Quantum Gravity at a Lifshitz Point

Ref. P. Horava, arXiv:0901.3775 [hep-th] ( c.f. arXiv:0812.4287 [hep-th] )

June 8th (2009)@KEK Journal ClubPresented by Yasuaki Hikida

INTRODUCTION

A renormalizable gravity theory

• String theory “small theory” of quantum gravity• Einstein’s theory is not perturbatively renormalizable

• A UV completion - Higher derivative corrections

• Unitarity problem

We need to include infinitely many number of counter term

Improves UV behavior

Ghost

Lifshitz-like points• Anisotropic scaling

• Dynamical critical systems– A Lifshitz scalar field theory ( z = 2 )

– A relevant deformation ( z = 1 )

• Desired gravity theory– Improved UV behavior with z > 1– Flow to Einstein’s theory in IR limit– Lorentz invariance may not be a fundamental property.

( z = 1 for relativistic theory )

Horava-Lifshitz gravity• Modified propagator ( z > 1 )

– UV behavior

• Improves UV behavior, power-counting renormalizable– IR behavior

• Flows to z=1, no higher time derivatives, no problems of unitarity• Horava-Lifshitz gravity

– Power-counting renormalizable in 3+1 dimensions– behaves as z=3 at UV and z=1 at IR

Plan of this talk

1. Introduction2. Lifshitz scalar field theory3. Horava-Lifshitz gravity4. Conclusion

LIFSHITZ SCALAR FIELD THEORY

Theories of the Lifshitz type• Lifshitz points– Anisotropic scaling with dynamical critical exponent z

• Action of a Lifshitz scalar

– Dynamical critical exponent z=2, Dimension– Ex. Quantum dimer problem, tricritical phenomena

• Detailed balance condition– Potential term can be derived from a variational principle

Ground-state wavefunction• Hamiltonian

• Ground state

HORAVA-LIFSHITZ GRAVITY

Fields, scalings and symmetries• ADM decomposition of metric

– Fields are • Scaling dimensions

• Foliation-preserving diffeomorphisms

Lagrangian (kinetic term)• Requirements– Quadratic in first time derivative– Invariant under foliation-preserving diffeomorphisms

• Dimensions of coupling constants

• Generalized De Witte metric of the space of metricsDimensionless at D=3, z=3

Extrinsic curvature of constant time leaves

for general relativity

Lagrangian (potential term)• Requirements– Independent of time derivatives– Invariant under foliation-preserving diffeomorphisms

• Dimensions of terms– Equal (UV) or less (IR) than– The choice of z=3 6th derivatives of spatial coordinates

• UV theory with detailed balance– To limit the proliferation of independent couplings

Gravity with z=2• Consider the Einstein-Hilbert action as W

– The potential term of this theory

– Flow from z=2 to z=1– Power-counting renormalizable at 2+1 dimensions– Could be used to construct a membrane theory

(cf. Horava, arXiv:0812.4287 )

Gravity with z=3• Consider the gravitational Chern-Simons as W

– The potential term of this theory

• The Cotton tensor

– Power-counting renormalizable at 3+1 dimensions• Short-distance scaling with z=3• A unique candidate for Eij with desired properties

Remarks• The Cotton tensor– Properties

• Symmetric and traceless• Transverse• Conformal with weight -5/2

– Plays the role of the Weyl tensor Cijkl in 3 dim.• Gravity with detailed balance– Action

– Ground state

Anisotropic Weyl invariance• The action may be conformal invariant since the Cotton

tensor is.• Decompose the metric as

• The action becomes

• At the action is invariant under

Local version of

Free-field fixed point• Kill the interaction– Set with keeping two parameters

• Expand around the flat background

– Gauge fixing :– Gauss constraint :

• Redefine the variables

Dispersion relations• The actions– Kinetic term

– Potential term

• Two special values of– : the scalar model H is a gauge artifact– : extra gauge symmetry eliminates H

• Dispersion relations– Transverse tensor modes :– A scalar mode for : It is desired to get

rid of this mode.

Relevant deformations• Deformations– Relax the detailed balance condition and add all marginal

and relevant terms – At IR lower dimension operators are important

• The Einstein-Hilbert action in the IR limit

• Differences– The coupling must be one.– The lapse variable N should depend on spatial coordinates.

Keeping detailed balance• Topological massive gravity

• The action

• The correspondence of parameters

CONCLUSION

Conclusion• Summary– Gravity theory with non-relativistic scaling at UV– Power-counting renormalizable with z=3, 3+1 dim.– Naturally flows to relativistic theory with z=1– Fixed codimension-one foliation

• Discussions– Horizon of black hole– Holographic principle

– Application to cosmology