most useful form is

SHD =

∫

d4x√−g

a1C2 + a2E + a3R + a4R2 ,

where C2(4) = R2µναβ − 2R2

αβ + 1/3 R2

is the square of the Weyl tensor in d = 4,

E = RµναβRµναβ − 4 RαβRαβ + R2

is integrand of the Gauss-Bonnet term (topological term in d=4 ).Ilya Shapiro, Lectures on QG from QFT perspective. Jena - Jan uary, 2015

Observation about higher derivatives.

The consistent theory can be achieved only if we include

SHD =

∫

d4x√−g

a1C2 + a2E + a3R + a4R2 ,

C2(4) = R2µναβ − 2R2

αβ + 1/3 R2 is the square of the Weyl tensor.

In quantum gravity such a HD term means massive ghost, thegravitational spin-two particle with negative kinetic ene rgy. Thisleads to the problem with unitarity, at least at the tree leve l.

In the semiclassical theory gravity is external and unitari ty of thegravitational S-matrix does not matter.

The consistency criterium includes: physically reasonabl esolutions and their stability under small perturbations.


The standard “old” approach to perturbative QG is based on th eassumptions:

• Quantization can be performed using the variables

gµν = ηµν + hµν . (1)

It is assumed that this choice of the quantum metric is the“right” one. This choice enables one to deal with thewell-defined object such as S - matrix.

• In case of quantum GR (1) corresponds to so-calledGaussian expansion of the action. The last means we can take

gµν = ηµν + κhµν . (2)

and then, for

SEH = − 1κ2

∫

d4x√−g R + 2Λ , κ2 = 16πG (3)

and the second order (Gaussian approximation) of the actionbecomes κ- independent.


• Finally, the expansion around the flat background means wehave Newton constant as a coupling.

In the recent (20 years) literature one can meet a lot of worksabout “non-Gaussian fixed point” in QG. Such a fixed pointwas originally conjectured by S. Weinberg as a hypothesis of“asymptotic safety”, which is an alternative to renormaliz ability.

• If being someday confirmed, such a discovery would meanthat we do not need to construct the quantum theory around flatspace. Instead, this would mean that the theory should beconstructed with some other background in mind.

• But then, what is the sense of the S - matrix approach,becomes unclear.


Further arguments:

• One should definitely quantize both matter and gravity, forotherwise the QG theory would not be complete.

• The diagrams with matter internal lines in a complete QG arebe exactly the same as in a semiclassical theory.

• This means one can not quantize metric without higherderivative terms in a consistent manner, since these terms areproduced already in the semiclassical theory.

• Indeed, most of the achievements in curved-space QFT arerelated to the renormalization of higher derivative vacuum terms,including Hawking radiation, Starobinsky inflation and oth ers.


Once again: what is bad in the higher-derivative gravity?

For the linearized gravity

gµν = ηµν + hµν , (1)

we meet

Gspin−2(k) ∼ 1m2

(

1k2 − 1

m2 + k2

)

, m ∝ MP .

This means that the tree-level spectrum includes masslessgraviton and massive spin-2 “ghost” with negative kineticenergy and huge mass.

• Interaction between ghost and gravitons may violate energyconservation in the massless sector (M.J.G. Veltman, 1963).

• In classical systems higher derivatives generate explodin ginstabilities at the non-linear level (M.V. Ostrogradsky, 1850).

• Without ghost one violates unitarity of the S -matrix.Ilya Shapiro, Lectures on QG from QFT perspective. Jena - Jan uary, 2015

Let us classify all assumptions we made to condemn higherderivative theory:

• One can draw conclusions about the theory of gravity usinglinearized gravity approximation. Furthermore, the S -matrix ofgravitons should be the central object of our interest.

•• Massless and massive (ghost) modes can be separated toperform the analysis of the particle content and stability.

••• Ostrogradsky instabilities or Veltman scattering arerelevant independent on the energy scale, in all cases theyproduce run-away solutions and the Universe “explodes”.

The standard conclusion is that one can treat higher derivat iveterms only as small corrections, in a perturbative framewor k.

The last assumption ••• is especially risky and should beverified. The safe criterion is the stability of low-energysolutions when the higher derivatives are present.


There is a simple way to check whether all these assumptionsare correctly working or not.

Let us take some higher derivative theory and verify the stab ilitywith respect to the linear perturbations on some, physicall yinteresting, dynamical background.

