Abstract
We consider Uncertainty Principles which take into account the role of gravity and the possible existence of extra spatial dimensions. Explicit expressions for such Generalized Uncertainty Principles in 4+n dimensions are given and their holographic properties investigated. In particular, we show that the predicted number of degrees of freedom enclosed in a given spatial volume matches the holographic counting only for one of the available generalizations and without extra dimensions.
Generalized uncertainty principle, extradimensions and holography
Fabio Scardigli^{I}; Roberto Casadio^{II}
^{I}CENTRA  Departamento de Fisica, Instituto Superior Tecnico, Av. Rovisco Pais 1, 1049001 Lisboa, Portugal
^{II}Dipartimento di Fisica, Università di Bologna and I.N.F.N., Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
ABSTRACT
We consider Uncertainty Principles which take into account the role of gravity and the possible existence of extra spatial dimensions. Explicit expressions for such Generalized Uncertainty Principles in 4+n dimensions are given and their holographic properties investigated. In particular, we show that the predicted number of degrees of freedom enclosed in a given spatial volume matches the holographic counting only for one of the available generalizations and without extra dimensions.
1 Introduction
During the last years many efforts have been devoted to clarifying the role played by the existence of extra spatial dimensions in the theory of gravity 1,2]. One of the most interesting predictions drawn from the theory is that there should be measurable deviations from the 1/r^{2} law of Newtonian gravity at short (and perhaps also at large) distances. Such new laws of gravity would imply modifications of those Generalized Uncertainty Principles (GUP's) designed to account for gravitational effects in the measure of positions and energies.
On the other hand, the holographic principle is claimed to apply to all of the gravitational systems. The existence of GUP's satisfying the holography in four dimensions (one of the main examples is due to Ng and Van Dam [3]) led us to explore the holographic properties of the GUP's extended to the braneworld scenarios [4]. The results, at least for the examples we considered, are quite surprising. The expected holographic scaling indeed seems to hold only in four dimensions, and only for the Ng and van Dam's GUP. When extra spatial dimensions are admitted, the holography is destroyed. This fact allows two different interpretations: either the holographic principle is not universal and does not apply when extra dimensions are present; or, on the contrary, we take seriously the holographic claim in any number of dimensions, and our results are therefore evidence against the existence of extra dimensions.
The fourdimensional Newton constant is denoted by G_{N} throughout the paper.
2 Linear GUP in four dimensions from micro black holes
In this Section we derive a GUP via a micro black hole gedanken experiment, following closely the content of Ref. [5].
When we measure a position with precision of order Dx, we expect quantum fluctuations of the metric field around the measured position with energy amplitude
The Schwarzschild radius associated with the energy DE,
falls well inside the interval Dx for practical cases. However, if we wanted to improve the precision indefinitely, the fluctuation DE would grow up and the corresponding R_{S} would become larger and larger, until it reaches the same size as Dx. As it is well known, the critical length is the Planck length,
and the associated energy is the Planck energy
If we tried to further decrease Dx, we should concentrate in that region an energy greater than the Planck energy, and this would enlarge further the Schwarzschild radius R_{S}, hiding more and more details of the region beyond the event horizon of the micro hole. The situation can be summarized by the inequalities
which, if combined linearly, yield
This is a generalization of the uncertainty principle to cases in which gravity is important, i.e. to energies of the order of Î_{p}. We note that the minimum value of Dx is reached for (DE)_{min} = Î_{p} and is given by (Dx)_{min} = 2 l_{p}.
2.1 Holographic properties
In this section, we investigate the holographic properties of the GUP proposed above. We shall estimate the number of degrees of freedom n(V) contained in a spatial volume (cube or "hypercube'') of size l. The holographic principle claims that n(V) scales as the area of the (hyper)surface enclosing the given volume, that is (l/l_{p})^{2+n} in 4+n dimensions.
