Contents

Fractal Quantum Space-Time

Fractal Quantum Space -Time

Leonardo Modesto

Perimeter Institute for Theoretical Physics,

31 Caroline St. N., Waterloo, ON N2L 2Y5, Canada

[12pt]

In this paper we calculated the spectral dimension of loop quantum gravity (LQG) using the scaling property of the area operator spectrum on spin-network states and using the scaling property of the volume and length operators on Gaussian states. We obtained that the spectral dimension of the spatial section runs from to , and under particular assumptions from to across a phase when the energy of a probe scalar field decreases from high to low energy in a fictitious time . We calculated also the spectral dimension of space-time using the scaling of the area spectrum operator calculated on spin-foam models. The main result is that the effective dimension is at the Planck scale and at low energy. This result is consistent with two other approaches to non perturbative quantum gravity: causal dynamical triangulation and asymptotically safe quantum gravity. We studied the scaling properties of all the possible curvature invariants and we have shown that the singularity problem seems to be solved in the covariant formulation of quantum gravity in terms of spin-foam models. For a particular form of the scaling (or for a particular area operator spectrum) all the curvature invariants are regular also in the Trans-Planckian regime.

1 Introduction

In past years many approaches to quantum gravity studied the fractal properties of the space-time. In particular in causal dynamical triangulation (CDT) [1] and asymptotically safe quantum gravity (ASQG) [2], a fractal analysis of the space-time gives a two dimensional effective manifold at high energy. In both approaches the spectral dimension is at small scales and at large scales.Recently the previous ideas have been applied in the context of non commutativity to a quantum sphere and -Minkowski [3] and in Loop Quantum Gravity [4]. The spectral dimension has been studied also in the cosmology of a Lifshitz universe [5]. Spectral analysis is a useful tool to understand the effective form of the space at small and large scales. We think that the fractal analysis could be also a useful tool to predict the behavior of the -point and -point functions at small scales and to attack the singularity problems of general relativity in a full theory of quantum gravity.

In this paper, we apply to loop quantum gravity (LQG) [6] [9] the analysis developed in the context of ASQG by O. Lauscher and M. Reuter [2]. In the context of LQG, we consider a spatial section, which is a manifold, and we extract the energy scaling of the metric in two different way from the area spectrum on the spin-network states and from the volume and length operators spectrum on Gaussian states. The result is the same until the Planck scale. We apply the same analysis to the space-time using the area spectrum that is suggested by the spin-foam models [7]. In the space-time case, the result will be consistent with the spectral dimension calculated in the different approach of non-perturbative quantum gravity [1], [2].

In LQG, the average metric defines an infinite set of metric at different scales labeled by . The metric average is over spin-network states, , where and is a diffeomorfism invariant length scale because is the Diff-invariant representation (in the paper we will consider also the average over Gaussian states obtaining the same result until the Planck scale). The length is typically of the order of , where is the momentum of a probe field which plays the rule of microscope. The metrics one for each scale refer to the same physical system, the “quantum spacetime”, but describe its effective metric structure on different scales. An observer using a “microscope” with a resolution will perceive the universe to be a Riemannian manifold with metric . We suppose at every fixed , is a smooth classical metric. But since the quantum spacetime is characterized by an infinity number of metrics , it can acquire very nonclassical and in particular fractal features.

In a somewhat simplified form, the construction of a quantum spacetime within LQG can be summarized as follows. We start from the Hilbert space of LQG and we calculate the expectation value of the metric operator at any scale , or for any representation . The quantum space-time is specified by the infinity of Riemannian metrics . An observer exploring the structure of the space-time using a microscope of resolution ( is the energy scale) will perceive the universe as a Riemannian manifold with the metric which is a fixed metric at every fixed scale , the quantum space-time can have fractal properties because on different scales different metrics apply. In this sense the metric structure on the quantum space-time is given by an infinite set of ordinary metrics labelled by or by the Diff-invariant length scale . In our analysis we will consider the expectation value as a smooth Riemannian metric because we can approximate any metric with a weave state which is a spin network state with a large number of links and nodes that reweave the space. Microscopically it is a Planck size lattice but, at macroscopic scale, it appears as a continuum smooth metric. Since we are interested to the fractal properties of the space (space-time) at different scales we suppose equal all the representations on the spin-network links that across the surface of a given tetrahedron in the dual triangulation (at a fixed scale). For this reason it is sufficient to analyze the metric scaling using an individual link. If denotes a weave state at the scale (the scale is defined such that all the representations that across a given surface are equal) and a weave state at the scale , we have

