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Quantum Gravity at a Lifshitz Point

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

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Quantum Gravity at a Lifshitz Point

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  1. 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

  2. INTRODUCTION

  3. A renormalizable gravity theory • String theory  “small theory” of quantum gravity • Einstein’s theoryis not perturbativelyrenormalizable • A UV completion - Higher derivative corrections • Unitarity problem We need to include infinitely many number of counter term Improves UV behavior Ghost

  4. 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 )

  5. 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

  6. Plan of this talk • Introduction • Lifshitz scalar field theory • Horava-Lifshitz gravity • Conclusion

  7. LIFSHITZ SCALAR FIELD THEORY

  8. 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

  9. Ground-state wavefunction • Hamiltonian • Ground state

  10. HORAVA-LIFSHITZ GRAVITY

  11. Fields, scalings and symmetries • ADM decomposition of metric • Fields are • Scaling dimensions • Foliation-preserving diffeomorphisms

  12. 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 Extrinsic curvature of constant time leaves Dimensionless at D=3, z=3 for general relativity

  13. 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

  14. 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 )

  15. 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 Eijwith desired properties

  16. 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

  17. 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

  18. Free-field fixed point • Kill the interaction • Set with keeping two parameters • Expand around the flat background • Gauge fixing : • Gauss constraint : • Redefine the variables

  19. 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.

  20. 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.

  21. Keeping detailed balance • Topological massive gravity • The action • The correspondence of parameters

  22. CONCLUSION

  23. 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

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