For the fourdimensional GUP considered in the previous section, Eq. (6), we find that this scaling does not occur. In fact, (Dx)_{min} ~ l_{p} and a cube of side l contains a number of degrees of freedom equal to
We then conclude that this GUP, obtained by linearly combining the quantum mechanical expression with gravitational bounds, does not imply the holographic counting of degrees of freedom.
3 Ng and Van Dam GUP in four dimensions
An interesting GUP that satisfies the holographic principle in four dimensions has been proposed by Ng and van Dam [3], based on Wigner inequalities about distance measurements with clocks and light signals [6].
Suppose we wish to measure a distance l. Our measuring device is composed of a clock, a photon detector and a photon gun. A mirror is placed at the distance l which we want to measure and m is the mass of the system "clock + photon detector + photon gun''. We call "detector'' the whole system and let a be its size. Obviously, we suppose
which means that we are not using a black hole as a clock. Be Dx_{1} the uncertainty in the position of the detector, then the uncertainty in the detector's velocity is
After the time T = 2 l/c taken by light to travel along the closed path detectormirrordetector, the uncertainty in the detector's position (i.e. the uncertainty in the actual length of the segment l) has become
We can minimize Dx_{tot} by suitably choosing Dx_{1},
Hence
Since T = 2 l/c, we have
This is a purely quantum mechanical result obtained for the first time by Wigner in 1957 [6]. From Eq. (13), it seems that we can reduce the error (Dx_{tot})_{min} as much as we want by choosing m very large, since (Dx_{tot})_{min }® 0 for m ® ¥. But, obviously, here gravity enters the game.
In fact, Ng and van Dam have also considered a further source of error, a gravitational error, besides the quantum mechanical one already addressed. Suppose the clock has spherical symmetry, with a > r_{g}. Then the error due to curvature can be computed from the Schwarzschild metric surrounding the clock. The optical path from r_{0} > r_{g} to a generic point r > r_{0} is given by (see, for example, Ref. [7])
and differs from the "true'' (spatial) length (r r_{0}). If we put a = r_{0}, l = r, the gravitational error on the measure of (l a) is thus
where the last estimate holds for l > a >> r_{g}.
If we measure a distance l > 2a, then the error due to curvature is
Thus, according to Ng and van Dam the total error is
This error can be minimized again by choosing a suitable value for the mass of the clock,
and, inserting m_{min} in Eq. (17), we then have
The global uncertainty on l contains therefore a term proportional to l^{1/3}.
3.1 Holographic properties
We now see immediately the beauty of the Ng and van Dam GUP: it obeys the holographic scaling. In fact in a cube of size l the number of degrees of freedom is given by
as required by the holographic principle.
4 Models with n extra dimensions
We shall now generalize the procedure outlined in a previous section to a spacetime with 4+n dimensions, where n is the number of spacelike extra dimensions [4]. The first problem we should address is how to relate the gravitational constant G_{N} in four dimensions with the one in 4+n, henceforth denoted by G_{(4+n)}.
This of course depends on the model of spacetime with extra dimensions that we consider. Models recently appeared in the literature mostly belong to two scenarios:

the ArkaniHamedDimopoulosDvali (ADD) model [1], where the extra dimensions are compact and of size L;

the RandallSundrum (RS) model [2], where the extra dimensions have an infinite extension but are warped by a nonvanishing cosmological constant.
A feature shared by (the original formulations of) both scenarios is that only gravity propagates along the n extra dimensions, while Standard Model fields are confined on a fourdimensional submanifold usually referred to as the braneworld .