(1.1)

On the right hand side of (1.1) we have a single link spin-network state at the scales and . We are rescaling together all the representations dual to a given triangulation and then we can consider a single face of a single tetrahedron. For this reason on the right hand side of (1.1) we have one single link spin-network duals to one face of one tetrahedron. In other words all the scaling properties are encoded in a single link graph if we are interesting to the scaling property of the metric and in particular to the fractal structure of the space (space-time). In the paper we will study the fractal properties of the spatial section of LQG also using the expectation value of the volume operator and length operator on Gaussian states. Those states can be treated as semiclassical until the Planck scale and then are useful for our intent.

The paper is organized as follows. In the first section we extract the information about the scaling property of the spatial section metric from the area spectrum of LQG and from the average of the volume operator on of the length operator on Gaussian states. Using the area operator spectrum in the context of spin-foam models we obtain the scaling properties of the metric in . In the second section we give a detailed review of the spectral dimension in diffusion processes. In the third section, we calculate explicitly the spectral dimension of the spatial section in LQG and of the space-time dimension. In the fourth section we show that the curvature invariant can be upper bounded using their scaling properties.

2 Metric Scaling from the Area Spectrum

In this section we extract the scaling property of the expectation value of the metric operator from the area spectrum obtained in LQG and Spin-Foams models.

2.1 Metric scaling in LQG

One of the strongest results of LQG is the quantization of the area, volume and recently length operators [10] [11]. In this section, we recall the area spectrum and we deduce that the energy scaling of the -metric of the spatial section. For a spin-network, , without edges and nodes on the surface we consider the area spectrum

(2.1)

where are the representations on the edges that cross the surface . Using (2.1), we can calculate the relation between the area operator average [10] for two different states of two different representations, and ,

(2.2)

We can introduce the length squared defined by and the infrared length squared . Using this definition, we obtain the scaling properties of the area eigenvalues. If is the area average at the scale and is the area average at the scale (with ), then we obtain the scaling relation

(2.3)
Figure 1: The first and the second pictures on the left represent two weave states with different loop’s density. The picture represents only loops but argument is valid for any spin-network. Any geometry can be approximate by a weave state. The third picture represents the part of a spin-network that across a given surface.

We restricted our analysis to the case when a single edge crosses the surface because of the argument exposed at the end of the introduction and that we will go now to reconsider. In our analysis we will consider the expectation value of the metric at the energy scale as a smooth Riemannian metric because we can approximate any metric with a weave state which is characterized (for example) by a large number of loop that reweave the space. Since we are interested to the fractal properties of the space at different scales we consider equal all the representations on the links that across a given surface at a fixed scale (see Fig.1). We suppose to have a spin-network (weave state) that approximates the metric at a given fixed scale. We concentrate our attention on a small region which is locally approximated by a single tetrahedron and we look at the scaling of the areas of such tetrahedron Fig.(3). That scaling is given by the scaling of the areas of its faces because in our approximation we do not care about the non commutativity of the metric. However, this approximation is a good approximation if we look at the scaling and then at the spectral properties of the space. Locally we suppose that any face of the tetrahedron is crossed by links with equal representations. In other words we can suppose to consider the spatial section as a 3-ball and triangulate it in a very fine way (the dual of the triangulation is a spin-network). Now we consider another 3-ball but at a smaller scale (also in this case the dual is another spin-network). Since we are considering the 3-ball at two different scales all the representations of the spin-network states will be rescaled of the same quantity (see Fig(2)). If we concentrate on an individual tetrahedron of the 3-ball triangulation we can extract its scaling considering just an area and then just a dual reppresentation (if all the representation of the weave state that cross that surface are equal). The representations involved in the spin-network will be different to approximate a 3-ball but the global scaling will be the same and this is what we will use to calculate the spectral dimension.

