Quantizing Horava-Lifshitz Gravity via Causal Dynamical Triangulations
Christian Anderson, Steven Carlip, Joshua H. Cooperman, Petr Horava, Rajesh Kommu, Patrick R. Zulkowski
QQuantizing Hoˇrava-Lifshitz Gravityvia Causal Dynamical Triangulations
Christian Anderson a ∗ , Steven J. Carlip b † , Joshua H. Cooperman b ‡ ,Petr Hoˇrava c,d § , Rajesh K. Kommu b ¶ , and Patrick R. Zulkowski c,d (cid:107) a Department of Physics, Harvard University, Cambridge, MA 02138 b Department of Physics, University of California, Davis, CA 95616 c Berkeley Center for Theoretical Physics and Department of Physics,University of California, Berkeley, CA 94720 d Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
Abstract
We extend the discrete Regge action of causal dynamical triangulations to include discrete versions ofthe curvature squared terms appearing in the continuum action of (2+1)-dimensional projectable Hoˇrava-Lifshitz gravity. Focusing on an ensemble of spacetimes whose spacelike hypersurfaces are 2-spheres, weemploy Markov chain Monte Carlo simulations to study the path integral defined by this extended discreteaction. We demonstrate the existence of known and novel macroscopic phases of spacetime geometry,and we present preliminary evidence for the consistency of these phases with solutions to the equationsof motion of classical Hoˇrava-Lifshitz gravity. Apparently, the phase diagram contains a phase transitionbetween a time-dependent de Sitter-like phase and a time-independent phase. We speculate that thisphase transition may be understood in terms of deconfinement of the global gravitational Hamiltonianintegrated over a spatial 2-sphere. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] (cid:107) [email protected] a r X i v : . [ h e p - t h ] N ov Connections
An intriguing body of evidence hinting at a deep connection between Hoˇrava-Lifshitz gravity and causaldynamical triangulations has recently accumulated in the literature. First, one of us demonstrated consis-tency of the spectral dimension computed in Hoˇrava-Lifshitz gravity with the spectral dimension measured incausal dynamical triangulations in 3 + 1 dimensions [32]. Benedetti et al then verified within causal dynam-ical triangulations [19] the prediction of Hoˇrava-Lifshitz gravity for the behavior of the spectral dimensionin 2 + 1 dimensions [32]. Next, Ambjørn et al exhibited the remarkable resemblance of the phase diagramof causal dynamical triangulations [2] to the phase diagram of Lifshitz matter systems [29] as exemplified bythe Lifshitz scalar field [34]. These authors and one of us also conjectured [2, 34] that the apparent tricriticalpoint of the former phase diagram could correspond to the tricritical limit of Hoˇrava-Lifshitz gravity withdynamical critical exponent z equal to the dimension of space. Then, both Benedetti et al and Ambjørn etal noted the compatibility of certain solutions of Hoˇrava-Lifshitz gravity with the minisuperspace model fitto the expectation value of the geometries emerging from causal dynamical triangulations [2, 19]. Recently,Sotiriou et al successfully fit the behavior of the spectral dimension of causal dynamical triangulations atintermediate scales to a dispersion relation derived from Hoˇrava-Lifshitz gravity [48]. Just days ago, Buddargued that the kinetic term in the semiclassical effective action of causal dynamical triangulations exhibitsa Hoˇrava-Lifshitz-like form [22].This mounting evidence motivated us to extend the Regge action—or, more precisely, the discrete path in-tegral measure—used in causal dynamical triangulations to include dependence on the broader class of termsappearing in the action of Hoˇrava-Lifshitz gravity. Our purpose is threefold: firstly, to test the applicabilityof causal dynamical triangulations to modified classical theories of gravitation; secondly, to explore quan-tum Hoˇrava-Lifshitz gravity with nonperturbative techniques; and, thirdly, to illuminate further the linksbetween Hoˇrava-Lifshitz gravity and causal dynamical triangulations. As an initial step toward these goals,we have begun to investigate an appropriate reduction of (2 + 1)-dimensional projectable Hoˇrava-Lifshitzgravity using causal dynamical triangulations. This model, though removed from the phenomenologicallyinteresting case of 3 + 1 dimensions, serves as a simplified yet nontrivial beginning for our research.After briefly introducing the formalisms of Hoˇrava-Lifshitz gravity and causal dynamical triangulations insection 2, we explain our adaptation of causal dynamical triangulations to Hoˇrava-Lifshitz gravity in section3. We present the results of our initial numerical studies—evidence for the existence of extended phases ofgeometry, the structure of these phases, and their relation to the classical solutions—in section 4. Finally, wesummarize our conclusions and discuss ongoing and future work in section 5. To streamline our presentation,we relegate to appendix A the construction of the relevant classical solutions of Hoˇrava-Lifshitz gravity andto appendix B certain geometric properties of causal dynamical triangulations. Hoˇrava-Lifshitz gravity is a field theory of the dynamical metric on spacetime manifolds that carry theadditional structure of a preferred foliation by spacelike hypersurfaces. The relevant set of gauge symmetriesis then the group Diff F ( M ) of diffeomorphisms of the spacetime manifold M that preserve the preferredfoliation F . The foliation structure leads to one important novelty: the possibility of anisotropic scalingwith a nontrivial dynamical scaling exponent z measuring the degree of anisotropy between space and time.In Minkowski spacetime the relativistic scaling is thus replaced by the anisotropic scaling t −→ ˜ t = b z t (2.1a) x −→ ˜ x = b x (2.1b)for constant b >
0. Starting with z > We have decided, by a majority vote among the coauthors of this paper, to follow the terminology commonly accepted inthe literature and refer to gravity models with anisotropic scaling as Hoˇrava-Lifshitz gravity, despite the dissenting vote of oneof us (PH). M ) of full spacetime diffeomorphisms. This procedure leads to a reformula-tion of the model as a specific scalar-tensor theory with higher-derivative interactions in which the spacelikehypersurfaces of constant scalar field dynamically determine the leaves of the foliation F . Such a relativisticrewriting is only equivalent to the original nonrelativistic formulation at the classical level with, moreover,subtle regularity conditions on the scalar field’s dynamics. At the quantum level the relativistic rewrit-ing immediately besets the theory with the notorious problem of time, whereas the nonrelativistic natureof the original formulation of Hoˇrava-Lifshitz gravity potentially renders the spacetime metric’s dynamicsmore directly compatible with quantum mechanics. Additionally, as we shall show, the original formulationof Hoˇrava-Lifshitz gravity is nicely suited to the framework of causal dynamical triangulations, which alsoutilizes a preferred foliation structure in the microscopic definition of the path integral for gravity.In a local smooth coordinate chart ( t, x ) adapted to the preferred foliation F , Diff F ( M ) consists of allreparametrizations of the form t −→ ˜ t = f ( t ) (2.2a) x −→ ˜ x = ζ ( t, x ) (2.2b)for arbitrary functions f and ζ . Note that, although reparametrizations of the space coordinates may betime-dependent, reparametrizations of the time coordinate must be space-independent. Given its preferredfoliation, the structure of Hoˇrava-Lifshitz gravity is naturally discussed in the Arnowitt-Deser-Misner for-malism [18]. In this formalism the spacetime metric tensor g ( t, x ) is decomposed into the metric tensor γ ( t, x ) on a spacelike hypersurface Σ of constant time coordinate t , the shift vector N ( t, x ), and the lapsefunction N ( t, x ) such that one can reassemble the standard line element asd s = − N ( t, x )d t + γ ij ( t, x ) (cid:2) d x i + N i ( t, x )d t (cid:3) (cid:2) d x j + N j ( t, x )d t (cid:3) . (2.3)Note, however, that recombining the spatial metric tensor, the shift vector, and the lapse function intothe spacetime line element (2.3) is somewhat misleading since, in the regime with z >
