Quantum gravity at a Lifshitz point

PETR HOŘAVA PHYSICAL REVIEW D 79, 084008 (2009)ture—that of a codimension-one foliation [13]. This foliationstructure F is to be viewed as a part of the topologicalstructure of M, before any notion of a Riemannian metricis introduced. The leaves of this foliation are the hypersurfacesof constant time. Coordinate transformationsadapted to the foliation are of the form~x i ¼ ~x i ðx j ;tÞ; ~t ¼ ~tðtÞ: (13)Thus, the transition functions are foliation-preserving diffeomorphisms.We will denote the group of foliationpreservingdiffeomorphisms of M by Diff F ðMÞ. In thelocal adapted coordinate system, the infinitesimal generatorsof Diff F ðMÞ are given byx i ¼ i ðt; xÞ; t ¼ fðtÞ: (14)We will simplify our presentation by further assuming thatthe spacetime foliation is topologically given byM ¼ R ; (15)with all leaves of the foliation topologically equivalent to afixed D-dimensional manifold .Differential geometry of foliations is a well-developedbranch of mathematics, and represents the proper mathematicalsetting for the class of gravity theories studiedhere. We will not review the geometric theory of foliationsin any detail here, instead referring the reader to [14–16].For example, there are two natural classes of functions thatcan be defined on a foliation: In addition to functions thatare allowed to depend on all coordinates, there is a specialclass of functions which take constant values on each leafof the foliation. We will call such functions ‘‘projectable.’’Foliations can be equipped with a Riemannian structure.A Riemannian structure compatible with our codimensiononefoliation of M consists of three objects: g ij , N i , and N,with N a projectable function; both N and N i transform asvectors under the reparametrizations of time. As pointedout above, these fields can be viewed as a decomposition ofa Riemannian metric on M into the metric g ij inducedalong the leaves, the shift variable N i , and the lapse field N.The generators of Diff F ðMÞ act on the fields viag ij ¼ @ i k g jk þ @ j k g ik þ k @ k g ij þ f _g ij ;N i ¼ @ i j N j þ j @ j N i þ _j g ij þ fN _ i þ f N_i ;N ¼ j @ j N þ fN _ þ f N: _(16)In [3], these transformation rules were derived by startingwith the action of spacetime diffeomorphisms on the relativisticmetric in the ADM decomposition, and taking thec !1 limit. We also saw in [3] that N i and N can benaturally interpreted as gauge fields associated with thetime-dependent spatial diffeomophisms and the time reparametrizations,respectively. In particular, since N is thegauge field associated with the time reparametrization fðtÞ,it appears natural to restrict it to be a projectable functionon the spacetime foliation F .If we wish instead to treat N as an arbitrary function ofspacetime, we have essentially two options. First, we canallow an arbitrary spacetime-dependent N as a backgroundfield, but integrate only over space-independent fluctuationsof N in the path integral. As the second option, wewill encounter situations in which N must be allowed to bea general function of spacetime, because it participates inan additional gauge symmetry. When that happens, we willintegrate over the fluctuations of N in the path integral. Anexample of such an extra symmetry is the invariance underanisotropic Weyl transformations discussed in Sec. II C 3below, and in Sec. 5.2 of [3].B. LagrangiansWe formally define our quantum field theory of gravityby a path integral,ZDgij DN i DN expfiSg: (17)Here Dg ij DN i DN denotes the path-integral measurewhose proper treatment involves the Faddeev-Popov gaugefixing of the gauge symmetry Diff F ðMÞ, and S is the mostgeneral action compatible with the requirements of gaugesymmetry (and further restricted by unitarity). As is oftenthe case, this path integral is interpreted as the analyticcontinuation of the theory which has been Wick rotated toimaginary time ¼ it.Our next step is to construct the action S compatible withour symmetry requirements. For simplicity, we will assumethat all global topological effects can be ignored, freelydropping all total derivative terms and not discussing possibleboundary terms in the action. This is equivalent toassuming that our space is compact and its tangentbundle topologically trivial. The refinement of our constructionwhich takes into account global topology andboundary terms is outside of the scope of the present work.1. The kinetic termThe kinetic term in the action will be given by the mostgeneral expression which is (i) quadratic in first timederivatives _g ij of the spatial metric, and (ii) invariant underthe gauge symmetries of foliation-preserving diffeomorphismsDiff F ðMÞ. The object that transforms covariantlyunder Diff F ðMÞ is not _g ij , but instead the second fundamentalformK ij ¼ 12N ð _g ij r i N j r j N i Þ: (18)This tensor measures the extrinsic curvature of the leavesof constant time in the spacetime foliation F . In terms ofK ij and its trace K g ij K ij , the kinetic term is given by084008-4