Figure 2: This picture represents the scaling of a 2-dimensional version of the 3-ball explained in the text. We consider a simplicial decomposition of the 2-ball and we rescale the radius of the ball. A rescaling of the radius of the 2-sphere corresponds to a rescaling of all the representations dual to the triangle’s area.
Figure 3: Scaling of the tetrahedron for different values of the representation or the length . In the picture is represented one tetrahedron for different values of the representation . We can consider this tetrahedron and its scaling as part of a simplicial decomposition.

We denote with a weave state at the scale and with a weave state at the scale . We can think those weave states to describe the 3-ball at two different scales. The scale is defined by . All the representations that crosses a surface of a single tetrahedron have the same value, as explained above. If and are two different scale and the number of links that across that surface we obtain

(2.4)

where , are two spin-network such that only one link of the graph crosses the surface we are considering; the spin-network are at the scale and respectively. In other words all the scaling properties are encoded in a single link graph when we are interesting to the scaling property of the metric and in particular to the fractal structure of the space.

The classical area operator can be related to the spatial metric in the following way. The classical area operator can be expressed in terms of the density triad operator, , and the density triad is related to the three dimensional triad by and . If we rescale the area operator by a factor , , consequently the density triad scales by the same quantities, . The triad instead, using the above relation, scales as and its inverse . The metric on the spatial section is related to the triad by and then it scales as , or, in other words, the metric scales as the area operator. Using (2.3), we obtain the following scaling for the metric

(2.5)

The scaling (2.5) is not an assumption if we restrict our attention to diagonal part of the metric (see the last part of this section); this assumption is justified because we are not interested to the non commutativity of the metric at fixed scale but instead to the the metric at different scales. We have a fixed manifold and also a fixed metric at any scale . Formula (2.5) provides a relation between two metrics at different scales and . If we want to explore the spatial section structure at a fixed length we should use a microscope of resolution or, in other words, we should use (for example) a probe scalar field of momentum (, this approximation can be justified using Riemann normal coordinates in a small region of the manifold. This approximation is related to the curvature of the manifold and not to the scaling properties of the metric). The scaling property of the metric in terms of can be obtained by replacing: , and , where is an infrared energy cutoff and is the Planck energy. The scaling of the metric as function of , and is,

(2.6)

In particular we will use the scaling properties of the inverse metric (see also below the last part of this section),

(2.7)

We define the scaling factor in (2.7), introducing a function : . From the explicit form of we have three different phases where the behavior of can be approximated as follows,

(2.8)

We consider to be constant for ; in particular we require that , . To simplify the calculations without modifying the scaling properties of the metric, we introduce the new function . The behavior of is exactly the same as in (2.8) but with better properties in the infrared limit which one useful in the calculations. We define here the scale function for future reference in the next sections,

(2.9)

We can make more clear the argument of this section in the following way. The metric is related to the density triad by . If we take a tetrahedron as our chunk of space (substantially this correspond to take four valent spin-networks and to identify the point with the node dual to the tetrahedron) the metric can be expressed in terms of the area of the faces and the angles of the tetrahedron

(2.10)

The area are three areas that shide a node and is the cosine of the angle between the normals to the face and . Because we are interested in the scaling of the metric we can consider an equilateral tetrahedron, , and then all the representations of the dual spin-network are equal. For the same reason we do not quantize the operators because they are related to the quantum anisotropy or non commutativity of the metric that does not contain information about the scaling (see Fig.3). Under those assumptions the operator is diagonal on the spin-network states, because the angular part is frozen, and reduces to

(2.11)

We indicate the spin-network with and calculate the expectation value of (2.10),

(2.12)

Since under our assumption, dictated from the physics we want to study it is simple to extract the determinant and obtain the spectrum of the inverse metric, we have:

(2.13)

The scaling of the metric is defined by looking on to representations and that define two different scales and calculating the following ratio,

(2.14)

We stress that the scaling is independent from the angular variables because we are interested in metrics at different scales and we do not take care of the different directions at a fixed scale.