1, different termscontributing to d s carry different scaling dimensions.The shift vector and the lapse function play the role of gauge fields associated with Diff F ( M ). Sincethe time coordinate reparametrizations (2.2a) are independent of the space coordinates, a natural choiceis to restrict the corresponding gauge field N ( t, x ) also to be only a function of the time coordinate. Wechoose to make this restriction, yielding the so-called projectable version of the theory. (See, for instance,[31, 20, 46, 52, 53].) Of course, we have interest in studying the more general nonprojectable version in whichthe lapse function is a spacetime field and of which the projectable version is a dynamical limit. The gaugesymmetries then permit new terms in the action involving spatial derivatives of the lapse function; findingthe appropriate realization of such terms in the framework of causal dynamical triangulations is beyond thepresent work’s scope. We instead focus on the projectable version of Hoˇrava-Lifshitz gravity, which has aparticularly clear translation into the language of causal dynamical triangulations.Employing the Arnowitt-Deser-Misner decomposition of the metric tensor, we now construct the actionof projectable Hoˇrava-Lifshitz gravity. We aim to build a power-counting renormalizable unitary classicaltheory of gravitation. Even in relativistic gravity, including higher curvature terms can render the theoryrenormalizable [26, 50]; typically, however, such terms come at the cost of sacrificing perturbative unitarityand propagating unphysical degrees of freedom [49]. These issues stem from the inclusion of higher temporalderivative terms, which necessarily accompany higher spatial derivative terms in a relativistic theory. Byonly permitting higher spatial derivatives, we can in principle avoid problems with perturbative unitarityand unphysical degrees of freedom. Working in the Arnowitt-Deser-Misner formalism allows for the ratherstraightforward inclusion of higher spatial derivatives and exclusion of higher temporal derivatives as wedesire in constructing the action of Hoˇrava-Lifshitz gravity.We build the general action in d spatial dimensions as the sum of a kinetic term quadratic in temporalderivatives and a potential term of mass dimension 2 d in spatial derivatives. The dynamical critical exponent z thus equals d . Accordingly, we form the kinetic term from invariants of ∂ t γ ( t, x ). This derivative aloneis not covariant under Diff F ( M ), but the extrinsic curvature tensor K ( t, x ) of a spacelike hypersurface Σsatisfies this criterion. With K ( t, x ) having components K ij ( t, x ) = 12 N ( t ) [ ∂ t γ ij ( t, x ) − ∇ i N j ( t, x ) − ∇ j N i ( t, x )] (2.4)3or the covariant derivative ∇ associated with γ ( t, x ), the most general kinetic term invariant under Diff F ( M )is 116 πG (cid:90) M d t d d x (cid:112) γ ( t, x ) N ( t ) (cid:2) K ij ( t, x ) K ij ( t, x ) − λK ( t, x ) (cid:3) . (2.5)Here the parameter λ arises from the generalized DeWitt supermetric compatible with the theory’s gaugesymmetries, and K ( t, x ) is the trace of the extrinsic curvature tensor K ( t, x ) [30, 31].We form the potential term from invariants of γ ( t, x ) and its spatial derivatives. The most generalpotential term is 116 πG (cid:90) M d t d d x (cid:112) γ ( t, x ) N ( t ) V [ γ ( t, x )] , (2.6)where V [ γ ( t, x )] is a scalar functional of the spatial metric tensor and its spatial derivatives up to order 2 d .Putting together (2.5) and (2.6), the action of projectable Hoˇrava-Lifshitz gravity is S HL [ g ( t, x )] = 116 πG (cid:90) M d t d d x (cid:112) γ ( t, x ) N ( t ) (cid:8) K ij ( t, x ) K ij ( t, x ) − λK ( t, x ) − V [ γ ( t, x )] (cid:9) . (2.7)Note that the coupling constant G is related but not equal to the Newton constant G N : the low energyNewton constant G N is determined by rescaling the time coordinate by an effective speed of light factorselected so that the term linear in the spatial Ricci scalar within V [ γ ( t, x )] is correctly normalized withrespect to the kinetic term [30, 31]. This rescaling also dictates how the constant term in V [ γ ( t, x )] relatesto the low energy cosmological constant Λ.For d >
2, with the anticipated z = d scaling at short distances, there is a proliferation of marginal andrelevant contributions to the potential term; for d = 2, however, the most general action containing only themarginal and relevant terms for z = 2 is quite compact: S HL [ g ( t, x )] = 116 πG (cid:90) M d t d x (cid:112) γ ( t, x ) N ( t ) (cid:2) K ij ( t, x ) K ij ( t, x ) − λK ( t, x ) − αR ( t, x ) + βR ( t, x ) − (cid:3) (2.8)with R ( t, x ) the Ricci scalar of the spatial metric tensor γ ( t, x ) and the coupling constant Λ related to thelow energy cosmological constant Λ by the rescaling described above. The potential term’s relative simplicityin (2 + 1)-dimensional projectable Hoˇrava-Lifshitz gravity further motivates our initially studying this case.The equations of motion (A.2) derived from variation of the action (2.8) with respect to the spatialmetric tensor are independent of the coupling constant β : for d = 2 the R term is a total derivative, andthe Gauss-Bonnet theorem determines its integral over a spacelike hypersurface Σ of the preferred foliationin terms of the Euler number of Σ. Similarly, the equations of motion (A.1) obtained from variation of theaction (2.8) with respect to the shift vector are also insensitive to β . The coupling constant β does, however,appear in the equation of motion derived from variation of the action (2.8) with respect to the lapse function.Since the lapse function only depends on the time coordinate, its equation of motion takes the form of aspatially integrated Hamiltonian constraint H ⊥ = 0 (2.9)for the global Hamiltonian H ⊥ = (cid:90) Σ d x (cid:112) γ ( t, x ) (cid:2) K ij ( t, x ) K ij ( t, x ) − λK ( t, x ) + αR ( t, x ) − βR ( t, x ) + 2Λ (cid:3) . (2.10)At this stage there is an additional choice to make beyond that of restricting to projectable Hoˇrava-Lifshitz gravity. Recall the distinction that arises between noncompact and compact spatial topology whenimposing the local Hamiltonian constraint in relativistic theories of gravity. In the former case, after imposingappropriate asymptotic fall-off conditions and accounting for possible boundary contributions, the zero mode H ⊥ of the local Hamiltonian constraint does not vanish on physical states. Instead, this mode is one of theconserved asymptotic charges, namely the total energy. In the latter case the zero mode H ⊥ of the localHamiltonian constraint vanishes on all physical states since the relativistic gauge symmetries mix it togetherwith the other gauge symmetry generators. 4igure 1: From left to right: the (3 , , , H ⊥ on physical states is not mandatory: eitherone may treat H ⊥ as a gauge symmetry generator and thus impose the condition H ⊥ = 0, or one maytreat H ⊥ as a conserved global charge measuring the energy levels of physical states in the Hilbert space.Of course, the second option requires us to check that this Hamiltonian’s spectrum is bounded from belowon physical states. This novel situation was explicitly encountered in [54] whose authors constructed frombosonic systems on a rigid lattice the renormalization group fixed points corresponding to the free field limitof Hoˇrava-Lifshitz gravity with z = 2 and z = 3. There the specific choices that one could make at the levelof the microscopic lattice Hamiltonian gave rise to the option of imposing or not imposing the condition H ⊥ = 0 for the effective low energy gravitons.When we discretize the action (2.8) in section 3 for use as a functional on simplicial manifolds, we findthat the second option—treating H ⊥ as a conserved global charge—appears more natural in the setting ofcausal dynamical triangulations. The geometric restrictions derived by employing simplices of fixed edgelengths suggest that the lapse function has been effectively set to a constant, though without leading in anydiscernible way to the imposition of the constraint (2.9). Also, the R term with coupling constant β inthe action (2.8) does not affect any observables for the topological reasons stated above. The value of H ⊥ does, however, depend on β , which suggests that enforcing the constraint (2.9) is not completely consistent.In section 4.3, where we compare the geometries emerging from our numerical simulations to the classicalsolutions for the constraint (2.9) not imposed, we find evidence supporting our selection of the second option.These comparisons are, moreover, not without weight: as briefly discussed in [34] and further demonstratedin [35], the classical phase diagram for the theory with variable lapse differs significantly from that of thepresent setting. The classical theory that we quantize in section 4 is thus defined by the action (2.8) forfixed lapse. The causal dynamical triangulations approach aspires to define the continuum limit of a quantum theory ofgravity by appealing solely to those nonperturbative tools applied with great success to the quantization oflocal gauge field theories. (See [38, 39] for reviews.) In particular, the approach invokes a lattice regularizationof the path integral for gravity and then utilizes the renormalization group to explore its continuum limit.In causal dynamical triangulations one thus attempts to define this path integral as the partition function ofa statistical model of dynamical geometry. (See [3, 16] for recent reviews.) As in several previous programsof lattice quantization of gravity, the statistical ensemble of geometries is comprised of simplicial manifoldstriangulated by a fixed set of ( d + 1)-simplices. In 2 + 1 dimensions this set consists of those 3-simplices ortetrahedra pictured in figure 1.The novel feature of causal dynamical triangulations lies in the imposition of an additional restriction onthe geometries permitted to enter the path integral: these simplicial manifolds must possess a global foliationby spacelike hypersurfaces of constant discrete time. The global foliation was originally introduced to enablea Wick rotation from Lorentzian to Riemannian signature. The causal structure of a Lorentzian triangulation5ould thus be faithfully retained in its Wick rotated version, only the latter triangulation being amenable tonumerical analysis [7, 8, 10]. A simplicial manifold is endowed with the foliation structure as follows: everyspacelike hypersurface, all of a chosen fixed topology, is triangulated with equilateral d -simplices, and thenthe vertices of adjacent spacelike hypersurfaces are connected by timelike edges so as to produce only the( d + 1)-simplices of the fixed set. Spacelike edges have length squared l SL = a , defining a lattice spacingfor the triangulation, and timelike edges have length squared l T L = − ηa for η > η corresponds to the parameter α typically used in the causal dynamical triangulations literature.More concretely, one aims to approximate the path integral for general relativity, Z [ γ ( t f , x ) | γ ( t i , x )] = (cid:90) γ ( t f , x ) γ ( t i , x ) D g ( t, x ) e iS EH [ g ( t, x )] , (2.11)by the path sum Z [Γ f | Γ i ] = (cid:88) T C T e iS R [ T ] . (2.12)In (2.11) the path integral is taken over all physically distinct spacetime metric tensors g ( t, x ) interpolatingbetween the initial and final boundary geometries specified by the spatial metric tensors γ ( t i , x ) and γ ( t f , x ),and S EH denotes the Einstein-Hilbert action S EH [ g ( t, x )] = 116 πG N (cid:90) M d t d d x (cid:112) − g ( t, x ) [ R ( t, x ) − . (2.13)In (2.12) the path sum is taken over all causal triangulations T interpolating between the initial and finalboundary geometries specified by the triangulations Γ i and Γ f [7, 8], and S R denotes the Regge action S R [ T ] = 18 πG N (cid:88) h ∈T A h δ h − Λ16 πG N (cid:88) s ∈T V s . (2.14)Here, h is a ( d + 1 − A h and deficit angle δ h , and V s is the spacetime volumeof a ( d + 1)-simplex [51]. The measure factor C T is the inverse of the order of the automorphism group of T , included to account for discrete symmetries arising in the structure of T .Before proceeding further with our discussion of causal dynamical triangulations, we make a few clarifyingremarks on the implications of the above prescription for regularizing the continuum path integral (2.11). Therestriction to simplicial manifolds admitting a preferred foliation effectively changes the path integral measureand integration domain. Even though we continue to employ the Regge action, these changes may resultin our model belonging to a universality class different from that of Euclidean dynamical triangulations. In1+1 dimensions, where the path sum (2.12) can be evaluated analytically, this is known to be the case [1]: bypreventing the birth of baby universes, the foliation requirement alters the critical exponents, indicating thatEuclidean and causal dynamical triangulations occupy different universality classes. In higher dimensionsthe same situation seems very likely to hold true; otherwise, we should expect our model to fall into one ofthe universality classes in which no smooth macroscopic limit exists, and the spacetime geometry exhibits abranched polymer or crumpled behavior.The change in the path integral measure and integration domain may also translate into a change in ourtheory’s effective action such that it no longer assumes the general relativistic form. Recall that, althoughwe may locally treat the choice of a preferred foliation as a gauge choice, we cannot generally make sucha choice globally. A gauge choice typically introduces a Jacobian into the path integral measure, and amismatch between the path integral measure and the gauge choice can lead to a breaking of the gaugesymmetry. Starting with the Regge action, the question of whether counterterms sensitive to the preferredfoliation are generated thus remains open. If such counterterms appear, then the resulting effective actionlikely corresponds to Hoˇrava-Lifshitz gravity at some particular values of its couplings. The Regge action’sbare coupling constants G N and Λ are only indirectly related to the continuum renormalized values of theNewton constant and the cosmological constant at long distances, so we must not naively identify the formerwith the latter.Continuing with our discussion of causal dynamical triangulations, the action (2.14) simplifies consider-ably for the simplicial manifolds contributing to our ensemble since only a small set of simplices with fixed6eometries is used to construct them. Still, the path sum (2.12) has resisted all attempts at analytical compu-tation for d >
1, so exploration of its properties has primarily employed numerical techniques. In particular,Markov chain Monte Carlo methods are used to simulate the path sum (2.12) [4, 5, 9, 10, 11, 12, 14, 15, 24, 40].To render the path sum (2.12) amenable to such analysis, we must first Wick rotate the real time action(2.14) to imaginary time, a well-defined process owing to the foliated structure of causal dynamical triangu-lations. We perform the Wick rotation by analytically continuing the parameter η in the lower half complexplane [8]. For our case of interest—2 + 1 dimensions with spacelike hypersurfaces having the topology of S and periodic time coordinate having the topology of the 1-sphere S —the Regge action becomes S ( E ) CDT = − k N + k N (2.15)after Wick rotation and application of various topological relations for η = 1 [9]. Here, N is the numberof vertices and N is the number 3-simplices in the triangulation T . The coupling constants k and k arerelated to the bare couplings G N and Λ as k = a G N (2.16a) k = a Λ48 √ πG N + a G N (cid:18) π cos − − (cid:19) , (2.16b)We have thus transformed the path sum (2.12) into the statistical partition function Z [Γ f | Γ i ] = (cid:88) T C T e − S ( E ) CDT [ T ] . (2.17)Computer simulations of the partition function (2.17) in both 2 + 1 and 3 + 1 dimensions have thusfar provided considerable support for the existence of an extended phase of geometry possessing not only asemiclassical limit on large scales, but also a quantum regime on small scales [4, 9, 10, 11, 12, 13, 14, 15, 19,24, 40]. In particular, the average observed geometry matches well that of (possibly deformed) Euclidean deSitter spacetime at both the classical and semiclassical levels [4, 9, 10, 11, 12, 13, 14, 15, 19, 24]. Furthermore,studies of the spectral dimension of this extended phase of geometry have revealed an apparent dimensionalreduction to effective 2-dimensionality on small scales [13, 19, 40]. This phenomena of dynamical dimensionalreduction—particularly, the extrapolated value of the minimal dimensionality—has elicited comparisons ofcausal dynamical triangulations to both the asymptotic safety approach and Hoˇrava-Lifshitz gravity. Inthe former theory the effective change in the spectral dimension apparently results from large anomalousdimensions near the nontrivial fixed point, while in the latter theory the spectral dimension flows to theshort distance value of 2 as a result of the anisotropic scaling near the Gaussian fixed point [32, 42]. Furtherevidence for such dimensional reduction has also surfaced in other approaches to the quantization of generalrelativity [23].As in Euclidean dynamical triangulations, additional phases of geometry also emerge [43]. At firstthese phases were