If all three assumptions are correct, we will observe rapidl ygrowing modes even for the low-energy (slowly changing, orlow-curvature) background.

If all but the last (energy-scale independence) assumption s areright, we are going to observe no growing modes until thetypical value of curvature becomes of the order of M2

P .


For the sake of generality, let us consider not only classica l HDterms, but also take into account the semiclassical correct ions,derived via conformal anomaly.

• Consider the anomaly-induced effective action of gravity.

• Derive gravitational waves in the quantum-corrected theor y.

• Verify the stability of classical solutions.

Program realized in:

J. Fabris, A. Pelinson, F. Salles, I.Sh., JCAP 02 (2012);F. Salles and I.Sh., Phys. Rev. D89 (2014), arXiv:1401.4583.


Semiclassical theory of classically conformal free fields.

The vacuum action in the general case has the form

Svac = SEH + SHD ,

SHD =

∫

d4x√−g

αC2 + βE + γR + δR2

,

whereE = R2

µναβ − 4R2αβ + R2 .

In the case of conformal theory, at the one-loop level, it issufficient to consider

Sconf . vac =

∫

d4x√−g

a1C2 + a2E + a3R

.


Conformal anomaly

Consider the theory which includes gµν and matter fields Φ.kΦ is the conformal weight of the field.

The Noether identity for the local conformal symmetry is[

− 2 gµνδ

δgµν+ kΦΦ

δ

δΦ

]

S(gµν , Φ) = 0 .

It means T µµ = 0 on shell

− 2√g

gµνδSvac(gµν)

δgµν= T µ

µ = 0 .

At quantum level Svac(gµν) is replaced by effective action (EA)

Γvac(gµν) = Svac(gµν) + Γvac(gµν) .


The theory of matter includes N0 conformal real scalars ,N1/2 Dirac spinors and N1 vectors .

The expression for divergences

Γdiv =1ε

∫

d4x√

g

β1C2 + β2E + β3R

.

where

β1

− β2

β3

=1

360(4π)2

3N0 + 18N1/2 + 36N1

N0 + 11N1/2 + 62N1

2N0 + 12N1/2 − 36N1

The anomalous trace is

〈T µµ 〉 = −

β1C2 + β2E + a′R

.

Useful conformal variables:

gµν = gµν · e2σ , ∆ = 2 + 2Rµν∇µ∇ν − 2

3R+

13(∇µR)∇µ .

√−g(E − 23R) =

√

−g(

E − 23R + 4∆4σ

)

.


Anomaly-induced action of vacuum

One can use 〈T µµ 〉 as an equation for the finite 1-loop EA

− 2√−ggµν

δ Γind

δgµν= 〈T µ

µ 〉 = − 1(4π)2

(

ωC2 + bE + cR)

. (∗)

Solution is straightforward Riegert, Fradkin & Tseytlin, PLB-1984.

The solution for the effective action is

Γind = Sc[gµν ] +1

(4π)2

∫

d4x√

−g ωσC2 (1)

+ bσ(

E − 23R)

+ 2bσ∆4σ − 112

(c +23

b)[R − 6(∇σ)2 − (σ)]2) ,

where Sc [gµν ] = Sc [gµν ] is the integration constant of (*). .The disadvantage is that this action is not covariant since i t isnot expressed in terms of original metric. gµν .


Introduce∫

x

=

∫

d4x√−g , ∆4 G(x , y) = δ(x , y).

An equivalent covariant solution has the form

Γind = Sc[gµν ] −3c + 2b36(4π)2

∫

d4x√

−g(x)R2(x)

+

∫∫

x y

[ω

4C2 +

b8

(

E − 23R)

]

xG(x , y)

(

E − 23R)

y .

One can rewrite this expression using auxiliary scalars,

Γind = Sc[gµν ] −3c + 2b36(4π)2

∫

x

R2(x) +

∫

x

12ϕ∆4ϕ− 1

2ψ∆4ψ

+a

8π√−b

ψC2 + ϕ

[√−b

8π

(

E − 23R)

− a

8π√−b

C2]

.

I.Sh. and A. Jacksenaev, Ph.L.B (1994).Ilya Shapiro, Lectures on QG from QFT perspective. Jena - Jan uary, 2015

An important application is the anomaly-induced inflation, orModified Starobinsky ModelFabris, Pelinson, Solà, I.Sh. et al. (1998-2003, 2011-2012).

