November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
Modern Physics Letters A
c© World Scientific Publishing Company
Planck Length and Cosmology
Xavier Calmet
Service de Physique Théorique, CP225
Boulevard du Triomphe
B-1050 Brussels
Belgium
xavier.calmet@ulb.ac.be
Received (Day Month Year)
Revised (Day Month Year)
We show that an unification of quantum mechanics and general relativity implies that
there is a fundamental length in Nature in the sense that no operational procedure would
be able to measure distances shorter than the Planck length. Furthermore we give an
explicit realization of an old proposal by Anderson and Finkelstein who argued that a
fundamental length in nature implies unimodular gravity. Finally, using hand waving
arguments we show that a minimal length might be related to the cosmological constant
which, if this scenario is realized, is time dependent.
Keywords: General Relativity; Quantum Mechanics; Cosmology.
PACS Nos.: 98.80.-k, 04.20.-q.
1. Introduction
The idea that a unification of quantum mechanics and general relativity implies
the notion of a fundamental length is not new1. However, it has only recently
been established that no operational procedure could exclude the discreteness of
space-time on distances shorter than the Planck length2. This makes the case for
a fundamental length of the order of the Planck length much stronger. It seems
reasonable to think that any quantum description of general relativity will have to
include the fact that measurement of distance shorter than the Planck length are
forbidden. It is notoriously difficult to build a quantum theory of gravity. Besides
technical difficulties the lack of experimental guidance, the Planck length being
so miniscule lP ∼ 10−33cm, is flagrant. In this work we shall however argue that
a fundamental length in nature, even if it is as small as the Planck scale may
have dramatic impacts on our universe. In particular, we will argue that it may be
related to the vacuum energy, i.e. dark energy and thus account for roughly 70% of
the energy of the universe.
We shall first present our motivation for a minimal length which follows from
quantum mechanics, general relativity and causality. We will then argue that a fun-
http://arxiv.org/abs/0704.1360v1
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
2 Xavier Calmet
damental length in nature may lead to unimodular gravity. If this is the case, the
cosmological constant is an integration parameter and is thus arbitrary. Finally, we
shall consider argument based on spacetime quantization to argue that the cosmo-
logical constant might not be actually constant but might be time dependent.
2. Minimal Length from Quantum Mechanics and General
Relativity
We first review the results obtained in ref.2. We show that quantum mechanics
and classical general relativity considered simultaneously imply the existence of a
minimal length, i.e. no operational procedure exists which can measure a distance
less than this fundamental length. The key ingredients used to reach this conclusion
are the uncertainty principle from quantum mechanics, and gravitational collapse
from classical general relativity.
A dynamical condition for gravitational collapse is given by the hoop
conjecture3: if an amount of energy E is confined at any instant to a ball of size
R, where R < E, then that region will eventually evolve into a black hole. We use
natural units where ~, c and Newton’s constant (or lP ) are unity. We also neglect
numerical factors of order one.
From the hoop conjecture and the uncertainty principle, we immediately deduce
the existence of a minimum ball of size lP . Consider a particle of energy E which
is not already a black hole. Its size r must satisfy
r ∼> max [ 1/E , E ] , (1)
where λC ∼ 1/E is its Compton wavelength and E arises from the hoop conjecture.
Minimization with respect to E results in r of order unity in Planck units or r ∼ lP .
If the particle is a black hole, then its radius grows with mass: r ∼ E ∼ 1/λC . This
relationship suggests that an experiment designed (in the absence of gravity) to
measure a short distance l << lP will (in the presence of gravity) only be sensitive
to distances 1/l.
Let us give a concrete model of minimum length. Let the position operator
x̂ have discrete eigenvalues {xi}, with the separation between eigenvalues either of
order lP or smaller. For regularly distributed eigenvalues with a constant separation,
this would be equivalent to a spatial lattice. We do not mean to imply that nature
implements minimum length in this particular fashion - most likely, the physical
mechanism is more complicated, and may involve, for example, spacetime foam or
strings. However, our concrete formulation lends itself to detailed analysis. We show
below that this formulation cannot be excluded by any gedanken experiment, which
is strong evidence for the existence of a minimum length.
Quantization of position does not by itself imply quantization of momentum.
Conversely, a continuous spectrum of momentum does not imply a continuous spec-
trum of position. In a formulation of quantum mechanics on a regular spatial lattice,
with spacing a and size L, the momentum operator has eigenvalues which are spaced
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
Xavier Calmet 3
by 1/L. In the infinite volume limit the momentum operator can have continuous
eigenvalues even if the spatial lattice spacing is kept fixed. This means that the
displacement operator
x̂(t)− x̂(0) = p̂(0) t
does not necessarily have discrete eigenvalues (the right hand side of (2) assumes
free evolution; we use the Heisenberg picture throughout). Since the time evolu-
tion operator is unitary the eigenvalues of x̂(t) are the same as x̂(0). Importantly
though, the spectrum of x̂(0) (or x̂(t)) is completely unrelated to the spectrum of
the p̂(0), even though they are related by (2). A measurement of arbitrarily small
displacement (2) does not exclude our model of minimum length. To exclude it, one
would have to measure a position eigenvalue x and a nearby eigenvalue x′, with
|x− x′| << lP .