viewed as unphysical, but current interpretations favor their role as further phases in thevicinity of a multicritical fixed point [2]. Specifically, in both 2 + 1 and 3 + 1 dimensions there exists aphase characterized by spacelike hypersurfaces that effectively decouple from one another [9, 12, 40], and in3 + 1 dimensions a second additional phase, distinguished by its large Hausdorff dimension, appears [12, 40].Recently, Ambjørn et al have argued that the transition between this last phase and the physical phaseis of second order [5]. This finding raises the possibility of rigorously defining a continuum limit of causaldynamical triangulations. We now derive a discrete form of the action (2.8) suitable for analysis with the techniques of causal dynamicaltriangulations. As above we assume a topology of S × S , primarily motivated by the relative ease ofnumerically analyzing such compact spacetimes. As in the lattice regularizations of local quantum fieldtheories [38, 39], the precise details of the discretization do not matter since universality ensures that manyof the details at the lattice spacing scale become irrelevant in the long distance limit. Our primary goal in7onstructing a discrete analogue of the action (2.8) is thus to build an action sufficiently specific so that thecontinuum limit lies in the universality class of Hoˇrava-Lifshitz gravity yet sufficiently generic so that thecontinuum limit does not depend on all of the renormalized coupling constants in this universality class. Ofcourse, we can only judge whether or not we have achieved these goals once we have thoroughly studied thequantum theory of the model defined below.Working from the continuum action (2.8), we allow two technical criteria to guide our construction ofits discrete version: first, the discrete action should manifestly reduce to the Regge action used in causaldynamical triangulations when the bare coupling constants λ and α assume their general relativistic values,and, second, the transfer matrix corresponding to the discrete action defined on the space of boundarygeometries should yield a well-defined Hamiltonian. These two criteria apply to the classical discrete actionconstructed below. In the quantum theory defined via the path integral based on this action, we naturallyanticipate a nontrivial relationship between the bare coupling constants and the renormalized couplingconstants. Depending on this relationship, the quantum theory’s long distance continuum limit may or maynot be relativistic.To implement the first criterion, we use the Gauss-Codazzi equation, R ( t, x ) = R ( t, x ) − (cid:2) K ( t, x ) − K ij ( t, x ) K ij ( t, x ) (cid:3) + Total Derivative , (3.1)to rewrite the action (2.8) as S HL [ g ( t, x )] = 116 πG (cid:90) M d t d x (cid:112) − g ( t, x ) [ R ( t, x ) − − λ πG (cid:90) M d t d x (cid:112) γ ( t, x ) N ( t ) K ( t, x ) − α πG (cid:90) M d t d x (cid:112) γ ( t, x ) N ( t ) R ( t, x ) (3.2)up to irrelevant boundary terms. We neglect the term in the action (2.8) with coupling constant β as itcontributes only an additive constant for the spacetime manifolds under consideration. In this form theaction (3.2) straightforwardly reduces to the Einstein-Hilbert action (2.13) when the coupling constants λ and α take on their general relativistic values of one and zero. We use the discrete action (2.15) for theEinstein-Hilbert portion of the action (3.2), leaving us the task of determining the discrete analogues of the K and R terms. We construct these terms below, postponing discussion of the second criterion. There exist well-established prescriptions for constructing the Ricci scalar and the trace of the extrinsiccurvature tensor in Regge calculus. The former is defined in terms of deficit angles δ h about ( d + 1 − h of a ( d + 1)-dimensional simplicial manifold [51]: (cid:90) M d t d d x (cid:112) − g ( t, x ) R ( t, x ) = 2 (cid:88) h ∈T A h δ h . (3.3)The latter is defined in terms of angles ψ h between the normal vectors to the two d -simplices intersecting atthe ( d + 1 − h within the spacelike hypersurface Σ of a ( d + 1)-dimensional simplicialmanifold [21, 28]: (cid:90) Σ d d x (cid:112) γ ( t, x ) K ( t, x ) = (cid:88) h ∈ Σ A h ψ h . (3.4)Technically, these curvatures have a distributional definition on the hinges, which complicates the construc-tion of curvature squared terms. In particular, simply taking the square of (3.3) or (3.4) to define the discreteversions of the R or K terms leads to a mathematically ill-defined continuum limit [6]. Accordingly, weadhere to the philosophy of [6, 27] when building discrete analogues of curvature squared terms, adoptingthe alternative scheme of volume sharing. As discussed in [6], we make the identification (cid:90) Σ d d x (cid:112) γ ( t, x ) R ( t, x ) −→ (cid:88) o ∈ O τ ( T ) V ( s ) o (cid:18) δ o V o V ( s ) o (cid:19) (3.5)8or a curvature scalar R ( t, x ), where o is the object assigned the curvature density, O τ ( T ) is the set of allobjects o on the spacelike hypersurface Σ labelled by discrete time coordinate τ in the triangulation T , V o isthe appropriate volume of the object o , and V ( s ) o is the share-volume of the object o , namely the volume ofall top-dimensional objects containing o . Using this scheme we now address the R and K terms in turn. R Term
For the R term there are clear choices for the objects o and the top-dimensional objects containing o : sincethe Ricci scalar characterizes the intrinsic geometry of a 2-dimensional spacelike hypersurface, the objects o are vertices v and the top-dimensional objects are spacelike triangles (cid:52) . This is completely consistent withthe usual prescription for the Ricci scalar in Regge calculus. For a vertex v the deficit angle is δ v = 2 π − π N (cid:52) ( v ) (3.6)for the number N (cid:52) ( v ) of spacelike triangles containing v , the volume is V v = 1 , (3.7)and the share-volume is V ( s ) v = (cid:88) (cid:52)⊃ v A (cid:52) = √ a N (cid:52) ( v ) (3.8)since all of the spacelike triangles are equilateral. Employing (3.5) and noting that the most natural dis-cretization of the time integral is (cid:90) d t N ( t ) −→ (cid:88) τ √ ηa, (3.9)we make the identification (cid:90) M d t d x (cid:112) γ ( t, x ) N ( t ) R ( t, x ) −→ (cid:88) τ (cid:88) v ∈ V τ ( T ) √ ηa (6 − N (cid:52) ( v )) N (cid:52) ( v ) . (3.10)up to multiplicative factors, where V τ ( T ) denotes the set of vertices v belonging to the spacelike hypersurfacelabeled by the discrete time coordinate τ . K Term
For the K term the choices of objects o and top-dimensional objects containing o are not as clear. Theextrinsic curvature captures in part how the spacelike hypersurface is embedded in the spacetime manifold.The discrete analogue of the K term must reflect this geometric information, requiring that we appropriatelyaccount for how tetrahedra connect to the spacelike hypersurface. This observation suggests that we taketetrahedra as the top-dimensional objects contributing to the share-volume. Furthermore, in the continuum K scales as an inverse length squared, implying that the objects o be spacelike triangles. We thus need toassign a deficit angle δ (cid:52) to a spacelike triangle. We largely follow the treatment of [37]. Consider a spacelikehypersurface of the triangulation T . For each spacelike triangle we may define a future-directed normalvector at the center of the (3 , e between two adjacent such spacelike triangles is the deficit of the anglethat the normal vector traces out as it is parallel transported from its own (3 , , , δ e = 1 i (cid:16) π − θ (3 , L − θ (2 , L N ↑ (2 , ( e ) (cid:17) (3.11)with θ (3 , L and θ (2 , L the Lorentzian dihedral angles about spacelike edges and N ↑ (2 , ( e ) the number of future-directed (2 , e . We give the values of θ (3 , L and θ (2 , L in appendix9igure 2: An embedding in three dimensions of two (3 , , , π − θ (3 , L − θ (2 , L as it is parallel transported across the commonedge.B. We thus assign to a spacelike triangle the deficit angle δ (cid:52) = 1 i (cid:16) π − θ (3 , L − θ (2 , L N ↑ (2 , ( (cid:52) ) (cid:17) , (3.12)which, we note, is reminiscent of a trace. The volume of a spacelike triangle is V (cid:52) = √ a , (3.13)and the share volume is V ( s ) (cid:52) = 4 V (3 , L + V (2 , L N ↑ (2 , ( (cid:52) ) (3.14)with V (3 , L and V (2 , L the Lorentzian 3-volumes of the respective tetrahedra. We give the values of V (3 , L and V (2 , L in appendix B. The share volume assumes this value since a given spacelike triangle has four (3 , N ↑ (2 , ( (cid:52) ) (2 , (cid:90) M d t d x (cid:112) γ ( t, x ) N ( t ) K ( t, x ) −→ (cid:88) τ (cid:88) (cid:52)∈ T SLτ ( T ) a (cid:12)(cid:12) π − θ (3 , L − θ (2 , L N ↑ (2 , ( (cid:52) ) (cid:12)(cid:12) V (3 , L + V (2 , L N ↑ (2 , ( (cid:52) ) (3.15)up to multiplicative factors, where T SLτ ( T ) denotes the set of spacelike triangles (cid:52) belonging to the spacelikehypersurface labelled by the discrete time coordinate τ .The discretization (3.15) of the K term does not, however, respect our second criterion. Following [8], toensure the existence of a well-defined Hamiltonian on the space of boundary geometries, we must make (3.15)time-reversal invariant. A straightforward calculation shows that this invariance guarantees positivity of thesquared transfer matrix, which, along with the transfer matrix’s symmetry, yields a well-defined Hamiltonian[8]. To realize time-reversal invariance, we add an analogous term for past-directed (2 , K term is (cid:88) τ (cid:88) (cid:52)∈ T SLτ ( T ) a (cid:12)(cid:12) π − θ (3 , L − θ (2 , L N ↑ (2 , ( (cid:52) ) (cid:12)(cid:12) V (3 , L + V (2 , L N ↑ (2 , ( (cid:52) ) + (cid:12)(cid:12) π − θ (3 , L − θ (2 , L N ↓ (2 , ( (cid:52) ) (cid:12)(cid:12) V (3 , L + V (2 , L N ↓ (2 , ( (cid:52) ) . (3.16)Note that we did not require such an adjustment for the R term since it depends only on the intrinsicgeometry of the spacelike hypersurface. 10 .2 Imaginary Time Action Putting together (3.10) and (3.16), our discrete action for Hoˇrava-Lifshitz gravity becomes S HL [ T ] = S CDT [ T ]+ 1 − λ πG (cid:88) τ (cid:88) (cid:52)∈ T SLτ ( T ) a (cid:12)(cid:12) π − θ (3 , L − θ (2 , L N ↑ (2 , ( (cid:52) ) (cid:12)(cid:12) V (3 , L + V (2 , L N ↑ (2 , ( (cid:52) ) + (cid:12)(cid:12) π − θ (3 , L − θ (2 , L N ↓ (2 , ( (cid:52) ) (cid:12)(cid:12) V (3 , L + V (2 , L N ↓ (2 , ( (cid:52) ) − α πG (cid:88) τ (cid:88) v ∈ V τ ( T ) √ ηa (6 − N (cid:52) ( v )) N (cid:52) ( v ) . (3.17)Wick rotating the action (3.17) to imaginary time, we find that S ( E ) HL [ T ] = − k N + k N + 1 − λ πG (cid:88) τ (cid:88) (cid:52)∈ T SLτ ( T ) a (cid:16) π − θ (3 , E − θ (2 , E N ↑ (2 , ( (cid:52) ) (cid:17) V (3 , E + V (2 , E N ↑ (2 , ( (cid:52) ) + (cid:16) π − θ (3 , E − θ (2 , E N ↓ (2 , ( (cid:52) ) (cid:17) V (3 , E + V (2 , E N ↓ (2 , ( (cid:52) ) + α πG (cid:88) τ (cid:88) v ∈ V τ ( T ) √ ηa (6 − N (cid:52) ( v )) N (cid:52) ( v ) . (3.18)We give the values of θ (3 , E , θ (2 , E , V (3 , E , and V (2 , E in appendix B. We use the action (3.18) for a = 1 and η = 1 in the path integral quantization analyzed below. The value of the length a has no a priori physicalmeaning, and any value of the parameter η > is permitted. Following the analyses of [4, 5, 9, 10, 11, 12, 14, 15, 24, 40], we explore the partition function Z [Γ f | Γ i ] = (cid:88) T C T e − S ( E ) HL [ T ] (4.1)employing Markov chain Monte Carlo methods. In our simulations so far, we have fixed the topology ofthe spacelike hypersurfaces to be that of S and have fixed the total number T of spacelike hypersurfaces,introducing a discrete time coordinate τ that enumerates these hypersurfaces. In the simulations reportedbelow, we have set T = 64. Additionally, we impose periodic boundary conditions on this time coordinate,endowing it with the topology of S . Consequently, the initial triangulated spacelike hypersurface Γ i and thefinal triangulated spacelike hypersurface Γ f of each causal triangulations T entering the partition function(4.1) are identified.Next, we hold the number N of tetrahedra in each triangulation T approximately fixed; otherwise, theweights e − S ( E ) HL [ T ] appearing in the partition function (4.1) may grow without bound, eventually causing ourcomputer code to crash. To implement the constraint of fixed number of tetrahedra, we add to the action(3.18) a term of the form (cid:15) | N − ¯ N | , where (cid:15) is a Lagrange multiplier and ¯ N is the target number oftetrahedra. Essentially, this Lagrange multiplier term modifies the value of the coupling constant k . In allof the simulations reported below, we have set (cid:15) = 0 .
02 and ¯ N to approximately 10200. These parametervalues typically yield a one percent variation in N over any given ensemble. This number of tetrahedra issufficiently large for physical effects clearly to outweigh finite size effects and sufficiently small for our limitedcomputing resources to survey a reasonable portion of the coupling constant space.Finally, we tune to a set of coupling constants { k , λ, α, k c ( k , λ, α ) } on the critical surface of the couplingconstant space defined by the partition function (4.1) for a fixed N . As our notation suggests, we first selectvalues for k , λ , and α and then tune to the associated critical value k c of k . The critical value k c is that11or which N remains approximately constant while a simulation runs. In this sense our model is only well-defined at the critical value: for any other value the number of tetrahedra either increases without bound orplummets to zero.With these conditions established, a simulation begins with the generation of an initial triangulationhaving the topology S × S composed of ¯ N tetrahedra. Using the Pachner moves adapted to causaldynamical triangulations, as described, for instance, in [8], we run a standard Metropolis algorithm togenerate an ensemble of spacetimes representative of the weighting defined by the partition function (4.1),sampling only after a period of thermalization. We sample the representative spacetimes generated everyone hundred sweeps, a single sweep comprising ¯ N attempted Pachner moves. Once collected, we estimatethe expectation values of observables as averages over the ensemble.Testing our code is a nontrivial matter: without a known nonperturbative quantization of (2 + 1)-dimensional projectable Hoˇrava-Lifshitz gravity, we possess no standard of comparison for our results. Ofcourse, this situation also pertains to causal dynamical triangulations formulated with the Regge action. Ourcode is a modified version of that reported in [40], which has yielded independent corroboration of the resultsof [9, 10, 11, 12, 13, 14, 15, 19]. We have run the modified code at the general relativistic values of the couplingconstants λ and α to check that we correctly reproduce the results of these references. In figure 3 we presentdepictions of two representative spacetimes—one in the physical phase and one in the decoupled phase ofcausal dynamical triangulations—generated by our code. In figure 4 we plot the ensemble average spectraldimensions—discussed further below—for the two ensembles to which the representative spacetimes in figure3 belong. Up to finite size effects currently under investigation, these measurements agree quantitativelywith those of [19, 24, 40]. In figure 8( a ) below we also show the ensemble average discrete 2-volume asa function of discrete time for the second of these two ensembles. This data, as well as the fit to it, areconsistent with the findings of [9, 24]. Beyond these checks, we rely on the plausibility of our new results asa test of our code’s correctness. (a) (b) Τ N S L Τ N S L Figure 3: Depictions of representative spacetimes showing the number N SL of spacelike triangles as a functionof discrete time τ . (a) Phase A ( k = 6 . k = 1 . λ = 1 . α = 0 .
00) (b) Phase C ( k = 1 . k = 0 . λ = 1 . α = 0 . Our model’s coupling constant space is 4-dimensional. Based on the phase structures of both (2 + 1)- and(3+1)-dimensional causal dynamical triangulations, we expect that for fixed N our model is only well-defined12
100 200 300 4000.00.51.01.52.02.53.0 Σ < d S > (a) Σ < d S > (b) Figure 4: The ensemble average spectral dimension (cid:104) d s (cid:105) as a function of diffusion time σ . (a) Phase A( k = 6 . k = 1 . λ = 1 . α = 0 .
00) (b) Phase C ( k = 1 . k = 0 . λ = 1 . α = 0 . N increases without bound while the lattice spacing a vanishes such thatthe product N a remains constant. Supposing that our model possesses a second order phase transition,at which we could define its continuum limit, this transition must be located at a phase boundary on thecritical surface.Now, a 3-dimensional subspace of largely unknown extent represents a formidably expansive space toexplore numerically. Accordingly, we have limited our initial investigations to the subspace of the couplingconstant space consisting of the λ − α plane near the origin for fixed k . Specifically, we set k = 1 .
00, selectvalues for λ and α , and then tune to the value of k on the critical surface. With λ = 1 .
00 and α = 0 .
