Renormalizable 4D Quantum Gravity and its Cosmological...
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UT-Komaba 12/2, 2009
Renormalizable 4D Quantum Gravity and
its Cosmological ImplicationsK. Hamada
http://research.kek.jp/people/hamada/
References1. Conformal Field Theory on R x S^3 from Quantized Gravity, Int. J.
Mod. Phys. A24 (2009) 3073—3110, arXiv:0811.1647[hep-th].2. Renormalizable 4D Quantum Gravity as a Perturbed Theory from CFT,
arXiv:0907.3969[hep-th].3. From CFT Spectra to CMB Multipoles in Quantum Gravity Cosmology,
arXiv:0908.0192[astro-ph] with S. Horata and T. Yukawa.
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Introduction
The goal of quantum gravity is to understandbeyond the Planck scale phenomena
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Historical Background
Problems of quantizing Einstein gravity
• Coupling constant has dimension• Space-time singularity exists• Action is not bounded from below
Not renormalizable
In order to resolve these problems, four-derivative actions are introduced
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Four-derivative quantum gravity
• Coupling constant becomes dimensionless power-counting renormalizable
• Action becomes bounded from below, but ghost mode appears
ghost mode
The models are classified by ideas how to tackle unitarity problem
• Lee-Wick-Tomboulis approach• Horava approach• CFT approach
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• Lee-Wick-Tomboulis approach (1970s) :
Consider resummed propagatorfor asymptotically free field theories ( )
real pole of ghost mode disappears(This idea is still effective at IR, but not so at UV)
• Horava approach :
Give up Lorentz sym. make ghosts non-dynamical
• CFT approach (our model) :
Use non-perturbative methods (CFT in UV limit)Conformal sym. make ghosts not gauge invariant
and remove space-time singularity
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Renomalizable 4D Quantum Gravity as a Perturbed Theory from CFT
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Renormalizable 4D Quantum Gravity
The Action (Weyl + Euler + Einstein) weight
conformally invariant (no R^2) Planck constant
“t” is a unique dimensionless gravitational coupling constant indicating asymptotic freedom
conformally flatAt high energies(Perturbation is defined
about this config.)“b” is not independent coupling, which is expanded by t
At the Einstein action dominates
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Wess-Zumino Integrability Condition
Conformal variation of effective action (=path integral over conf. mode)
conformal anomaly( UV divergence)WZ integrability condition
(Last three termsare trivial)require Weyl and Euler combinations
Determine the typeof UV divergences(= bare action)
R^2 is forbidden no R^2 divergences(renormalizability is non-trivial !)
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What Means No R^2 ActionIn four-derivative models, R^2 action is commonlyintroduced as a kinetic term of the conformal mode
Therefore, no R^2 action means thatthere is no coupling constant for this mode
Non-perturbative treatment of the conformal mode is required
• Kinetic term of the conformal mode is inducedfrom the measure quantum mechanically
• Renormalization factor of this mode is unity: and so on
I will show these properties explicitly using dimensional regularization
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The Induced Action
Jacobian to preserve diff. inv.= WZ action for conformal anomaly
Practical measure definedon the background
Lowest term of S (=Riegert action) is coupling-independent
cf. Liouville action
Dynamics of conformal mode is induced from the measure
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Relationship between 2 and 4 dim.4DQG2DQG
modified
Euler density
relation
Conf. inv. diff. op.
WZ action
Liouville action Riegert action
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Conformal Symmetry as Diffeomorphism Invariance
Background-metric independence
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Diffeomorphism Invariance: gauge parameter
Mode decomposition
coupling const.
traceless
no coupling const.
Conformal mode and traceless tensor mode are completely decoupled
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Gauge Symmetry at t = 0 (1)
Introduce the gauge parameterand take the limit with leaving finite
Usual gauge symmetry of the Weyl action
gauge-fixed as usual
cf. This is similar to gauge symmetry of vector field
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Gauge Symmetry at t = 0 (2)Take the gauge parameter to be a conformal Killing vector:
Lowest term in the traceless-mode transformation vanishes!
