Quantum Gravity at a Lifshitz Point Ref. P. Horava, arXiv:0901.3775 - - PowerPoint PPT Presentation

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 Club Presented 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. Introduction
• 2. Lifshitz scalar field theory
• 3. Horava-Lifshitz gravity
• 4. 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 metrics

Dimensionless 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