00 thiscorresponds to a point in the physical phase of causal dynamical triangulations for the Regge action.Within this region we have generated forty-seven ensembles of representative spacetimes, each for adifferent set { k , λ, α, k c ( k , λ, α ) } of the coupling constants. We display in figure 5 the critical surface andthe associated phase structure as ascertained thus far. Our explorations indicate the existence of threephases: a phase contiguous with the physical phase of causal dynamical triangulations for the Regge actionthat we call phase C; a phase emerging for sufficiently large values of λ that we call phase D; and a phaseemerging for sufficiently large values of α that we call phase E. We are not entirely certain that phasesD and E are distinct: our measurements might instead indicate modulation of a single phase across therelevant region in the λ − α plane, a possibility under active investigation. On the other hand, we are quitecertain that phases D and E are not artifacts: our simulations exhibit the characteristic lengthening of theautocorrelation time near the phase boundaries, and the geometric properties of phases D and E persistunder an increase in the total number of spacelike hypersurfaces. The presence of these novel phases ofextended geometry counts as the first explicit indication that the spatial curvature squared terms in theaction (3.18) exert a significant effect as opposed to being renormalized to irrelevance. This nicely matchesthe scaling behavior of the curvature squared terms expected from the analytic approach to Hoˇrava-Lifshitzgravity.In figure 6 we depict spacetimes representative of each of the three phases. Specifically, we plot thenumber N SL of spacelike triangles—a measure of the discrete 2-volume—as a function of discrete time τ .Representative spacetimes in phase C are characterized by a single correlated accumulation of tetrahedraspread across a substantial range of τ . Representative spacetimes in phase D are characterized by anintermittent series of accumulations of tetrahedra each spread across a small range of τ . Representativespacetimes in the phase E are characterized by a moderately uniform distribution of tetrahedra spreadacross the entire range of τ . We now present preliminary evidence suggesting that the ensemble average geometries in all three phasesshow signs of being both physical and semiclassical. This evidence stems from analyses of two geometricobservables associated with our ensembles of representative spacetimes: the spectral dimension as a functionof diffusion time and the discrete 2-volume as a function of discrete time. Based on these analyses, we drawcertain comparisons to the relevant solutions of the classical theory now in imaginary time presented in13 a) (b)
Figure 5: (a) The critical surface as explored thus far. (b) The critical surface projected onto the λ - α planeshowing phases C, D, and E respectively in blue circles, magenta squares, and orange diamonds.appendix A. The spectral dimension of a space measures its effective dimensionality as experienced by a random walker.We determine the ensemble average spectral dimension by the method described, for instance, in [19].Specifically, we directly measure the return probability P r ( σ ) as a function of diffusion time σ for eachrepresentative spacetime in a given ensemble. We then compute the ensemble average spectral dimension as (cid:104) d s ( σ ) (cid:105) = − (cid:104) P r ( σ ) (cid:105) d ln σ (4.2)employing an appropriate discretization of the derivative.In figure 7 we display plots of the ensemble average spectral dimension as a function of diffusion timefor each of the three phases. Before interpreting these plots, we should comment on the evident bifurcationfor small values of diffusion time. This effect reflects the discrete nature of our triangulated spacetimes:random walks of even and of odd lengths yield diverging estimates for the spectral dimension on scalessufficiently short in comparison to the discretization scale. Increasing the number of tetrahedra comprisingeach spacetime pushes the bifurcation scale towards smaller diffusion times, revealing the physical nature ofthe spectral dimension on such scales.Now compare the plots of figure 7 to those of ordinary causal dynamical triangulations in figure 4. Allthree of the Hoˇrava-Lifshitz spectral dimensions much more closely resemble that of figure 4(b)—for thephysical phase of causal dynamical triangulations—than that of figure 4(a)—for the decoupled phase ofcausal dynamical triangulations. The resemblance between the plots of figures 7(a) and 4(b) is particularlyclose, as we might expect, since the two ensembles are both in phase C. In particular, the spectral dimensionsshown in figure 7 all peak at values between 2 and 3 for small diffusion times and then gradually decay forlarge diffusion times. We expect this decay at large diffusion times: these spacetimes are compact and havecurvature.For (2 + 1)-dimensional causal dynamical triangulations the spectral dimension in the physical phasereaches 3 for ensembles characterized by larger values of N [19, 40]. Such measurements provide a key pieceof evidence for the semiclassical nature of the phase’s ensemble average geometry. This suggests that themaxima of the spectral dimension plots in figures 4(b) and 7(a) are depressed by finite size effects. Assumingthat such effects are comparable for our ensembles in phases D and E, we are led to conclude that the14 a) (b) (c) Τ N S L Τ N S L Τ N S L Figure 6: Depictions of representative spacetimes showing the number N SL of spacelike triangles as a functionof discrete time τ . (a) Phase C ( k = 1 . k = 0 . λ = 0 . α = 0 .
50) (b) Phase D ( k = 1 . k = 0 . λ = 3 . α = 0 .
00) (c) Phase E ( k = 1 . k = 0 . λ = 1 . α = 2 . N in the hope of resolving these issues. Nevertheless, we maintain that the similaritiesin form between the spectral dimensions depicted in figure 7 and in figure 4(b) hint at the physicality andsemiclassicality of phases C, D, and E. -Volume The foliated structure of causal triangulations allows for the measurement of certain quantities as functionsof the discrete time coordinate. The number N SL of spacelike triangles on a given spacelike hypersurface isone such quantity. Although the value of N SL on a single spacelike hypersurface is not physically meaningful,the set { N SL (Σ τ ) } Tτ =1 does contain physical information.For an ensemble in phase C, there exist straightforward techniques for performing a coherent ensembleaverage of { N SL (Σ τ ) } Tτ =1 . (See, for instance, [15].) These techniques rely on the characteristic feature ofphase C spacetimes: the single accumulation of tetrahedra as depicted in figures 3(b) and 6(a). Intuitively,each method functions to align an appropriately defined center of this accumulation with the central valueof the discrete time coordinate. Once accomplished for all of the representative spacetimes in an ensemble,the coherent ensemble average of { N SL (Σ τ ) } Tτ =1 is defined by the discrete timewise average.We employ a method that we call equal discrete 2-volume splitting. The name refers to the algorithmfor appropriately defining the center of the accumulation of tetrahedra. Formally, given a representativespacetime, form all divisions D i of that spacetime into two sets each of T spacelike hypersurfaces, maintaining15
100 200 300 4000.00.51.01.52.02.53.0 Σ < d S > (a) Σ < d S > (b) Σ < d S > (c) Figure 7: The ensemble average spectral dimension (cid:104) d s (cid:105) as a function of diffusion time σ . (a) Phase C( k = 1 . k = 0 . λ = 0 . α = 0 .
50) (b) Phase D ( k = 1 . k = 0 . λ = 3 . α = 0 .
00) (c) PhaseE ( k = 1 . k = 0 . λ = 1 . α = 2 . D = (cid:110)(cid:110) Σ , . . . , Σ T (cid:111) , (cid:110) Σ T +1 , . . . , Σ T (cid:111)(cid:111) (4.3a) D = (cid:110)(cid:110) Σ , . . . , Σ T +1 (cid:111) , (cid:110) Σ T +2 , . . . , Σ (cid:111)(cid:111) (4.3b)... D T = (cid:110)(cid:110) Σ T , . . . , Σ T − (cid:111) , (cid:110) Σ T , . . . , Σ T − (cid:111)(cid:111) . (4.3c)Then select the two particular divisions ¯ D eq and ˜ D eq that most nearly equalize the discrete 2-volume summedover the T spacelike hypersurfaces in each set of the division: the quantity (cid:88) τ ∈ { i,...,i + T − } N SL (Σ τ ) − (cid:88) τ ∈ { i + T ,...,i − } N SL (Σ τ ) (4.4)is minimized for both ¯ D eq and ˜ D eq . Note that ¯ D eq and ˜ D eq only differ in the order of their two sets. Nextrelabel the discrete time coordinate over the set of values {− T , − T + 1 , . . . , T − , T } so that the values {− T , . . . , − } label the first set of spacelike hypersurfaces and the values { , . . . , T } label the second set ofspacelike hypersurfaces in both ¯ D eq and ˜ D eq . Finally choose the division for which the values of N SL (Σ − )and N SL (Σ ) are greatest. With all of the spacetimes in an ensemble so aligned, perform a discrete timewiseaverage of N SL over the representative spacetimes.In figure 8 we show the results of the equal discrete 2-volume splitting average for two ensembles in phaseC. The data points indicate the equal discrete 2-volume splitting average value of { N SL (Σ τ ) } Tτ =1 ; the lightvertical bars indicate one standard deviation of error. The thin curve is a one parameter fit of the datapoints within the central accumulation of tetrahedra to the functional form N SL ( τ ) = 2 π (cid:104) N (3 , (cid:105) ˜ s (cid:104) N (1 , (cid:105) / cos (cid:32) τ ˜ s (cid:104) N (1 , (cid:105) / (cid:33) , (4.5)which is a discretization of the 2-volume as a function of global time of Euclidean de Sitter spacetime. (See[24] and [15] for a derivation of this discretization in 2 + 1 and 3 + 1 dimensions, respectively.) Substantialevidence already demonstrates that, for the general relativistic values of λ and α , the ensemble averagegeometry in phase C closely approximates Euclidean de Sitter spacetime [4, 9, 10, 11, 12, 13, 14, 15, 19, 24].The plot in figure 8(b) provides the first evidence that the ensemble average geometry also has this propertyfor the broader range of nonrelativistic values of both λ and α .Hoˇrava-Lifshitz gravity admits Euclidean de Sitter spacetimes: for λ > and α = 0, the solutions (A.10)and (A.11) coincide in their description of such spacetimes. Note, however, that for relatively small values of16 a) (b) Figure 8: The ensemble average number (cid:104) N SL (cid:105) of spacelike triangles as a function of discrete time τ . (a)Phase C ( k = 1 . k = 0 . λ = 1 . α = 0 .
00) with ˜ s = 0 .
46 (b) Phase C ( k = 1 . k = 0 . λ = 0 . α = 0 .