2.2 Metric Scaling in Spin-Foams

We can repeat the scaling analysis above in the case of a four dimensional spin-foam model. In the spin-foam models framework the starting point is a simplicial decomposition of the space-time in -simplexes. Any simplex is made of tetrahedron and we can consider the area operator associated with whatever face of a general tetrahedron of the decomposition. The face can be directed in any direction and then can be space-like or time-like. The area operator commutes with all the constraints and then is a good observable. The result useful for our aim is that in the context of spin-foams models we can have three possible area spectrum: , and . In the first case, when the area eigenvalues are [7], the scaling of the metric is

(2.15)

where . Given the explicit form of the scaling in (2.15), we introduce the new scaling function,

(2.16)

The infrared modification, introduced by hand, does not change the high energy behavior of the scaling function and we can take in the calculations. A different ordering in the area operator quantization can give a different spectrum [7], [8]. The scaling function in this case is

(2.17)

where we have introduced the usual infrared modification: . We can consider also in the same scaling of the spatial section, this corresponds to the matching of the area spectrum that comes from the spin-foam model with the kinematical area spectrum of LQG. The result is (2.9) specialized to four dimension,

(2.18)

3 Metric Scaling in LQG from Gaussian States

In this section we extract the scaling of the metric using a recent result of Bianchi [11]. In [11] has been calculated the expectation value of the volume operator and in particular of the length operator on a gaussian state. The author consider a -valent monochromatic spin-network (the valence of the node is four and all the representations are equal: ) and the state introduced by Rovelli and Speziale [13],

(3.1)

where , and is the basis state associated to the intertwining tensor for the representations , that are equal in our particular case. This state has good semiclassical geometric properties and the interested reader is invited to consult the original paper for the details [13]. The expectation value of the volume and the length operator on (3.1) is

(3.2)

This behavior is correct also for small values of as it is evident from the plots in Fig.4. There is a new version of Gaussian states that confirm this result for any value of the representation . The choice of monochromatic representation is not restrictive but it is necessity because we are interesting to the scaling property of the metric at different scales and the scale is defined by . The results in (3.2) suggests the following scale of the metric, of the metric,

(3.3)

where and are two different representation that satisfy the relation and , . The scaling of the metric for a test field of momentum is

(3.4)

This result coincides with the scaling obtained on spin-network states for .

Figure 4: The two plots represent respectively the expectation value of the volume and length operators on Gaussian states. The expectation value of the volume is well fitted with the dashed line by the function and the length operator expectation value by the function it is relevant to observe that the perfect matching until Planck scales (in the plots ).

4 The spectral dimension

In this section we determine the spectral dimension of the quantum space and the quantum space-time. This particular definition of a fractal dimension is borrowed from the theory of diffusion processes on fractals [14] and is easily adapted to the quantum gravity context.

Let us study the diffusion of a scalar test (probe) particle on a -dimensional classical Euclidean manifold with a fixed smooth metric . The corresponding heat-kernel giving the probability for the particle to diffuse from to during the fictitious diffusion time satisfies the heat equation

(4.1)

where denotes the scalar Laplacian: . The heat-kernel is a matrix element of the operator ,

(4.2)

In the random walk picture its trace per unit volume,

(4.3)

has the interpretation of an average return probability. (Here denotes the total volume.) It is well known that possesses an asymptotic expansion (for ) of the form . For an infinite flat space, for instance, it reads for all . Thus, from the knowledge of the function , one can recover the dimensionality of the target manifold from the -independent logarithmic derivative

(4.4)

This formula can also be used for curved spaces and spaces with finite volume provided is not taken too large.

In quantum gravity it is natural to replace by its expectation value on the spin-network states : (we will calculate also the spectral dimension of the spatial section on Gaussian states and in that case the expectation value is over the state of section (3)). Symbolically,

(4.5)

The third relation is not an equality but an approximation because it is valid only in the case that the metric operator is diagonal on the state considered. This is not true in general on spin-network states but, in the case we are interested in the scaling properties of the metric, we will not consider the non diagonal terms that are related to the non commutativity (or angular part) of the metric. In particular we do not quantize the angular part of the metric. This assumption justifies (4.5). On the other part on gaussian states (4.5) is a good approximation in the large limit. For the space-time spectral dimension the scalar product in (4.5) is the physical scalar product that defines the dynamics. In our context the dynamics is defined by the spinfoam models [7].

