Spectral dimension of Horava-Snyder spacetime and the Ad S 2 × S 2 momentum space
aa r X i v : . [ h e p - t h ] A ug Spectral dimension of Horava-Snyder spacetime and the
AdS × S momentum space F.A. Brito and E. Passos
Departamento de F´ısica, Universidade Federal de Campina Grande,Caixa Postal 10071, 58109-970 Campina Grande, Para´ıba, Brazil (Dated: November 26, 2018)We show that the UV-regime at the Lifshitz point z = 3 is equivalent to work with a momentamanifold whose topology is the same as that of an AdS × S space. According to Snyder’s theory,curved momentum space is related to non-commutative quantized spacetime. In this sense, ouranalysis suggests an equivalence between Horava-Lifshitz and Snyder’s theory. PACS numbers: 04.60.Nc, 11.30.Cp
I. INTRODUCTION
There is the possibility of understanding the quantum gravity aspects by studying the spectral dimension of thespacetime as considered by Horava and Ambjorn [1–4]. One of the best way of applying such investigations is throughthe diffusion equation. The diffusion process can be seen as an way of a diffusing particle to probe the spectraldimension. It happens that the dimension seen by the particle can change along its diffusing process. It may evenbecome fractal as in polymeric chains. In this letter we show that the spectral dimension of a curved momentum spacegives the same result as in the Horava-Lifshitz gravity [1, 2]. In the latter case, for a 3+1-dimensional spacetime,i.e., D = 3, the spectral dimension flows continuously from d s = 2 at z = 3 to d s = 4 at z = 1 as one goesfrom small to large distances. Although in the Horava-Lifshitz gravity the spacetime is continuous the behavior ofthe spectral dimension agrees with the Ambjorn’s CDT quantum gravity [3, 4] that is based on a four-dimensionaldiscrete spacetime — see also [5–7]. In our study we show that equivalently one can curve the momentum space toget the same spectral dimension in both Horava-Lifshitz gravity and CDT quantum gravity. This seems not to besurprising since it is well-known long ago [8, 9] that curved momentum space leads to discrete (and non-commutative)spacetime. Furthermore, as well discussed in [10] and anticipated by Snyder [8, 9], a quantized spacetime leadsto spacetime uncertainty relations that follow from the algebra of coordinate operators describing the coordinates inquantum spacetime. Such uncertainties introduces limitations in the accuracy of localization of spacetime events invery short distances (or very high energies) — see below. This is the regime where a quantum theory of gravity shouldbe implemented. In the Horava-Lifshitz gravity this regime is understood as a quantum theory at Lifshitz point z = 3.Another interesting candidate to quantum gravity is Doubly Special Relativity (DSR) which also develops spacetimenoncommutativity that one can be shown through the use of a Snyder type algebra — see [11]. II. THE DIFFUSION EQUATION AND THE CURVED SPACE MOMENTUM
The spectral dimension can be understood in terms of a diffusion equation. The diffusion time is regarded as thescale responsible to probe the manifold in study. At small diffusion time the dimension of a curved manifold coincideswith the spectral dimension. At sufficiently large diffusion time they start to be different. In our investigations weassume the spacetime to be a flat manifold. This is because the spectral UV/IR flow in Horava-Lifshitz theory shouldstill be true for curved spacetime as shown in [12].One can consider the diffusion equation as ∂∂σ ρ ( x , τ ; x ′ , τ ′ ; σ ) = (cid:18) ∂ ∂τ + ∇ (cid:19) ρ ( x , τ ; x ′ , τ ′ ; σ ) (1)whose solution is ρ ( x , τ ; x ′ , τ ′ ; σ ) = Z dω d D k (2 π ) D +1 e iω ( τ − τ ′ )+ i k · ( x − x ′ ) e − σ ( ω + | k | ) (2)This solution enables us to find the average return probability P ( σ ) ≡ ρ ( x , τ ; x ′ , τ ′ ; σ ) (cid:12)(cid:12)(cid:12) x = x ′ ; τ = τ ′ given by P ( σ ) = Z dω d D k (2 π ) D +1 e − σ ( ω + | k | ) = Cσ ( D +1) / . (3)The spectral dimension is given by d s = − d ln P ( σ ) d ln σ = D + 1 , (4)where C is some nonzero constant. In this case the spectral dimension coincides with the topological dimension ofthe R D +1 spacetime [1].In the Horava-Lifshitz gravity one modifies the UV behavior of theory by using the Lifshitz scaling such that | k | → | k | z , being z = 3 the Lifshitz