Conformal symmetry (on )
15 generators of conformal algebra
generated from Riegert and Weyl actions quantum mechanically
fixed by physical state conditions (=WdW eq.)
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For example, conformally coupled scalar field action satisfies
on flat backgroundby conformal Killing vectors
In the same way, the kinetic terms of the vector-field action, the Weyl action are invariant under the conformaltransformations, respectively.
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The Perturbation about CFT
CFT + perturbations (by single “t”)
This model :
cf. Early 4-derivative models in 1970’s
Free + perturbations (by two couplings)
Non-perturbative (conformal mode is treated exactly)
perturbative (all modes are treated in perturbation)
Conformal symmetry mixespositive- and negative-metric modes
light on unitarity in strong gravity
Gauge symmetry in UV limit doesnot mix gravitational modes at all
ghosts appear
graviton picture
Riegert + Weyl
R^2 + Weyl
See ref.1 in detail
[Remark: Lee and Wick’s idea corresponds to mixing all modes by interactions ]
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Dimensional Regularization and Renormalization
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On Dimensional RegularizationDimensional regularization
• manifestly diffeomorphism invariant• can compute higher-loop corrections
cf. DeWitt-Schwinger method (exactly 4 dim. method)
one-loop order
conformal anomaly
heat kernel
guarantee
conformal anomaly comes fromDOF between D and 4 dimensions
and
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Renormalizable action Euclidean sign.
coupled with QEDBare action D-dimensional WZ integrability
Renormalization factors
( )
conformal mode is not renormalizedbecause this mode has no its own coupling constant
Ward-Takahashi identity
Ambiguity is fixed !
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Conformal Anomaly (WZ action)Residues are functionsof renormalized coupling
beta functionBare action vertices and counterterms
ordinary countertermsfor gauge field
new vertices and new counterterms
Bare Weyl action Wess-Zumino actionfor conformal anomaly
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Laurent expansion of b
Euler term
counterterms
new WZ actions andnew counterterms
Dynamics of conformalmode is induced
Positive constant
Kinetic term (Riegert action) is induced
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Beta functions
no t^2 correctionResidues b_n
Hathrell, Ann.Phys.142(1982)34
( corrections from diagrams with internal gravitational lines)
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Non-renormalization of Conformal Mode( )
+ = UV finite
z: infinitesimal fictitious mass (IR regularization)Not gauge invariant cancel out !
propagator
[Remark : Einstein action cannot be considered as the mass termdue to the existence of exponential factor of conformal mode]
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Vertex function ( ) of e^6Two-point function of e^4
These are renormalized bythe condition
Two-point function of e^6was also checked.
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AF and Running Coupling Constant
whereAsymptotic Freedom
New dynamical scale:Physical momentum: with
: momentum defined on the flat background
(# Conf. anomaly is necessary to preserve diff. inv.)
In UV limit, CFT about , and thus no singularityIn IR limit, conformal symmetry is completely broken at
turn to Einstein gravity
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Summary of Conformal AnomalyConformal anomalies are divided into two groups:
1. Coupling-independent part (= Riegert action)guarantee conf. sym., or CFT at t = 0, against the name of ‘anomaly’quantum diff. inv. (= background-metric indep.)
