Quantum gravity at a Lifshitz point

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PETR HOŘAVA PHYSICAL REVIEW D 79, 084008 (2009)lieved to be asymptotically free [32–34]. As we mentionedin the Introduction, the asymptotic freedom of (68) wouldseem to make this theory an excellent candidate for solvingthe problem of quantum gravity in 3 þ 1 dimensions, wereit not for one persistent flaw: After the Wick rotation to 3 þ1 dimensions, the spectrum of physical states containsghosts which violate unitarity in perturbation theory.Our construction of z ¼ 4 theory in 4 þ 1 dimensionsbenefits from the asymptotic freedom of the fourdimensionalhigher-curvature theory (68), but avoids thepitfall of its perturbative nonunitarity. Indeed, we are onlyinterested in the four-dimensional action W in theEuclidean signature, in order to construct the 4 þ 1 dimensionalaction (70).The only remaining coupling-constant renormalizationin the high-energy limit of the theory in 4 þ 1 dimensionsis the renormalization of . However, is not an independentcoupling associated with interactions; instead, it survivesin the noninteracting limit, and parametrizes a familyof free-field fixed point as and are sent to zero. In thisrespect, the quantum behavior of this theory would be verysimilar to the behavior in quantum critical Yang-Millsstudied in [9], which inherits asymptotic freedom fromrelativistic Yang-Mills in four dimensions.Setting ¼ 0 in (68) and ¼ 1=4 in (70) would lead toa theory which exhibits an additional gauge invariance,acting on the fields asg ij ! expf2ðt; xÞgg ij ; N ! expf4ðt; xÞgN;N i ! expf2ðt; xÞgN i :(71)These are the local anisotropic Weyl transformations withz ¼ 4.B. z ¼ 4 gravity in 3 þ 1 dimensionsIn three dimensions, the action of Euclidean gravityquadratic in the curvature tensor isW ¼ 1 MZd 3 px ffiffiffi g ðRij R ij þ R 2 Þ: (72)As in four dimensions, there are again only two independentterms in W, but for a different reason: When D ¼ 3,the Riemann tensor is determined in terms of the Riccitensor, and the Weyl tensor vanishes identically. The twocouplings M and M= are now dimensionful, of dimension1. In power counting, this makes the theory described by(72) super-renormalizable. When we use W to generate thepotential term S V for z ¼ 4 gravity in 3 þ 1 dimensions,we consequently end up with a theory whose action againhas the form (70), now with W given by (72) and in 3 þ 1dimensions, where it is super-renormalizable by powercounting. As in all the previous examples with variousvalues of z, relevant deformations flow the theory to z ¼1 in the infrared.Such super-renormalizable terms can also be added toour z ¼ 3 theory of gravity described in (38). These termswill give spatial dynamics to the conformal factor of thespatial metric, improving the short-distance properties ofthe propagator for the scalar mode H of the metric, restoringpower-counting renormalizability in the case when H ispresent as a physical field.C. The case of z ¼ 0: Ultralocal gravityIn the Hamiltonian formulation of general relativity, theHamiltonian is given by a sum of constraints,H ¼ Z d D xðNH ? þ N i H i Þ: (73)Notably, the algebra of the Hamiltonian constraintsH ? ðxÞ and H i ðxÞ in general relativity is not a true Liealgebra—in particular, the constraints do not form thenaively expected algebra of spacetime diffeomorphisms.Instead, the structure ‘‘constants’’ of the commutator ofH ? ðxÞ with H ? ðyÞ are field dependent, because theycontain the components of the spatial metric.In [35], an alternative theory of gravity was proposed, inwhich the constraints do form a Lie algebra. In this theory,the commutators of H i with themselves and with H ? arethe same as in general relativity, but the problematic fielddependentcommutator of H ? ðxÞ with H ? ðyÞ is simplyreplaced by zero. This symmetry can be viewed either as acontraction of the symmetries formed by the Hamiltonianconstraints of general relativity, or as a contraction of thealgebra of infinitesimal spacetime diffeomorphisms. Thecontracted symmetry algebra respects a dimension-onefoliation of spacetime by a congruence of timelike curves.This congruence can be used to identify the points of spaceat different times; as a result, the spacetime in this theory ofgravity carries a preferred structure of absolute space.The theory of gravity that realizes this symmetry structureis known as the ‘‘ultralocal theory’’ of gravity. It isinteresting to note that ultralocal gravity fits naturally intoour framework of gravity models with anisotropic scalingand nontrivial dynamical exponents z Þ 1. As shown in[35], the required symmetries force the action of the ultralocaltheory to be of the same form S ¼ S K S V as thetheories considered here, with the potential term containingonly the cosmological constant,S V ¼ Z dtd D px ffiffiffi g ; (74)and no curvature-dependent terms. There is a clear way tointerpret (74) in our framework of gravities with anisotropicscaling: The value of z can be read off as one-half ofthe number of derivatives appearing in S V . This is equivalentto declaring (74) to be of the same dimension as thekinetic term S K . Either way, this approach suggests that theultralocal theory corresponds formally to the limiting caseof z ! 0.084008-12