Consider a Cosmological Model based on the action

Stotal = − M2P

16π

∫

d4x√−g (R + 2Λ) + Smatter + Svac + Γind .

Equation of motion in physical time dt = a(η)dη

¨aa+

3a ˙aa2 +

a2

a2 −(

5 +4bc

)

aa2

a3 − 2k(

1 +2bc

)

aa3

− M2P

8πc

(

aa+

a2

a2 +ka2 − 2Λ

3

)

= 0 ,

where k = 0,±1, Λ is the cosmological constant.


Particular solutions Starobinsky, Ph.L.B., 1980

a(t) =

ao exp(Ht) , k = 0ao cosh(Ht) , k = 1ao sinh(Ht) , k = −1

,

Hubble parameter H =MP√−32πb

(

1 ±√

1 +64πb

3Λ

M2P

)1/2

.

Perturbations of the conformal factor:

σ(t) → σ(t) + y(t).

The criterion for a stable inflation

c > 0 ⇐⇒ N1 <13

N1/2 +118

N0 ,

in agreement to Starobinsky (1980).

Remark. The value of Λ and the choice of k = 0, ±1 do notinfluence the stability condition.


Unstable case:

N1 >13

N1/2 +118

N0 .

One can chose initial conditions such that the Universe perf ormssufficient inflation and then gracefully exits to the FRW stag e.

The Starobinsky model is based on the unstable case

M.V. Fischetti, J.B. Hartle & B.L. Hu, PRD20 (1979);• A.A. Starobinski, Ph.L.B 91 (1980) 99;V.F. Mukhanov & G.V. Chibisov, JETP Lett. 33 (1981); JETP (1982);A.A.Starobinski, Let.Astr.Journ. 9 (1983);A. Vilenkin, PRD 32 (1985);P. Anderson, PRD 28 (1983).

Stable case: The Universe starts inflation from an arbitraryposition at the phase plane, anomaly-induced inflation “kil ls”any matter content in a dozen of Planck times.

Pelinson, Takakura, I.Sh. NPB (PS) - 2003.


Simple test of the Modified Starobinsky Model.

Pelinson, I.Sh. & Takakura IRGA-NPB(PS)- 2003.,

Consider late Universe, k = 0, H0 =√

Λ/3.

Only photon is active, N0 = 0 , N1/2 = 0 , N1 = 1 .

Graviton typical energy is H0 ≈ 10−42 GeV , =⇒ all massiveparticles (even neutrino) mν ≥ 10−12 GeV decouple fromgravity. c < 0 =⇒ today inflation is unstable.

Stability for the small H = H0 case: H → H0 + const · eλt

λ3 + 7H0λ2 +

[

(3c − b)4H02

c− M2

P

8πc

]

λ − 32πbH03 + M2

PH0

2πc= 0 .

The solutions are λ1 = −4H0 , λ2/3 = −32

H0 ± MP√

8π|c|i .

Λ > 0 efficiently protects low-energy solution bfromhigher-derivative instabilities in this case.


Consider unstable case, matter (or radiation) dominatedUniverse, close to the classical FRW solution. Then

¨aa+

3a ˙aa2 +

a2

a2 −(

5 +4bc

)

aa2

a3 − 2k(

1 +2bc

)

aa3

− M2P

8πc

(

aa+

a2

a2 +ka2 − 2Λ

3

)

= − 13c

ρmatter ,

The terms of the first line, of quantum origin, behave like 1/t4.

The second line terms, of classical origin, behave like 1/t2.

After certain time the “quantum” terms become negligible.

Conclusion:Classical solutions are very good approximate solutions of thetheory with quantum corrections and/or higher derivatives ,as far as we deal only with the conformal factor dynamics.


Stability & Gravitational Waves

As far as classical action and quantum, anomaly-induced ter m,both have higher derivatives, an important question is whet herthe stability of classical solutions in cosmology holds or n ot.

Consider small perturbation

gµν = g0µν + hµν , hµν = δ gµν ,

where g0µν = 1, −δij a2(t), µ = 0, 1, 2, 3 and i = 1, 2, 3.

hµν(t ,~r) =

∫

d3k(2π)3 ei~r ·~k hµν(t , ~k) .

Using the conditions ∂i hij = 0 and hii = 0 , together withthe synchronous coordinate condition hµ0 = 0, we arrive at theequation for the tensor mode

Fabris, Pelinson and I.Sh., NPB (2001);Fabris, Pelinson, Salles and I.Sh., JCAP (2011).