point at which gravity is renormalizable via power counting [1, 2]. The IRregime of the theory is recovered at z = 1. In this proposal one has P ( σ ) = Z dω d D k (2 π ) D +1 e − σ ( ω + | k | z ) = Cσ ( Dz +1) / . (5)The spectral dimension is now given by d s = Dz + 1 , (6)that is the most important result found in [1]. Of course, in this setup on has to change the Laplacian ∆ of thediffusion equation (1) in the form ∇ → ∇ z . In a general curved manifold the formula (4) should be replaced by[12] n = 2 lim λ →∞ d ln N ∆ ( λ ) d ln λ , (7)where N ∆ ( λ ) counts the number of eigenvalues, with multiplicity less than λ , of the Laplacian ∆ on a closed Rie-mannian manifold M of dimension n . In this case the spectral dimension presents the same flow as developed in flatspacetime, that is n ≡ d s . Thus, n = 2 at UV regime flows to n = 4 at IR regime — see [12] for further details.However, for the sake of simplicity, in the following we maintain our investigations in flat spacetime.Notice we can rewrite the first equation in (5) in terms of a new D -dimensional momentum variable | k | → | p | /z to find P ( σ ) = Z dωd D p (2 π ) D +1 f ( | p | , z ) e − σ ( ω + | p | ) , (8)where we define the momentum dependent function as f ( | p | , z ) = c | p | α , α = Dz − D. (9)In order to determine the spectral dimension on influence of the momentum dependent function, let us first obtainthe average return probability in the form P ( σ ) = 4 πc Z dωd p (2 π ) D +1 | p | D − α e − σ ( ω + | p | ) = Cσ ( D + α +1) / . (10)In this case, the spectral dimension of the spacetime is given by d s = D + α + 1 = Dz + 1 , (11)that is precisely the result (6) for the spectral dimension in the Horava-Lifshitz gravity. For a 3+1-dimensionalspacetime we have D = 3, in this case the spectral dimension flows continuously from ds = 2 at z = 3 to ds = 4 at z = 1 as one goes from small to large distances.Now we argue that one can also make a flow between the UV and IR regime by considering a curved momentumspace and keeping the theory fixed in z = 3. Thus the anisotropic rescaling of the momentum element volume madein (8) is now given by f ( | p | , z ) ≡ c | p | α p | det G | , (12)where G µν ( p ) is the metric in the momentum space [8, 9].Assume for a theory in 3+1 dimensions in the UV regime ( p → ∞ ) we have f ( | p | , z ) ∼ | p | − , that is p | det G | → const . This precisely happens to the volume element of an AdS × S momentum space given by the metric ds = − | p | Λ dω + Λ | p | d p + Λ d Ω , (13)where | p | = p + p + p and Λ is the AdS and S momentum radius. Thus the theory in UV regime have an AdS × S curved momentum space. It is well-know long ago that the momentum space with constant curvature suchas AdS (or dS ) space has a non-commutative spacetime counterpart [8, 9]. We shall turn to this point shortly.To make the metric to flow continuously to the IR regime one can use a more general metric such as the four-dimensional Reissner-Nordstr¨om black hole metric on the momentum space ds = − (cid:18) | p | Λ (cid:19) dω + (cid:18) | p | Λ (cid:19) − (cid:0) d p + | p | d Ω (cid:1) , (14)that becomes flat in the IR regime ( p →
0) where one recovers f ( p , z ) →
1. Notice that we have taken the metric ofan extremal Reissner-Nordstr¨om black hole solution of a four-dimensional spacetime to lead to a black role solutioninto the four-dimensional space momentum earlier discussed by making use of the following suitable change (cid:16) r r (cid:17) ± → (cid:16) p Λ (cid:17) ∓ . (15)Thus, in general we have a curved momentum volume element rescaled by the function f ( | p | , z ) ≡ c | p | α +2 (cid:18) | p | Λ (cid:19) − = (cid:18) | p | Λ (cid:19) − , (16)where we have considered z = 3, D = 3, that means α = −
2, and c = 1. It is worth noticing that for | p | Λ ≪ f ( | p | , z ) ≡ exp (cid:16) − | p | Λ (cid:17) .Thus, as expected, for the function f ( | p | , z ) = (cid:16) | p | Λ (cid:17) − into the formula (8), the spectral dimension (4) flowsfrom ds = 2 to ds = 4 as one goes from the UV ( σ →
0) to IR ( σ → ∞ ) regime as the Fig. 1 shows. This has thesame behavior found in the Ref. [3]. FIG. 1: The flow of the spectral dimension from ds = 2 to ds = 4 as one goes from UV ( σ →
0) to IR ( σ → ∞ ). Let us now focus on the
AdS part of the curved momentum space. Firstly, notice that we recover the
