Quantum gravity at a Lifshitz point

QUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)(October 2008). I wish to thank the organizers for theirhospitality, and the participants for stimulating discussions.This work has been supported by NSF GrantNo. PHY-0555662, DOE Grant No. DE-AC03-76SF00098, and the Berkeley Center for TheoreticalPhysics.[1] S. Weinberg, in General Relativity. An Einstein CentenarySurvey, edited by S. W. Hawking and W. Israel(Cambridge University Press, Cambridge, England, 1980).[2] E. S. Fradkin and A. A. Tseytlin, Phys. Rep. 119, 233(1985).[3] P. Hořava, arXiv:0812.4287, [J. High Energy Phys. (to bepublished)].[4] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49,435 (1977).[5] S.-K. Ma, Modern Theory of Critical Phenomena(Benjamin, New York, 1976).[6] S. Sachdev, Quantum Phase Transitions (CambridgeUniversity Press, Cambridge, England, 1999).[7] E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 11, 255 (1941); 11, 269(1941).[8] R. M. Hornreich, M. Luban, and S. Shtrikman, Phys. Rev.Lett. 35, 1678 (1975).[9] P. Hořava, arXiv:0811.2217.[10] E. Ardonne, P. Fendley, and E. Fradkin, Ann. Phys. (N.Y.)310, 493 (2004).[11] P. M. Chaikin and T. C. Lubensky, Principles ofCondensed Matter Physics (Cambridge University Press,Cambridge, England, 1995).[12] The idea that Lorentz invariance might be an emergentsymmetry has a long history, going back at least to thepioneering paper [46].[13] In differential geometry, a codimension-q foliation F on ad-dimensional manifold M is defined as M equippedwith an atlas of coordinate systems ðy a ;x i Þ a ¼ 1; ...q,i ¼ 1; ...d q, such that the transition functions take therestricted form ð~y a ; ~x i Þ¼ð~y a ðy b Þ; ~x i ðy b ;x j ÞÞ. The generaltheory of foliations is reviewed e.g. in [14–16].[14] H. B. Lawson, Jr., Bull. Am. Math. Soc. 80, 369 (1974).[15] C. Godbillon, Feuilletages (Birkhauser, Boston, 1991).[16] I. Moerdijk and J. Mrčun, Introduction to Foliations andLie Groupoids (Cambridge University Press, Cambridge,England, 2003).[17] J. Zinn-Justin, Nucl. Phys. B275, 135 (1986).[18] J. Zinn-Justin and D. Zwanziger, Nucl. Phys. B295, 297(1988).[19] J. W. York, Phys. Rev. Lett. 26, 1656 (1971).[20] J. W. York, Phys. Rev. Lett. 28, 1082 (1972).[21] J. W. York, J. Math. Phys. (N.Y.) 13, 125 (1972).[22] G. Parisi and Y.-S. Wu, Sci. Sin. 24, 483 (1981).[23] M. Namiki, Stochastic Quantization (Springer, New York,1992).[24] M. Le Bellac, F. Mortessagne, and G. Batrouni,Equilibrium and Non-Equilibrium Statistical Thermodynamics(Cambridge University Press, Cambridge,England, 2004).[25] E. Witten, arXiv:gr-qc/0306083.[26] In contrast, for some theories with detailed balance,0 expf W=2g does represent a physical normalizableground-state wave function. Examples include the Lifshitzscalar theory (as discussed, for example, in [10]), and thequantum critical Yang-Mills with z ¼ 2 in 4 þ 1 dimensions[9].[27] S. Deser, R. Jackiw, and S. Templeton, Phys. Rev. Lett. 48,975 (1982).[28] S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. (N.Y.)140, 372 (1982).[29] S. Deser and Z. Yang, Classical Quantum Gravity 7, 1603(1990).[30] B. Keszthelyi and G. Kleppe, Phys. Lett. B 281, 33 (1992).[31] The main difference between (58) and topologically massivegravity stems from the fact that here we are onlyinterested in the Euclidean-signature version of (58), withthe real action W. In topologically massive gravity, theEuclidean action W is interpreted as the Wick rotation ofthe real action from the physical signature 2 þ 1, leadingto a slightly different reality condition on W, with w 2purely imaginary. There has been a recent resurgence ofinterest in topological massive gravity, initiated by [47];see also [48].[32] K. S. Stelle, Phys. Rev. D 16, 953 (1977).[33] E. S. Fradkin and A. A. Tseytlin, Nucl. Phys. B201, 469(1982).[34] E. S. Fradkin and A. A. Tseytlin, Nucl. Phys. B203, 157(1982).[35] C. Teitelboim, in General Relativity and Gravitation,edited by A. Held (Plenum Press, New York, 1980), Vol. 1.[36] C. J. Isham, Proc. R. Soc. A 351, 209 (1976).[37] V. A. Belinsky, I. M. Khalatnikov, and E. M. Lifshitz, Adv.Phys. 19, 525 (1970).[38] V. A. Belinsky, I. M. Khalatnikov, and E. Lifshitz, Adv.Phys. 31, 639 (1982).[39] M. Henneaux, Bull. Soc. Math. Belg. 31, 47 (1979).[40] D. T. Son, Phys. Rev. D 78, 046003 (2008).[41] K. Balasubramanian and J. McGreevy, Phys. Rev. Lett.101, 061601 (2008).[42] C. J. Isham, arXiv:gr-qc/9210011.[43] K. V. Kuchař, in Winnipeg 1991, General Relativity andRelativistic Astrophysics, 1991.[44] This picture might change if we allow sufficiently singularfoliations, for example, if such singularities turn out to berequired for a consistent summation over spacetime topologies.[45] P. W. Anderson, Basic Notions of Condensed MatterPhysics (Addison-Wesley, Reading, MA, 1984).[46] S. Chadha and H. B. Nielsen, Nucl. Phys. B217, 125(1983).[47] W. Li, W. Song, and A. Strominger, J. High Energy Phys.04 (2008) 082.[48] E. Witten, arXiv:0706.3359.084008-15