Quantum Gravity at a Lifshitz Point Ref. P. Horava, arXiv: [hep-th] ( c.f. arXiv: [hep-th] ) June 8 th 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  6 th 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: )

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 E ij with desired properties

Remarks The Cotton tensor – Properties Symmetric and traceless Transverse Conformal with weight -5/2 – Plays the role of the Weyl tensor C ijkl 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