AdS × S geometry (13) as the ‘near-horizon’ limit | p | → ∞ of the metric (14) in momentum space. As firstly showed bySnyder the AdS momentum space is related to a non-commutative spacetime. In the following we shall make a shortdiscussion on this important result in order to adapt it to our set up.The
AdS part of the momentum space with the
AdS × S geometry satisfies p − | p | + p = Λ , p = 1 a η η , | p | ≡ p r = 1 a η ′ η , p = 1 a ηη , Λ = 1 a , η ′ = | ~η | , (17)where a ( which is in the same footing as the Planck length λ P given in [10]) is a natural unit of length of the quantizedspacetime and the variables η , ..., η satisfy the quadratic form that defines a four-dimensional space with constantcurvature − η = η − η ′ − η , (18)where η ′ = η + η + η . This allows us to write commutation relations for a non-commutative spacetime whosecoordinates are operators with the following structure[ˆ t, ˆ r ] = i a M r , M r = ˆ r p + ˆ t p r , ˆ r = i a (cid:18) η ∂∂η ′ − η ′ ∂∂η (cid:19) , ˆ t = i a (cid:18) η ∂∂η + η ∂∂η (cid:19) . (19)The commutation relation between the radial coordinate and its conjugate momentum is now given by[ˆ r , p r ] = i (1 + a p r ) . (20)Notice that as a → | p | → ∞ where this space becomes curved with AdS × S geometry. III. IR/UV (4D/2D) TRANSITION INTO LOOP MOMENTA
As in the previous discussions on the spectral dimension, we can also rewrite the following integral momenta inHorava-Lifshitz theory in terms of a new momentum variable | k | → | p | /z such as Z dωd D − k ω − | k | z − M → Z d D p f ( | p | , z ) p − M , (21)for f ( | p | , z ) given as in Eq. (9). Notice there is no difference in the physical description between the original andtransformed integrals above. As a consequence we could change our way of facing the physics described by the integralof l.h.s. by looking into the integral of the r.h.s. recognizing it as the description of a process where a modified loopmomenta via the function f ( | p | , z ) is now present and the propagator is kept as the usual one. The mainly differenceis that now the four-dimensional theory in the IR regime, i.e., f ( | p | , z ) = (cid:16) | p | Λ (cid:17) − → | p | →
0, looks to be atwo-dimensional theory in the UV regime, i.e., f ( | p | , z ) = (cid:16) | p | Λ (cid:17) − ∝ | p | − as | p | → ∞ — see below.Here we show how to proceed in order to modify the usual loop momenta to get processes in the Horava-Lifshitztheory at UV-regime. Let us now apply the momentum dependent function f ( | p | , z ) in a loop momenta integral thathave quadratic divergence by power counting in a 3+1-dimensional spacetime in the folowing process given by the“tadpole” iλ Z d p p − M → iλ Z d p f ( | p | , z ) p − M → πc iλ Z dω Z ∞ d p | p | D − α ω − | p | − M ) UV → πc iλ Z dω Z ∞ d p ω − | p | − M ) (22)Notice that in the ultraviolet regime the integral (22) changes its quadratic divergence to logarithmic divergence.One should note that the integral into loop momenta is quite similar to an integral of a two-dimensional theory. Thisis in accord with the spectral flow observed above.We shall not attempt to write down here a Lagrangian for a field theory with the UV-completion considered inHorava-Lifshitz theory, but our analysis suggests that it should be a renormalizable field theory that in UV regimebehaves like a two-dimensional theory. This has a close connection with gravity in two dimensions. This is becausetwo-dimensional gravity can be simply described in terms of a renormalizable two-dimensional field theory such asLiouville theory [13]. IV. DISCUSSIONS
In this letter we have found that in a curved momenta space with asymptotic
AdS × S geometry one may have thesame physics of Horava theory. Furthermore, if we allow ourselves to speculate a bit more, the would be holographiccorrespondence AdS /CF T in the momentum space would lead to a non-commutative conformal field theory in aone-dimensional spacetime, that may correspond to a non-commutative conformal ‘quantum mechanics’. However isnot clear at all which symmetries are present in both momentum space and spacetime in the present case. A pointin this direction is the fact that in crystallography the cubic lattice is identical to the reciprocal lattice, but furtherstudies in this direction should be addressed elsewhere. Acknowledgements.