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 8 th (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 We need to include infinitely many number of counter term • A UV completion - Higher derivative corrections Improves UV behavior • Unitarity problem Ghost

Lifshitz-like points • Anisotropic scaling ( z = 1 for relativistic theory ) • 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.

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 Extrinsic curvature of constant time leaves • Dimensions of coupling constants Dimensionless at D =3, z =3 • Generalized De Witte metric of the space of metrics 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: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 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