Given , the spectral dimension of the quantum space or space-time is defined in analogy with (4.4):

(4.6)

The fictitious diffusion process takes place on a “manifold” which, at every fixed scale , is described by a smooth Riemannian metric . While the situation appears to be classical at fixed , nonclassical features emerge since at different scales different metrics apply. The metric depends on the scale at which the spacetime structure is probed by a fictitious scalar field.

In quantum geometry the equation (4.1) is replaced with the expectation value on the spin-network states,

(4.7)

where . We denote the scaling of the metric operator by a general function that we will specify case by case farther on in the paper,

(4.8)

where we have shorten, and .

The nonclassical features are encoded in the properties of the diffusion operator. We define the covariant Laplacians corresponding to the metrics and by and , respectively at the scale and . We extract the scaling of the Laplacian operators from the behavior of the metric at different scales

(4.9)

We suppose the diffusion process involves (approximately) only a small interval of scales near over which the expectation value of the metric does not change much then the corresponding heat equation contains the for this specific, fixed value of the momentum scale :

(4.10)

The equation (4.10) is exactly (4.7) where we suppressed the index , and introduced the Laplacian at the scale in terms of the Laplacian at the scale .

Denoting the eigenvalues of by and the corresponding eigenfunctions by , we have the following eigenvalue equation for the Laplacian

(4.11)

Using (4.11) the equation (4.10) is solved by

(4.12)

Proof 1 of (4.12). We want to obtain using the definition given at the beginning of this section. Using (4.9) and (4.11) the solution of (4.10) is:

(4.13)

Proof 2 of (4.12). We show below that the left hand side and the right hand side of (4.10) are equal.

From the knowledge of the propagation kernel (4.13) we can time-evolve any initial probability distribution according to , where is the determinant of . If the initial distribution has an eigenfunction expansion of the form we obtain,

(4.14)

Proof of (4.14).

(4.15)

From second to third line we used the weave function normalization property:

(4.16)

If the ’s are significantly different from zero only for a single eigenvalue , we are dealing with a single-scale problem and then we can identify . However, in general the ’s are different from zero over a wide range of eigenvalues. In this case we face a multiscale problem where different modes probe the spacetime on different length scales.

If is the Laplacian on the flat space, the eigenfunctions are plane waves with momentum , and they probe structures on a length scale of order . Hence, in terms of the eigenvalue the resolution is . This suggests that when the manifold is probed by a mode with eigenvalue it “sees” the metric for the scale . Actually the identification is correct also for a curved space because the parameter just identifies the scale we are probing. Therefore we can conclude that under the spectral sum of (4) we must use the scale which depends explicitly on the resolving power of the corresponding mode. In eq. (4.12), can be interpreted as . Thus we obtain the traced propagation kernel

(4.17)

It is convenient to choose as a macroscopic scale in a regime where there are not strong quantum gravity effect.
Proof of (4.17).

(4.18)

We have used (4.12) and (4.9) from the third to the forth line, (4.16) in the last line.

We assume for a moment that is an approximately flat metric. In this case the trace in eq. (4.17) is easily evaluated in a plane wave basis:

(4.19)

The dependence from in (4.19) determines the fractal dimensionality of spacetime via (4.6). In the limits and where we are probing very large and small distances, respectively, we obtain the dimensionalities corresponding to the largest and smallest length scales possible. The limits and of are determined by the behavior of for and , respectively.

The quantum gravity effects stop below some scale energy that we denoted by and we have . In this case (4.19) yields , and we conclude that the macroscopic spectral dimension is . In the next section we apply the introduced ideas to the spatial section in LQG and to the space-time in the covariant spin-foam formulation of quantum gravity.

The result we will find about the hight energy spectral dimension are of general character. The above assumption that is flat was not necessary for obtaining the spectral dimension at any fixed scale. This follows from the fact that even for a curved metric the spectral sum (4.17) can be represented by an Euler-Maclaurin series which always implies (4.18) as the leading term for .
Proof of (4.19).