50) with ˜ s = 0 . ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææàààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààà ìììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììì òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò - - -
10 0 10 20 300100200300400500 Τ < N S L > Figure 9: The ensemble average number (cid:104) N SL (cid:105) of spacelike triangles as a function of discrete time τ for fourensembles in phase C differing in their respective values of the coupling constant λ : blue circles ( k = 1 . k = 0 . λ = − . α = 0 . k = 1 . k = 0 . λ = 0 . α = 0 . k = 1 . k = 0 . λ = 1 . α = 0 . k = 1 . k = 0 . λ = 2 . α = 0 . α , the solutions (A.10) and (A.11) do not deviate too markedly from Euclidean de Sitter spacetime. Giventhe presence of finite size effects and the inherent error in our measurements, one of these solutions for α (cid:54) = 0may well fit better the ensemble average geometry in phase C. Our current data are not sufficiently detailedfor us to make a conclusive statement. Furthermore, there exists some evidence [19] that the ensembleaverage geometry in (2 + 1)-dimensional causal dynamical triangulations is a deformed version of Euclideande Sitter spacetime. Supposing that this is the case, the process of taking the continuum limit must generatesome additional terms besides those in the Regge action.The two plots in figure 8 quite closely resemble one another—in both spatial and temporal extents of thecentral accumulation of discrete 2-volume—even though that of figure 8(a) is for general relativistic valuesof λ and α while that of figure 8(b) is for nonrelativistic values of λ and α . To dispel the suspicion thatthe K and R terms are renormalized to irrelevance in phase C, we present in figure 9 four plots of theequal discrete 2-volume splitting average of { N SL (Σ τ ) } Tτ =1 each for a different value of λ . Clearly, varying λ affects the ensemble average geometry that emerges.For phases D and E we currently do not know how to coherently average { N SL (Σ τ ) } Tτ =1 over an ensemble.In an effort to determine a method, we computed the ensemble average power in the discrete Fourier transformof { N SL (Σ τ ) } Tτ =1 , the results of which we display in figure 10. As these plots show, there is no notableperiodicity present in these ensemble’s average geometry since virtually all of the power falls in the zerofrequency mode. This lack of periodicity may in fact be indicative of the semiclassical nature of phase E.With α > Ansatz (A.4) whenthe squared scale factor a ( t ) has the constant value (cid:112) α . The depiction in figure 6(c) of a representativespacetime in phase E resembles to a certain extent a spacetime with approximately constant scale factor,and the plot of figure 10(b) reinforces this interpretation. Moreover, this accords with our expectation thatas currently devised the Markov chain Monte Carlo simulations converge on spacetimes globally minimizing17he action (3.18). (a) (b) Figure 10: The ensemble average power (cid:104)| c | (cid:105) as a function of discrete frequency ν in the discrete Fouriertransform of the number N SL of spacelike triangles as a function of discrete time τ . (a) Phase D ( k = 1 . k = 0 . λ = 3 . α = 0 .
00) (b) Phase E ( k = 1 . k = 0 . λ = 1 . α = 2 . { N SL (Σ τ ) } Tτ =1 between theensemble in phase C associated with figure 6(a) and the ensemble in phase E associated with figure 6(c).Specifically, for each representative spacetime of this ensemble in phase E, we calculate the discrete timeaveraged deviation ∆ N SL of the discrete 2-volume from the mean. The ensemble average (cid:104) ∆ N SL (cid:105) provides areasonable measure of the deviations from a time-independent geometry. For this ensemble in phase C, wecalculate the deviations (cid:104) ∆ N SL (cid:105) min and (cid:104) ∆ N SL (cid:105) max of the discrete 2-volume from the mean for the spacelikehypersurfaces of minimal and of maximal (cid:104) N SL (cid:105) within the central accumulation of tetrahedra. The values (cid:104) ∆ N SL (cid:105) min and (cid:104) ∆ N SL (cid:105) max provide a reasonable measure of the range of deviations from a Euclidean deSitter geometry. We find that (cid:104) ∆ N SL (cid:105) max (cid:104) N SL (cid:105) max (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C = 0 . < (cid:104) ∆ N SL (cid:105)(cid:104) N SL (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E = 0 . < (cid:104) ∆ N SL (cid:105) min (cid:104) N SL (cid:105) min (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C = 0 .
78 (4.6)and that (cid:113) (cid:104) ∆ N SL (cid:105) max (cid:112) (cid:104) N (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C = 0 . > (cid:113) (cid:104) ∆ N SL (cid:105) (cid:112) (cid:104) N (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E = 0 . > (cid:113) (cid:104) ∆ N SL (cid:105) min (cid:112) (cid:104) N (cid:105) min (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C = 0 . . (4.7)To achieve a more faithful comparison, we have considered the above two ratios instead of the deviationsthemselves. Although not definitive, the fact that the ratios for phase E fall between those for phase C lendscredence to our interpretation of the semiclassical nature of phase E.Supposing that further analysis supports the conjecture that time-independent geometries dominatephase E, how could one then interpret the C-E phase transition? A possible answer involves the globalgravitational Hamiltonian H ⊥ of (2.10). Recall that, in our implementation of Hoˇrava-Lifshitz gravity intothe framework of causal dynamical triangulations, we chose not to impose the condition (2.9) as a constraint,instead treating H ⊥ as a conserved global charge, essentially the total energy. In our statistical ensemblewhether H ⊥ vanishes or has a nontrivial spectrum on physical states thus becomes a question of dynamics.If the dynamics enforce the vanishing of H ⊥ on physical states, then we might interpret this phenomenonas dynamical confinement of the gravitational charge. A similar phenomenon has recently been discoveredin (2 + 1)-dimensional relativistic chiral gravity [44]. There not just one gravitational charge but an infinitehierarchy of conserved chiral charges are confined, that is, dynamically vanish on all physical states of finiteenergy.Now consider the hypothetical phase in which H ⊥ is dynamically confined. On spacetime geometries of theFriedmann-Lemaˆıtre-Robertson-Walker type, the Hamiltonian constraint equation becomes the Friedmannequation for the scale factor. The Friedmann equation precludes the ground state geometry from beingtime-independent, forcing a cosmological, de Sitter-like evolution of the universe. In the context of causaldynamical triangulations, we observe this behavior in phase C.18n the hypothetical deconfined phase, on the other hand, the situation is more reminiscent of a typicalcondensed matter system: the total Hamiltonian simply measures the system’s energy levels with the groundstate identified as the lowest energy state, typically static. We apparently observe this behaviour in ourphase E. Accordingly, we speculate that the C-E phase transition may be viewed as the deconfinement ofthe gravitational charge H ⊥ . Motivated by a suite of striking similarities between Hoˇrava-Lifshitz gravity and causal dynamical triangula-tions, we constructed a discrete version of the action for (2 + 1)-dimensional Hoˇrava-Lifshitz gravity adaptedto the formalism of causal dynamical triangulations. Using this action in Markov chain Monte Carlo simula-tions of the corresponding path integral, we found significant evidence for the existence of extended phasesof geometry approximating certain classical solutions of Hoˇrava-Lifshitz gravity. Quantum Hoˇrava-Lifshitzgravity thus appears amenable to and compatible with the techniques of causal dynamical triangulations.Since we have only just initiated this study, the prospects for further research are expansive. In thenear term we have three primary goals for ongoing research. First, we plan to map more extensively ourmodel’s coupling constant space to determine the extent of the phases so far discovered and to identifythe locations of phases not yet discovered. Specifically, we hope to ascertain how the curvature squaredterms affect the decoupled phase of causal dynamical triangulations. Second, we wish to further our analysisof the semiclassical natures of phases C, D, and E along the lines of [4, 15, 19, 24]. Third, we want toestablish the orders of our model’s phase transitions. In this direction we have attempted to determine orderparameters for the C-D and C-E phase transitions; unfortunately, none of our trial order parameters haveyet