In the ADD case the link between G_{N} and G_{(4+n)} can be fixed by comparing the gravitational action in four dimensions with the one in 4+n dimensions. The spacetime topology in such models is , where is the usual fourdimensional spacetime and Â^{n} represents the extra dimensions of finite size L. The spacetime brane has no tension and therefore the action S_{(4+n)} can be written as
where , are the projections on of R and g. Here L^{n} is the "volume'' of the extra dimensions and we omitted unimportant numerical factors. On comparing the above expression with the purely fourdimensional action
we obtain
The RS models are more complicated. It can be shown [2] that for n = 1 extra dimension we have G_{(4+n)} = s^{1}G_{N}, where s is the brane tension with dimensions of length^{1} in suitable units. The gravitational force between two pointlike masses m and M on the brane is now given by
where the correction to Newton law comes from summing over the extra dimensional graviton modes in the graviton propagator [2]. However, since Eq. (24) is obtained by perturbative calculations, not immediately applicable to a nonperturbative structure such as a black hole, we shall consider only the ADD scenario in this paper. To be more precise, from tabletop tests of the gravitational force one finds that n > 2 in ADD [1,8]. On the other hand, black holes with mass M << s^{1} are likely to behave as pure fivedimensional in RS [9], therefore our results for n = 1 should apply to such a case.
5 Ng and Van Dam GUP in 4+n dimensions
Ng and van Dam's derivation can be generalized to the case with n extra dimensions. The Wigner relation (13) for the quantum mechanical error is not modified by the presence of extra dimensions and we just need to estimate the error dl_{C} due to curvature.
We ought not to consider micro black holes created by the fluctuations DE in energy, as in Section 2, but we have rather to deal with (more or less) macroscopic clocks and distances. This implies that we have to distinguish four different cases:
1. 0 < L < r_{g} < a < l;
2. 0 < r_{(4+n)} < L < a < l;
3. 0 < r_{(4+n)} < a < L < l;
4. 0 < r_{(4+n)} < a < l < L;
where r_{(4+n)} is the Schwarzschild radius of the detector in 4+n dimensions, and of course r_{g} = r_{(4)}. The curvature error will be estimated (as before) by computing the optical path from a º r_{0} to l º r. Of course, we will use a metric which depends on the relative size of L with respect to a and l, that is the usual fourdimensional Schwarzschild metric in the region r > L, and the 4+n dimensional Schwarzschild solution in the region r < L (where the extra dimensions play an actual role).
In cases 1. and 2 the length of the optical path from a to l can be obtained using just the fourdimensional Schwarzschild solution and the result is given by Eq. (19).
In cases 3. and 4. we instead have to use the Schwarzschild solution in 4+n dimensions [11],
at least for part of the optical path. In the above,
and A_{n+2} is the area of the unit (n+2)sphere, that is
Besides, we note that, for n = 0,
that is, C coincides in four dimensions with the Schwarzschild radius of the detector. The Schwarzschild horizon is located where (1 C/r^{n}^{+1}) = 0, that is at r = C^{1/(n+1)}º r_{(4+n)}, or
where
is an unimportant numerical factor.
Since measurements can be performed only on the brane, to the uncertainty Dx in position we can still associate an energy given by Eq (1). The corresponding Schwarzschild radius is now given by Eq. (29) with m = DE/c^{2} and the critical length such that Dx = r_{(4+n)} is the Planck length in 4+n dimensions,
For sake of simplicity (because a(0) = 1 and in any case a(n) ~ 1) we define the Planck length in 4+n dimensions as
The energy associated with l_{(4+n)} is analogously the Planck energy in 4+n dimensions,
where Î_{p} is the Planck energy in four dimensions given in Eq. (4).
In case 3. we obtain the length of the optical path from a to l by adding the optical path from a to L and that from L to l. We must use the solution in 4+n dimensions for the first part, and the fourdimensional solution for the second part of the path,
It is not difficult to show that from r_{(4+n)} < L (which holds in cases 3. and 4.) we can infer
Now, suppose a^{n}^{+1} >> C = , that is a >> r_{(4+n)}, so that we are not doing measures inside a black hole. Then r_{g} < r_{(4+n)} << a < L < l and
The error caused by the curvature (when a < L < l) is therefore linear in m,
We recall that the curvature error in four dimensions does not contain the size of the clock. On the contrary, this error in 4+n dimensions depends explicitly on the size a of the clock and on the size L of the extra dimensions. Hence the total error is given by
where J = 2 (l/c)^{1/2} and K is defined above. This error can be minimized with respect to m,
Finally,
where we used the definition of J and K.