(

2f1 +f22

) ....

h +[

3H(

4f1 + f2)

+ 4f1 + f2] ...

h +[

3H2(

6f1 +f22− 4f3

)

+H(

16f1+92

f2)

+6H(

f1−f3)

+2f1+12

(

f2+f0+f4ϕ)

+32

f4Hϕ− 23

f5ϕ2]

h

−(

4f1 + f2) ∇2h

a2 +[

H(

4f1 − 6f3)

− 21HH(1

2f2 + 2f3

)

− H(3

2f2 + 6f3

)

+3H2(

4f1+12

f2−4f3)

−9H3(f2+4f3)

+H(

4f1+32

f2)

+32

f4ϕ(

3H2+H)

+H(

3f4ϕ+32

f0−2f5ϕ2)+12

f4...ϕ −4

3f5ϕϕ

]

h−[

H(

4f1+f2)

+4f1+f2] ∇2h

a2

+[

5f4H...ϕ +f4

....ϕ −

(

36HH2 + 18H2 + 24HH + 4...

H)

(

f1 + f2 + 3f3)

−HH(

32f1 + 36f2 + 120f3)

− 8H(

f1 + f2 + 3f3)

−H2(

4f1 + 6f2 + 24f3)

−4H(

f1+f2+3f3)

−9f4ϕ(

H3+HH)

+f4ϕ(

3H2+5H)

−H3(

8f1+12f2+48f3)


+f5ϕ2(1

2H2 +

13

H)

+23

f5Hϕϕ− 16

f5ϕ2 +13

f5ϕ...ϕ]

h + f0[

2H + 3H2]

h

+[

H(

2f1 +12

f2)

+2H(

f1 + f2 +3f3)

−12

(

f2 + f4ϕ+ f0 + 3f4Hϕ)

− 13

f5ϕ2] ∇2h

a2 +[

2f1 +12

f2] ∇4h

a4 = 0 ,

where the f - terms are defined as

f0 = −MP2

16π; f1 = a1 + a2 −

b + ω

2√−b

ϕ+ω

2√−b

ψ ;

f2 = −2a1 − 4a2 +ω + 2b√

−bϕ− ω√

−bψ;

f3 =a1

3+ a2 −

3c + 2b36

− 3b + ω

6√−b

ϕ+ω

6√−b

ψ;

f4 = −4π√−b

3; f5 =

12.


Qualitative results were achieved by using both

I. Analytical methods. We can approximately treat allcoefficients as constants. There is a mathematically consis tentway to check when (and whether) it works. And, with theWolfram’s Mathematica software, manipulating our equatio n isnot too difficult, indeed.

II. Numerical methods. Modern methods to study the CMBspectrum (CMBEasy software) can be pretty well applied togravitational waves and give the result which is perfectlyconsistent with the output of method II.

Net Result: The stability does not actually depend on quantumcorrections. It is completely defined by the sign of the class icalcoefficient a1 of the Weyl-squared term. The sign of this termdefines whether graviton or ghost has positive kinetic energ y!


We can distinguish the following two cases:

• The coefficient of the Weyl-squared term is a1 < 0 Then


(

1k2 − 1

m2 + k2

)

, m ∝ MP ,

there are no growing modes up to the Planck scale, ~k2 ≈ M2P .

For the dS background this is in a perfect agreement withStarobinsky, Let.Astr.Journ. (in Russian) (1983);Hawking, Hertog and Real, PRD (2001).

• The classical coefficient a1 > 0 , also G < 0 .


(

1m2 + k2 − 1

k2

)

, m ∝ MP .

and there are rapidly growing modes at any scale.


Last discussion concerning ghosts

In fact, the result is natural. The anomaly-induced quantumcorrection is O(R3

....) and O(R4....), Until the energy is not of the

Planck order of magnitude, these corrections can not compet ewith classical O(R2

....) - terms.

For a1 < 0 there are no growing tensor modes in the higherderivative gravity on cosmological backgrounds.

Massive ghosts are present only in the vacuum state. We just d onot observe them “alive” until the typical energy scale is be lowthe Planck mass.

• All in all, massive ghosts do not pose real danger below thePlanck scale. Above MP we need new ideas to fight ghosts.


quantum gravity from the qft perspective · classical gravity – newton’s law, ~f 12 = − gm1...

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