2. Coupling-dependent part (= ordinary conf. anomaly)beta functionviolate conf. sym., as the name
The coupling ‘t’ measures a degree of deviation from CFT
In any case, these conformal anomalies are necessaryto preserve quantum diffeomorphism invariance
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Reproduce Hathrell’s Result of Conf. AnomalyHathrell, Ann.Phys.142(1982)34;
Ann.Phys.139(1982)136Counterterms used by Hathrell
He carried out 3-loop computations of conformal anomaly in curved space for various matter fields
and found the following relationship between the residues b and c:
(universal independent of matter contents)
D dim. Gravitational action
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Quantum Gravity Cosmology
From CFT spectra to CMB multipoles
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From CFT to Einstein Theory
+ Riegert = CFT Einstein theory
Evolution of the universe is described as a function of running coupling
Space-time transition
At very high energies beyond the Planck scale
CFT, no space-time singularity
At dynamical energy scale( )
(breaking of conformal invariance)
Running coupling increases according to inflationary expansion
Tim
e ev
olut
ion
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Inflation induced by Quantum Gravity
Running coupling diverges
End at dynamical time
Inflation is driven by Riegert + Einstein system
K. H, S. Horata, and T. Yukawa, Phys. Rev. D74 (2006) 123502
-70
-60
-50
-40
-30
-20
-10
0
10
-2 -1 0 1 2 3
inflation
start
Einsteinphase
Inflation starts at Planck time
RGE :
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Einstein Phase( )Low energy effective theory of gravity(=derivative expansion about Einstein theory)
tree + 1-loop tree
cf. chiral perturbation theory
Using Einstein Equationhigher-derivative terms are reduced to be one
1-loop correction :If one taks phenomenological parameter to be positivethis term becomes irrelevant at low energies
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Number of e-foldings
Inflation era
Friedmann era
Expansion of the universe
Evolutional Scenario
(~4000Mpc)Resolve the horizon problemcan be observed through CMB
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Evolution of fluctuation (From CFT to CMB)
CMB spectrum is computed using cosmological perturbation theory,which is consistent with WMAP
Planck phenomena (CFT) space-time transition (big bang) today
From Planck length tocosmological distance
293059 101010 +=
inflation Friedmann
Spectrum at transition point
inflation 0.00
0.10
0.20
-2
-1
0
1 10-3
10-2
0.00
0.10
0.20
Bardeen Potential Φ(b1=10, m=0.0156)
proper time, log10(τ/τp)
k [Mpc-1]
58.0
58.5
59.0
59.5
60.0 10-3
10-2
1 × 10-4
3 × 10-4
5 × 10-4
proper time τ
k [Mpc-1]
0
2000
4000
6000
1 10 100 500 1500
wmap 5yrsacbar2008
inflation
Scale-inv. spectrum at Planck time
Amplitude decreases during inflationResolve the flatness problem
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Conclusion and Discussion
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Renormalizable 4D quantum gravity was formulated as a perturbed theory from CFT, in which it is essential that there is no R^2 action.
The conformal mode is treated non-perturbatively so that conformal symmetry becomes exact quantum mechanically at the vanishing coupling limit.
Using dimensional regularization, I computed higher order corrections, and then showed that the conformal mode is not renormalized.
Quantum gravity scenario of inflation is constructed, in which the conformal mode serves for the scalar field (=inflaton).
It is proposed that the primordial spectrum of the universe originates from conformal invariance.
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The conformal invariance forces us change the aspect of space-time at very high energies above the Planck scale, where a traditional S-matrix description is not adequate at all. Consequently, this requires a new prescription to deal with negative-metric modes in the context of CFT.
on R x S^3
Conformal Algebra = SO(4,2) ( )etc.
Special conf. transf.
Negative-metric creation mode mixes with positive-metric creation mode through special conformal transformation
negative-metric mode does not appear independently as a gauge-inv. state
See ref. 1 in detail
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Since conformal symmetry mixes positive-metric and negative-metricmodes of the field, we cannot consider these modes separatelyand thus the field acts as a whole in physical quantities.
Physical states diffeomorphism invariant fieldsEx. scalar curvature
(not each modes)
“Real fields”
This suggests that the correctness of the overall sign of the gravitational action (not the sign of each mode) is significantfor unitarity.