In case 4., the optical path from a to l can be obtained by using simply the Schwarzschild solution in 4+n dimensions. We get
Suppose now, as before, that a^{n}^{+1} >> C = , that is a >> r_{(4+n)} (i.e. our clock is not a black hole). We then have
If the distance we are measuring is, at least, of the size of the clock (l > 2 a), we can write
The error caused by the curvature is therefore (when a < l < L)
Here we again note that the curvature error in 4+n dimensions explicitly contains the size of the clock. The global error can be computed as before
where C is linear in m. Minimizing dl_{tot} with respect to m can be done in perfect analogy with the previous calculation. The result is
We note that the expression (40) coincides in the limit L ® a with Eq. (19) (taking l > 2 a), while, in the limit L ® l, we recover from Eq. (40) the expression (46) (of course, supposing also that l > 2 a).
5.1 Holographic properties
We finally examine the holographic properties of Eq. (46) for the GUP of Ng and van Dam type in 4+n dimensions. We just consider the expression in Eq. (46) because it also represents the limit of Eq. (40) for L ® l and l > 2 a. Moreover, for n = 0, Eq. (46) yields the fourdimensional error given in Eq. (19).
Since we are just interested in the dependence of n(V) on l and the basic constants, we can write
We then have that the number of degrees of freedom in the volume of size l is
and the holographic counting holds in fourdimensions (n = 0) but is lost when n > 0. In fact we do not get something as
as we would expect in 4+n dimensions. Even if we take the ideal case a ~ l_{(4+n)} we get
and the holographic principle does not hold for n > 0.
6 Concluding remarks
In the previous Sections, we have shown that the holographic principle seems to be satisfied only by uncertainty relations in the version of Ng and van Dam and for n = 0. That is, only in four dimensions we are able to formulate uncertainty principles which predict the same number of degrees of freedom per spatial volume as the holographic counting. This could be evidence for questioning the existence of extra dimensions. Moreover, such an argument based on holography could also be used to support the compactification of string theory down to four dimensions, given that there seems to be no firm argument which forces the low energy limit of string theory to be fourdimensional (except for the obvious observation of our world). In this respect, we should also say that the cases 3. and 4 of Section 5 do not have a good probability to be realized in nature since, if there are extra spatial dimensions, their size must be shorter than 10^{1} mm [8]. Therefore, cases 1. and 2. of Section 5 are more likely to survive the test of future experiments.
A number of general remarks are however in order. First of all, we cannot claim that our list of possible GUP's is complete and other relations might be derived in different contexts which accommodate for both the holography and extra dimensions. Further, one might find hard to accept that quantum mechanics and general relativity enter the construction of GUP's on the same footing, since the former is supposed to be a fundamental framework for all theories while the latter can be just regarded as a (effective) theory of the gravitational interaction. We might agree on the point of view that GUP's must be considered as "effective'' (phenomenological) bounds valid at low energy (below the Planck scale) rather than "fundamental'' relations. This would in fact reconcile our result that four dimensions are preferred with the fact that string theory (as a consistent theory of quantum gravity) requires more dimensions through the compactification which must occur at low energy, as we mentioned above. Let us also note that general relativity (contrary to usual field theories) determines the spacetime including the causality structure, and the latter is an essential ingredient in all actual measurements. It is therefore (at least) equally hard to conceive uncertainty relations which neglect general relativity at all. This conclusion would become even stronger in the presence of extra dimensions, since the fundamental energy scale of gravity is then lowered [1,2] (possibly) within the scope of present or nearfuture experiments and the gravitational radius of matter sources is correspondingly enlarged [10].
A final remark regards cases with less than four dimensions. Since Einstein gravity does not propagate in such spacetimes and no direct analogue of the Schwarzschild solution exists, one expects a qualitative difference with respect to the cases that we have considered here. For instance, a pointlike source in three dimensions would generate a flat spacetime with a conical singularity and no horizon [12]. Consequently, one does expect that the usual Heisenberg uncertainty relations hold with no corrections for gravity.