distinguished between these adjacent phases. If our speculation about the confinement-deconfinementnature of the C-E phase transition is correct, then the appropriately defined ground state energy may act asan order parameter.In the long term the richness of the literature on Hoˇrava-Lifshitz gravity provides a host of directionsfor continuing research. First of all, recall that the simplest versions of (2 + 1)-dimensional Hoˇrava-Lifshitzgravity, both projectable and nonprojectable, possess a single propagating scalar degree of freedom. This isin stark contrast to the case of not only (2 + 1)-dimensional general relativity, but also the nonrelativisticgenerally covariant version of Hoˇrava-Lifshitz gravity constructed in [33]. This scalar mode has generatedconsiderable controversy with regard to its phenomenological viability [46, 52], but its full dynamics, espe-cially around the stable ground state, remains poorly understood. The (2 + 1)-dimensional theory shouldprovide a clear window into the scalar mode’s dynamics since the complications of propagating tensor modesare absent. By studying the nonperturbative dynamics of the scalar mode using causal dynamical trian-gulations, we hope to shed light on this issue. Our first challenge is the identification of an observable incausal dynamical triangulations that measures the number of local propagating degrees of freedom or thatat least distinguishes between the absence and presence of local propagating degrees of freedom. Given therelative ease of extracting correlation functions of observables associated with the spacelike hypersurfaces ofcausal triangulations, we are currently working to understand the behavior of our model’s conformal modein relation to the scalar mode. We hope that such an investigation might illuminate the nature of the scalardynamics.Relatedly, perturbations of projectable Hoˇrava-Lifshitz gravity about (3 + 1)-dimensional Minkowskispacetime generate instabilities [41]. This finding is of course not surprising: Minkowski spacetime is notthe ground state of the commonly analyzed models. The situation is much improved on the background ofde Sitter spacetime [25, 53]. As the coupling constant λ approaches unity, however, the higher derivativeterms become relevant, leading to a breakdown of the linearized analysis. The authors of [45, 53] havesuggested that a nonperturbative Vainshtein mechanism might take effect, rendering this limit continuousto the general relativistic values of the coupling constants. Potentially, we could uncover this transitionbehavior in Markov chain Monte Carlo simulations. Indeed, the critical surface depicted in figure 5(a)provides a small but intriguing piece of evidence for that possibility: the apparent geometric feature alongthe λ = 1 direction.To assess the phenomenological viability of at least the simple version of Hoˇrava-Lifshitz gravity on whichwe have focused our study, we need to study the renormalization group flow of the coupling constants for the19urpose of comparing the long distance behavior to that of general relativity. In [36] Henson proposes a coarsegraining procedure for causal dynamical triangulations on which one could try to base a renormalizationgroup procedure. If this scheme proves apt – a question that we are currently exploring – then we plan toemploy the procedure to study the renormalization group flows of our model. Ultimately, we would like toexplore the general conjecture formulated by several groups [2, 34] suggesting that Hoˇrava-Lifshitz gravityand causal dynamical triangulations belong to the same universality class. This circumstance would neatlyexplain the remarkable resemblances between these two approaches to the quantization of gravity. Acknowledgments
We wish to thank Jan Ambjørn, Dario Benedetti, Diego Blas, Ted Jacobson, Renate Loll, Charles Melby-Thompson, Oriol Pujol`as, Kevin Schaeffer, Sergey Sibiryakov, Thomas Sotiriou, Lewis Tunstall Garcia-Huidobro, Matt Visser, and Silke Weinfurtner for illuminating discussions at various stages of this work. C.Anderson acknowledges support from National Science Foundation under REU grant PHY-1004848 at theUniversity of California, Davis. S. J. Carlip, J. H. Cooperman, and R. K. Kommu acknowledge supportfrom Department of Energy under grant DE-FG02-91ER40674. P. Hoˇrava and P. R. Zulkowski acknowledgesupport from National Science Foundation under Grant PHY-0855653, Department of Energy under GrantDE-AC02-05CH11231, and the Berkeley Center for Theoretical Physics.
A On the Classical Equations of Motion
We collect here the classical equations of motion stemming from the action (2.8) for constant lapse and theirsolutions relevant to the Markov chain Monte Carlo simulations reported above. In Riemannian signatureand for spacelike hypersurfaces having the topology of S , the equations of motion are0 = ∇ i π ij ( t, x ) (A.1)and0 = − (cid:112) γ ( t, x ) ∂ t (cid:104)(cid:112) γ ( t, x ) π ij ( t, x ) (cid:105) + 12 γ ij ( t, x ) (cid:2) K kl ( t, x ) K kl ( t, x ) − λK ( t, x ) − αR ( t, x ) + 2Λ (cid:3) − K il ( t, x ) K jl ( t, x ) + 2 λK ( t, x ) K ij ( t, x ) + 2 α ∇ i ∇ j R ( t, x ) − α ∇ R ( t, x ) γ ij ( t, x )+ 12 (cid:2) ∇ l N i ( t, x ) π jl ( t, x ) + ∇ l N j ( t, x ) π il ( t, x ) − π ij ( t, x ) ∇ l N l ( t, x ) (cid:3) (A.2)where π ij ( t, x ) = K ij ( t, x ) − λK ( t, x ) γ ij ( t, x ) . (A.3)The momentum constraint (A.1) results from variation of the shift vector, and the metric equations of motion(A.2) result from variation of the spatial metric tensor. Again we exclude the nonlocal integral constraint(2.9) arising from variation of the lapse.We seek solutions to the equations of motion (A.1) and (A.2) in the form of the Friedmann-Lemaˆıtre-Robertson-Walker Ansatz γ ( t, x ) = a ( t )ˆ γ ( x ) , (A.4)for spatially homogeneous and isotropic spatial metric tensor γ ( t, x ) and identically vanishing shift vector N ( t, x ). Here, ˆ γ ( x ) is the metric tensor on the round 2-sphere, and a ( t ) is the scale factor. Under theseassumptions the momentum constraint (A.1) is trivially satisfied. The metric equations of motion (A.2) aresatisfied if and only if 1 a ( t ) d a ( t )d t = α λ −
1) 1 a ( t ) − Λ2 λ − , (A.5)which implies that (cid:26) dd t (cid:20) a ( t ) − (2 λ − C (cid:21)(cid:27) = − α λ − C (2 λ − − λ − (cid:20) a ( t ) − C (2 λ − (cid:21) (A.6)20or constant of integration C . Letting u ( t ) = a ( t ) − C (2 λ − , (A.7)(A.6) becomes (cid:20) d u ( t )d t (cid:21) + 4Λ2 λ − u ( t ) = − α λ − C (2 λ − , (A.8)which has the solution u ( t ) = (cid:115) C (2 λ − − α
2Λ cos (cid:32) (cid:114) Λ2 λ − t + δ (cid:33) (A.9)for a second constant of integration δ . In terms of a ( t ), the solution (A.9) is a ( t ) = C (2 λ − (cid:115) C (2 λ − − | α |
2Λ cos (cid:32) (cid:114) Λ2 λ − t + δ (cid:33) (A.10)for positive values of α and a − ( t ) = C (2 λ − (cid:115) C (2 λ − + | α |
2Λ cos (cid:32) (cid:114) Λ2 λ − t + δ (cid:33) (A.11)for negative values of α . We assume that Λ > a − ( t ) whereas the solution(A.10) has no such restrictions. Also note that for vanishing α the solutions (A.10) and (A.11) both reduceto that of Euclidean de Sitter spacetime. B On the Geometry of Causal Dynamical Triangulations
For the tetrahedra depicted in figure 1, the Lorentzian dihedral angles between spacelike and timelike facesare θ (3 , L = π i log (cid:18) √ η + 1 √ √ η + 1 (cid:19) (B.1a) θ (2 , L = i log (cid:32) η + 3 − √ √ η + 14 η + 1 (cid:33) (B.1b) θ (1 , L = π i log (cid:18) √ η + 1 √ √ η + 1 (cid:19) , (B.1c)and the Lorentzian 3-volumes are V (3 , L = 112 (cid:112) η + 1 a (B.2a) V (2 , L = 16 √ (cid:112) η + 1 a (B.2b) V (1 , L = 112 (cid:112) η + 1 a . (B.2c)As previously mentioned, Wick rotation consists in analytically continuing η in the lower half complexplane [8]. If the argument of a square root becomes negative as a result of the Wick rotation, then we replace21t by the negative of the argument multiplied by − i . The Lorentzian dihedral angles (B.1) are thus continuedto their respective Euclidean values θ (3 , E = π − cos − (cid:18) √ η − √ √ η − (cid:19) (B.3a) θ (2 , E = cos − (cid:18) η − η − (cid:19) (B.3b) θ (1 , E = π − cos − (cid:18) √ η − √ √ η − (cid:19) , (B.3c)and the Lorentzian 3-volumes are thus continued to their respective Euclidean values V (3 , E = − i (cid:112) η − a (B.4a) V (2 , E = − i √ (cid:112) η − a (B.4b) V (1 , E = − i (cid:112) η − a . (B.4c)Note that the magnitude of η must be greater than to prevent the tetrahedra from becoming degenerate. References [1] J. Ambjørn, J. Correia, C. Kristjansen, and R. Loll. “On the relation between Euclidean and Lorentzian2D quantum gravity.”
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