Naively, two-point function of the “real field” is expected to be positive,because the Riegert and Weyl actions have the correct sign boundedfrom below and thus the path integral is well-defined future problem
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Appendix ICosmology
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Evolution equation for inflation Dynamical factor
Coupling is small Running coupling diverges
Inflation starts at Planck time
End at dynamical time 0
0.5
1
1.5
2
2.5
0 20 40 60 80 100 120
H, ρ
proper time,τ
ρH
FriedmanInflationEinsteinphase
Energy conservation
Energy shift = big bang
matter density
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Bardeen’s gravitational potentials
Evolution equation for gravitational potentials Dynamical factor
Constraint equation initially
finally
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Scale Invariant SpectrumInitial condition = two-point function of conformal mode
In Fourier space
Delta functionIn Fourier space
for GUT modelsHarrison-Zel’dovich-Peebles spectrum
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Appendix IIConformal Algebra
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Canonical Quantization on R x S^3R x S^3 background metric ( mode-expansions become simple)
Isometry of S^3 = SU(2)xSU(2)
Tensor harmonics that belongs to rep. with
Laplacianon S^3
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Conformally Coupled Scalar FieldThe action on R x S^3
dispersion relation
Mode expansion
Scalar harmonics
Quantization Wigner D function
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Conformal Algebra on R x S^3The generator of conformal algebra
15 conformal Killing vectors on R x S^3
Time translation:
Rotation on S^3:
Special conformal:
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Hamiltonian 1
15 generators S^3 rotation 6
Special conf. + dilatation transf. 4+4=8[=4 vectors of SO(4)]
Conformal algebra on R x S^3
6 generators of SU(2)xSU(2)
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Stress-tensor
The 15 generators of conformal algebra
SU(2)xSU(2) Clebsch-Gordan coeff. of SSS type
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Quantization of Conformal Mode
Mode expansion
Riegert action
Reduce to second order action by introducing new variable
(quantized on )
Dirac quantization
where
Scalar harmonics on S^3
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Hamiltonian
Special conformal transformation mixes positive- and negative-metric creation modes
4+4 generators of special conf. transf. SU(2)xSU(2) Clebsch-Gordan
Conformal Algebra SO(4,2)6 rotation generator on S^3
Casimir effect on R x S^3
etc.
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Traceless Tensor Fields
Take transverse gauge by using the four gauge parameters
Traceless tensor mode is decomposed as
Gauge-fixed Weyl action at t=0
radiation gauge+Furthermore, we take
residual gauge DOF = conformal symmetry
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Vector harmonics = rep. withTensor harmonics = rep. with (polarizations)
Transverse-traceless tensor mode
Transverse vector mode
Commutators
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: STT type: STV type: SVV type
SU(2)^2 CG coeff.
The generators of conformal algebra
Conformal symmetry mixes all modes in tensor field
up to field-dep. gauge transf.
Emphasize that negative-metric modes are necessary to formthe close algebra of conformal symmetry quantum mechanically
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Physical State ConditionsConfomal symmetry = diffeomorphism invariance
Physical state condition = Wheeler-DeWitt equation
Consider composite creation op. R_n satisfyingvacuum state
then
pure imaginary
“Real” states, such as scalar curvature
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Building Block R_n for Scalar FieldCommutator of Q_M and creation mode
Q_M-invariant creation operator is only
Consider a bilinear form
Q_M invariant J=L and
Thus, Q_M invariant operator in scalar field sector is given by
Z_2 symmetryX -X
Here,
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Building Blocks for Conformal ModeNo creation mode that commute with Q_M
Consider Q_M invariant bilinear forms, which are given by
where
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Building Blocks of Physical States
U(1) gauge fieldsScalar fields
Conformal mode
Traceless tensor fields
Classified by using crossing symmetry
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Physical states Diffeomorphism invariant fields
Conformal fields = “real fields” with even derivatives
Level (= dressed identity operator)n=0
n=2
n=4
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On Positivity of Two-Point FunctionPhysical states Diffeomorphism invariant fields
Conformal fields = “real fields” with even derivatives(level of building blocks are even)
Positivity means b_1 > 0 (right sign of WZ action)
At large b_1 limit, scalar curvature operator can be written by
Using the correlation function
Initial spectrum of the universe