References
[1] N. ArkaniHamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429, 263 (1998); Phys. Rev. D 59, 0806004 (1999); I. Antoniadis, N. ArkaniHamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 436, 257 (1998).
[2] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); Phys. Rev. Lett. 83, 4690 (1999).
[3] Y.J. Ng and H. van Dam, Mod. Phys. Lett. A 9, 335 (1994); Mod. Phys. Lett. A 10, 2801 (1995); Phys. Lett. B 477, 429 (2000); Y.J. Ng, Phys. Rev. Lett. 86, 2946 (2001).
[4] F. Scardigli and R. Casadio, Class. Qu. Grav. 20, 3915 (2003).
[5] F. Scardigli, Phys. Lett. B 452, 39 (1999).
[6] E.P. Wigner, Rev. Mod. Phys. 29, 255 (1957); H. Salecker and E.P. Wigner, Phys. Rev. 109, 571 (1958).
[7] L.D. Landau and E.M. Lifshitz, The classical theory of fields (Pergamon Press, Oxford, 1975).
[8] C.D. Hoyle et al., Phys. Rev. Lett. 86, 1418 (2001).
[9] S.B. Giddings, E. Katz, and L. Randall, JHEP 0003, 023 (2000); J. Garriga and T. Tanaka, Phys. Rev. Lett. 84, 2778 (2000).
[10] P.C. Argyres, S. Dimopoulos, and J. MarchRussell, Phys. Lett. B 441, 96 (1998).
[11] R.C. Myers and M.J. Perry, Ann. Phys. 172, 304 (1986).
[12] In three dimensions with negative cosmological constant one also has the BTZ black hole which forms when two pointlike particles collide provided certain initial conditions are satisfied. For a recent review, see Ref. [13].
[13] D. Birmingham, I. Sachs, and S. Sen, Int. J. Mod. Phys. D 10, 833 (2001).
Received on 23 December, 2004
 [1] N. ArkaniHamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429, 263 (1998);
 Phys. Rev. D 59, 0806004 (1999);
 I. Antoniadis, N. ArkaniHamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 436, 257 (1998).
 [2] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999);
 Phys. Rev. Lett. 83, 4690 (1999).
 [3] Y.J. Ng and H. van Dam, Mod. Phys. Lett. A 9, 335 (1994);
 Mod. Phys. Lett. A 10, 2801 (1995);
 Phys. Lett. B 477, 429 (2000);
 Y.J. Ng, Phys. Rev. Lett. 86, 2946 (2001).
 [4] F. Scardigli and R. Casadio, Class. Qu. Grav. 20, 3915 (2003).
 [5] F. Scardigli, Phys. Lett. B 452, 39 (1999).
 [6] E.P. Wigner, Rev. Mod. Phys. 29, 255 (1957);
 H. Salecker and E.P. Wigner, Phys. Rev. 109, 571 (1958).
 [7] L.D. Landau and E.M. Lifshitz, The classical theory of fields (Pergamon Press, Oxford, 1975).
 [8] C.D. Hoyle et al., Phys. Rev. Lett. 86, 1418 (2001).
 [9] S.B. Giddings, E. Katz, and L. Randall, JHEP 0003, 023 (2000);
 J. Garriga and T. Tanaka, Phys. Rev. Lett. 84, 2778 (2000).
 [10] P.C. Argyres, S. Dimopoulos, and J. MarchRussell, Phys. Lett. B 441, 96 (1998).
 [11] R.C. Myers and M.J. Perry, Ann. Phys. 172, 304 (1986).
 [13] D. Birmingham, I. Sachs, and S. Sen, Int. J. Mod. Phys. D 10, 833 (2001).
Publication Dates

Publication in this collection
06 Sept 2005 
Date of issue
June 2005
History

Received
23 Dec 2004