KEK 5/25、2009

Conformal Invariance and

Quantum Gravity

by K. Hamada

•K. H., Conformal Field Theory on R x S^3 from Quantized Gravity,[arXiv:0811.1647]

•K. H., S. Horata and T. Yukawa, “Focus on Quantum Gravity Research”(Nova Science Publisher, NY, 2006), Chap.1

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The goal of quantum gravity

is to understand the Planck scale phenomena

The starting point of quantum gravity is to give up graviton picture!

Quantum gravity = quantization of space-time

= quantization of graviton

Key idea

Conformal invariance/Background metric independence no scale and no singularity

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Conformally Invariant Actions

Square of Weyl tensor

This tensor will play a significant role in the early universe where it will be vanishing as required for the inflation

Euler density

These actions are conformally invariant.

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Renormalizable Quantum GravityThe Action (Weyl + Euler + Einstein)

conformally invariant (no R^2) Planck constant

“t” is a unique dimensionless gravitational coupling constant indicating asymptotic freedom

(conformally flat)

[ “b” is expanded by t, but the lowest is a coupling-independent constant ]

[ dominate the Einstein action ]

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The Perturbation about CFTThis model :

Conformal symmetry mixes positive- and negative-metric modes

light on unitarity in strong gravityCFT + perturbations

Non-perturbative (conformal mod is treated exactly)

cf. Early 4-th order models in 1970’sGauge symmetry does not

mix gravitational modes at allnon-unitary

Free + perturbations

perturbative (all modes are treated in perturbation)

graviton picture

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Dynamics of The Conformal Mode

Lowest term of S is coupling-independent

Jacobian=Wess-Zumino actionfor conformal anomaly

cf. Liouville action

Dynamics of conformal mode is induced from the measure!

Conformal Field Theory from Quantized Gravity

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Diffeomorphism Invariance: gauge parameter

Mode decomposition

coupling const.

traceless

no coupling const.

Conformal mode and traceless mode are 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 in the radiation gauge+

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 )

fixed by physical state conditionsOther fields:

to remove conformal-mode dependence

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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 and so on are invariant under the conformal transformations respectively.

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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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Stress-tensor

The 15 generators of conformal algebra

SU(2)xSU(2) Clebsch-Gordan coeff. of SSS type

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Conformal Algebra on R x S^3

Conformal algebra 15 generators: Hamiltonian

: S^3 rotation

: special conf. + dilatation transf.[=4 vectors of SO(4)]

6 generators of SU(2)xSU(2)

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The Conformal ModeWess-Zumino action on R x S^3

Rewrite into the second order form with constraint

Dirac quantizationMode expansion

where

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where

The generators of conformal algebra ( )

Casimir effecton R x S^3

Special conformal transformation mixes positive- and negative-metric creation modes

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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

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 tensor modes.

Emphasize that negative-metric modes are indispensable to formthe close algebra of conformal symmetry quantum mechanically

Physical States of Quantum Gravity

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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 the 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

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ExampleLevel

(= dressed identity operator)n=0

n=2

n=4

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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

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Evolution scenario

inflation

CFT

baryogenesiscorrelation length:

Number of e-foldings scale factor

K.H., Minamizaki, SugamotoarXiv:0708.2127[hep-ph]

Planck length at Planck time

grows up tothe Hubble distance

today today

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ConclusionThe conformal invariance forces us change the aspect of space-time at 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 which can be carried out by making use of the conformal symmetry.

Quantum diffeomorphism symmetry in 4D quantum gravity was examined.It was shown that conformal symmetry is equal to diffeomorphism invariance. Physical state condition was solved. Unitarity issue in strong gravity was discussed in the context of CFT.

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Effective action

Beta function conf. anomaly

Running coupling const.

physical momentum

Asymptotic freedom comoving momentum

At high energy beyond Planck scale

Singularities with divergent Riemann curvatureare excluded quantum mechanically

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Integrability and actionbare action(conf. anomaly)Conformal variation of effective action

(=path integral by conf. mode)

Integrability condition

Weyl action and Euler combination (no R^2)Integrable Action

asymptotically free dimensionless coupling constant