1N ð@ 2g ij r i N j r j N i Þ2 ffiffiffi Wp Gg ijk‘ ¼ 0; (75)g k‘the full equations of motion are automatically satisfied.While the full equations of motion are of second order intime derivatives and of order 2z in spatial derivatives, thesimpler equation (75) has its degree reduced by half. (Thisargument is reminiscent of the BPS condition in supersymmetrictheories.) A simple class of solutions to (75) canQUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)Historically, the ultralocal theory of gravity has beenstudied for at least two additional reasons, besides thecontext of [35]:now be obtained by setting N ¼ 1, N i ¼ 0, and takingg ij ¼ g ij ðxÞ to be an arbitrary (-independent) solutionof the equations of motion,(i) Ultralocal gravity was proposed by Isham in [36], inWan attempt to introduce a new formal expansion¼ 0;gparameter into general relativity. In [36], the suggestedij(76)expansion parameter was the coefficient infront of the scalar curvature term in S V , equal to onein the potential of general relativity and set equal tozero in the ultralocal theory.of the D-dimensional theory whose action is W. Clearly,this solution can be trivially continued back to real time,and represents a real static solution of the full theory.In particular, let us assume that the Euclidean action W(ii) Ultralocal theory is relevant for early universe cosmologyin general relativity, because it captures the situation is rather generic, and does not pose a very strongis such that it has the Euclidean AdS D as a solution. Thisdynamics of Friedmann-Robertson-Walker solutionsin the so-called ‘‘velocity dominated’’ earlyrestriction on W. With this assumption, the D þ 1 dimensionaltheory will have a classical solution which is thestages after the big bang, as was first shown by direct product of the time dimension and AdS D ,Belinsky, Khalatnikhov, and Lifshitz [37,38].Unfortunately, the z ! 0 limit is rather singular, and theN ¼ 1; N i ¼ 0;program outlined in (i) was never very successful. As tog ij dx i dx j ¼ d 2 þ sinh 2 d 2 D 1(ii), the embedding of ultralocal gravity into our framework(77)of gravity with anisotropic scaling raises the possi-The boundary of this solution is S D 1 R. The isometries:bility of interpreting the cosmological evolution as a flow,from z Þ 1 in the early universe to z ¼ 1 observed now.It is remarkable that, even though the action of ultralocalof the Euclidean AdS D induce conformal symmetries in theboundary. In addition, there is one more bulk isometry,given by time translations. Thus, the full symmetries aretheory is not invariant under all spacetime diffeomorphisms,the theory exhibits ‘‘general covariance’’ [35,39]:SOðD; 1ÞR: (78)In particular, the number of local symmetry generators perspacetime point is D þ 1, i.e., the same as in generalrelativity.These symmetries suggest that such a gravity theory in thebulk can serve as a possible holographic dual of dynamicalfield theories which are already critical in the static limit.Such problems are often encountered in the theory ofD. Bulk-boundary correspondence in gravity at a dynamical critical phenomena. Starting with a universalityLifshitz pointclass of a static critical system in D 1 spatial dimensions,the time-dependent dynamics of the system in DThe availability of gravity models with nontrivial valuesdimensions can also exhibit criticality, with the characteristicproperty of ‘‘critical slowing down’’ of time-of the dynamical critical exponent z can enhance thespectrum of examples of dualities between gravity in thedependent correlation functions. One given static universalityclass can belong to several different dynamical uni-bulk and field theory on the boundary. This could beparticularly relevant for understanding gravity duals ofversality classes. In particular, one universal characteristicnonrelativistic CFTs.of the dynamics is given by the critical exponent z.After the Wick rotation of the z ¼ 3 theory in 3 þ 1If we study such a dynamical critical system on R D ,dimensions to imaginary time , the action of this theoryit will exhibit the anisotropic scaling symmetry given bywas rewritten in a simple form (40) with the use of anauxiliary field B ij (10), with i ¼ 1; ...D 1. Another possibility is to put. The same rewriting applies to a muchthis system on S D 1 R, with the spatial slices of thebroader class of gravity models which satisfy detailedfoliation given by S D 1 of a fixed radius. On such abalance with some D-dimensional action W, such as thefoliation, the scale symmetry (10) is absent, since it wouldz ¼ 4 models discussed above. Using this formalism, wechange the radius of the sphere. However, the system stillcan find a large class of classical solutions of such theories,exhibits the symmetries of conformal transformations ofsimply by noting that if the following equation holds,S D 1 and time translations. Thus, the conformal symmetriesleft unbroken by the foliation are precisely the bulkisometries (78) of the AdS D R solution of gravity theorywith anisotropic scaling.Following [40,41], a nonrelativistic version of the AdS/CFT correspondence has indeed received a lot of attentionrecently. The focus in this area has been primarily on theCFTs with nontrivial values of z which exhibit conventionalrelativistic gravity duals. It is natural to broaden thisframework, and free the gravity side of the duality of the084008-13
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Quantum gravity at a Lifshitz point

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