Quantum gravity at a Lifshitz point

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PETR HOŘAVA PHYSICAL REVIEW D 79, 084008 (2009)pffiffiffig E ij ¼ W½g k‘Š(27)g ijfor some action W. The two copies of E ij in (26) arecontracted by G ijk‘ , the inverse of the De Witt metric(22). Loosely borrowing terminology from nonequilibriumdynamics, we will say that theories whose potential is ofthe form (26) with (27) for some W satisfy the ‘‘detailedbalance condition.’’In the context of condensed matter, the virtue of thedetailed balance condition is in the simplification of therenormalization properties. Systems which satisfy the detailedbalance condition with some D-dimensional actionW typically exhibit a simpler quantum behavior than ageneric theory in D þ 1 dimensions. Their renormalizationcan be reduced to the simpler renormalization of the associatedtheory described by W, followed by one additionalstep—the renormalization of the relative couplings betweenthe kinetic and potential terms in S. Examples ofthis phenomenon include scalar fields [17] or Yang-Millsgauge theories [9,18]. Investigating the precise circumstancesunder which this ‘‘quantum inheritance principle’’holds for gravity systems will be important for understandingthe quantum properties of gravity models with nonrelativisticvalues of z.Since we are primarily interested in theories which arespatially isotropic, W must be the action of a relativistictheory in Euclidean signature. (Obvious generalizations totheories with additional spatial anisotropies are clearlypossible, but will not be pursued in this paper.) In [3], atheory of gravity in D þ 1 dimensions satisfying the detailedbalance condition was constructed, with W theEinstein-Hilbert actionW ¼ 1 2 WZd D px ffiffiffi g ðR 2W Þ: (28)The potential S V of this theory takes the formS V ¼ 28 4 WZdtd D x ffiffiffipg NR ij 12 Rgij þ W g ij G ijk‘ R k‘ 12 Rgk‘ þ W g k‘ : (29)At short distances, the curvature term in W dominates over W , and the resulting potential S V is quadratic in thecurvature tensor: The theory exhibits anisotropic scalingwith z ¼ 2 in the UV. Turning on W in W leads to lowerdimensionterms in S V which dominate at long distances,and the theory undergoes a classical flow to z ¼ 1 in theIR. The anisotropic scaling in the UV shifts the criticaldimension of this theory, which is now renormalizable bypower counting in 2 þ 1 dimensions. In dimensions higherthan 2 þ 1, the theory with potential (29) is merely a lowenergyeffective field theory, and can be expected to breakdown at the scale set by the dimensionful coupling W .Here we are interested in constructing a theory whichsatisfies detailed balance, and exhibits the short-distancescaling with z ¼ 3 leading to power-counting renormalizabilityin 3 þ 1 dimensions. Therefore, E ij must be of thirdorder in spatial derivatives. As it turns out, there is a uniquecandidate for such an object: the Cotton tensorC ij ¼ " ik‘ r k ðR j 1‘ 4 Rj ‘Þ: (30)This tensor not only exhibits all the required symmetries, italso follows from a variational principle.1. Properties of the Cotton tensorThe Cotton tensor enjoys several symmetry propertieswhich may not be immediately obvious from its definitionin (30):(i) It is symmetric and traceless,C ij ¼ C ji ; g ij C ij ¼ 0: (31)(ii) It is transverse (or covariantly conserved),r i C ij ¼ 0: (32)(iii) It is conformal, with conformal weight 5=2. Moreprecisely, under local spatial Weyl transformationsg ij ! expf2ðxÞgg ij ; (33)it transforms asC ij ! expf 5ðxÞgC ij ; (34)with no terms containing derivatives of ðxÞ.The Cotton tensor plays an important role in geometry.Recall that in dimensions D>3, the property of conformalflatness of a Riemannian metric is equivalent to the vanishingof the Weyl tensor C ijk‘ , defined as the completelytraceless part of the Riemann tensor:1C ijk‘ ¼ R ijk‘D 2 ðg ikR j‘ g i‘ R jk g jk R i‘1þ g j‘ R ik ÞþðD 1ÞðD 2Þ ðg ikg j‘ g i‘ g jk ÞR:(35)In D ¼ 3, however, the Weyl tensor vanishes identically,and another object has to take over the role in the criterionof conformal flatness of 3-manifolds. This object is theCotton tensor, of third order in spatial derivatives.The Cotton tensor also plays an important role in physics.In the initial value problem of the Hamiltonian formulationof general relativity, it is natural to ask what set ofinitial conditions can be freely specified for the metric andits canonical momenta, without violating the constraintpart of Einstein’s equations. It was shown by York [19–21] that the Cotton tensor plays a central role in answeringthis question. The correct initial conditions are set by084008-6
QUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)specifying the values of two tensors with the symmetries ofthe Cotton tensor: One related to the initial value for theconformal structure of the spatial metric, and the otherspecifying the initial value of the conjugate momenta.For this reason, C ij is often referred to as the ‘‘Cotton-York tensor’’ in the physics literature.Lastly, the Cotton tensor follows from a variationalprinciple, with actionW ¼ 1 Zw 2 ! 3 ð Þ: (36)Here w 2 is a dimensionless coupling, and! 3 ð Þ¼Trð ^ d þ 2 3^ ^ Þ " ijk ð m i‘ @ ‘j km þ 2 3 n i‘ ‘ jm m kn Þd3 x (37)is the gravitational Chern-Simons term, with theChristoffel symbolsijktreated as known functionals ofthe metric g ij , and not as independent variables. Thevariation of (36) with respect to g ij yields the vanishingof the Cotton tensor as the equations of motion.Without any loss of generality, we will assume that thecoupling w 2 is positive; its sign can be changed by flippingthe orientation of the 3-manifold . Unlike in Chern-Simons gauge theories with a compact gauge group, thecoupling constant of Chern-Simons gravity in 2 þ 1 dimensionsis not quantized, as a result of the absence oflarge gauge transformations. In our framework, however,we are only interested in the action of a theory in threedimensions in ‘‘imaginary time,’’ and require that thisEuclidean action be real. This is to be contrasted with theconventional interpretation of the three-dimensional theory,which involves analytic continuation to real time in2 þ 1 dimensions, and imposes slightly different realityconditions on the action.2. z ¼ 3 gravity with detailed balanceHaving reviewed some of the properties of the Cottontensor, we can now write down the full action of our z ¼ 3gravity theory in 3 þ 1 dimensions:S ¼ Z dtd 3 x ffiffiffi p 2g N 2 ðK ijK ij K 2 2 Þ2w 4 C ijC ij¼ Z dtd 3 x ffiffiffi p 2g N 2 ðK ijK ijK 2 Þ 22w 4 r i R jk r i R jk r i R jk r j R ik 18 r iRr i R:(38)As a result of the uniqueness of the Cotton tensor, theaction given in (38) describes the most general z ¼ 3gravity satisfying the detailed balance condition, modulothe possible addition of relevant terms, which will bediscussed in Sec. II E.We can demonstrate that after the Wick rotation toimaginary time, this action can be written—up to a totalderivative—as a sum of squares,S ¼ i Z dd 3 x ffiffiffi p 2g N 2 ðK ijK ij þ K 2 Þþ 22w 4 C ijC ij¼ 2i Z dd 3 x ffiffiffi p 1g N K ij2w 2 C ij 1 G ijk‘ K k‘2w 2 C k‘ ; (39)First, C ij G ijk‘ C k‘ ¼ C ij C ij because C ij is traceless. As tothe cross terms K ij G ijk‘ C k‘ , they can be written as a totalderivative,1 Zdd 3 px ffiffiffi g NKij G ijk‘ Cw 2 k‘¼ 1 Zdd 3 p2w 2 x ffiffiffi g ð _gij r i N j r j N i ÞC ij¼ Z dd 3 px ffiffiffi Wg _g ij þ 1 g ij w 2 r iðN j C ij Þ¼ Z dd x 3 L _ þ 1 w 2 @ pffiffiffiið g Nj C ij Þ ;where we used the transverse property (32)ofC ij , and L isthe Lagrangian density of the action W in (36).Introducing an auxiliary field B ij , it is convenient torewrite the imaginary-time action asS ¼ 2i Z dd 3 x ffiffiffi p 1g N B ij K ij2w 2 C ijB ij G ijk‘ B k‘ : (40)This form of the action, with all terms at least linear in theauxiliary field B ij and with the linear term proportional to agradient flow equation, is symptomatic of theories satisfyingthe detailed balance condition in the context ofcondensed-matter systems, in particular, in the theory ofquantum and dynamical critical phenomena [4,5], stochasticquantization [22,23], and nonequilibrium statisticalmechanics [24].In that condensed-matter context, the property of detailedbalance often has one interesting implication. If aquantum critical system in D þ 1 dimensions satisfies detailedbalance with some W in D dimensions, the partitionfunction of the theory described by W yields a naturalsolution of the Schrödinger equation of the theory in D þ1 dimensions, which plays the role of a candidate groundstatewave function. Similarly, in nonequilibrium statisticalmechanics and dynamical critical phenomena, the correspondingstatement is essentially the Wick rotation of thiscorrespondence to imaginary time: The partition functionof the D-dimensional theory defined by W represents anequilibrium state solution of the dynamical theory withdetailed balance in D þ 1 dimensions.084008-7
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Quantum gravity at a Lifshitz point

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