Many minimum length arguments are obviated by the simple observation of the
minimum ball. However, the existence of a minimum ball does not by itself preclude
the localization of a macroscopic object to very high precision. Hence, one might
attempt to measure the spectrum of x̂(0) through a time of flight experiment in
which wavepackets of primitive probes are bounced off of well-localised macroscopic
objects. Disregarding gravitational effects, the discrete spectrum of x̂(0) is in princi-
ple obtainable this way. But, detecting the discreteness of x̂(0) requires wavelengths
comparable to the eigenvalue spacing. For eigenvalue spacing comparable or smaller
than lP , gravitational effects cannot be ignored, because the process produces min-
imal balls (black holes) of size lP or larger. This suggests a direct measurement of
the position spectrum to accuracy better than lP is not possible. The failure here
is due to the use of probes with very short wavelength.
A different class of instrument, the interferometer, is capable of measuring dis-
tances much smaller than the size of any of its sub-components. Nevertheless, the
uncertainty principle and gravitational collapse prevent an arbitrarily accurate mea-
surement of eigenvalue spacing. First, the limit from quantum mechanics. Consider
the Heisenberg operators for position x̂(t) and momentum p̂(t) and recall the stan-
dard inequality
(∆A)2(∆B)2 ≥ − 1
(〈[Â, B̂]〉)2 . (3)
Suppose that the position of a free mass is measured at time t = 0 and again at a
later time. The position operator at a later time t is
x̂(t) = x̂(0) + p̂(0)
. (4)
We assume a free particle Hamiltonian here for simplicity, but the argument can be
generalized2. The commutator between the position operators at t = 0 and t is
[x̂(0), x̂(t)] = i
, (5)
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
4 Xavier Calmet
so using (3) we have
|∆x(0)||∆x(t)| ≥ t
. (6)
We see that at least one of the uncertainties ∆x(0) or ∆x(t) must be larger than
of order
t/M . As a measurement of the discreteness of x̂(0) requires two position
measurements, it is limited by the greater of ∆x(0) or ∆x(t), that is, by
t/M ,
∆x ≡ max [∆x(0),∆x(t)] ≥
, (7)
where t is the time over which the measurement occurs andM the mass of the object
whose position is measured. In order to push ∆x below lP , we take M to be large. In
order to avoid gravitational collapse, the size R of our measuring device must also
grow such that R > M . However, by causality R cannot exceed t. Any component of
the device a distance greater than t away cannot affect the measurement, hence we
should not consider it part of the device. These considerations can be summarized
in the inequalities
t > R > M . (8)
Combined with (7), they require ∆x > 1 in Planck units, or
∆x > lP . (9)
Notice that the considerations leading to (7), (8) and (9) were in no way specific
to an interferometer, and hence are device independent. In summary, no device
subject to quantum mechanics, gravity and causality can exclude the quantization
of position on distances less than the Planck length.
3. Minimal Length and Unimodular Gravity
General relativity is a scaleless theory:
SGR =
−gR(g) (10)
varying this action with respect to the metric gµν leads to the well-known Einstein
equations. The action (10) is invariant under general coordinate transformations
and this may seem at odd with the notation of a minimal or fundamental length
in nature. This may suggest that a quantum mechanical description of general
relativity will fix the measure of Einstein-Hilbert action
−g to some constant
linked to the fundamental length. In that case one is led to unimodular gravity:
SGR =
d4xR(g) (11)
with the constraint
−g = constant which implies that only variation of the metric
which respect this contraint may be considered. This is basically the argument made
by Anderson and Finkelstein 4 in favor of a unimodular theory of gravity.
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
Xavier Calmet 5
There may be different ways to implement a minimal length in a theory, but
we shall concentrate on one approach based on a noncommutative spacetime which
indeed leads to a unimodular theory of gravity. Positing a noncommutative relation
between e.g. x and y implies ∆x∆y ≥ |θxy| ∼ l2, with [x̂, ŷ] = iθxy and where l
is the minimal length introduced in the theory. This also implies that a spacetime
volume is quantized ∆V ≥ l4.
One of the motivations to consider a noncommutative spacetime is that the non-
commutative relations for the coordinates imply the existence of a minimal which
can be thought of being proportional to the square root of the vacuum expecta-
tion value of θµν i.e. lmin ∼
θ. If this length is fundamental it should not de-
pend on the observer. Assuming the invariance of this fundamental length, one can
show that there is a class of spacetime symmetries called noncommutative Lorentz
transformations5 which preserve this length. It has recently been shown6, that there
are also general coordinate transformations ξµ(x̂) that leave the canonical noncom-
mutative algebra invariant and thus conserve the minimal length:
[x̂µ, x̂ν ] = iθµν , (12)
where θµν is constant and antisymmetric. They are of the form: ξµ(x̂) = θµν∂νf(x̂),
where f(x̂) is an arbitrary field. The Jacobian of these restricted coordinate trans-
formations is equal to one. This implies that the four-volume element is invariant:
d4x′ = d4x. These noncommutative transformations correspond to volume preserv-
ing diffeomorphisms which preserve the noncommutative algebra. A canonical non-
commutative spacetime thus restricts general coordinate transformations to volume
preserving coordinate transformations. These transformations are the only coordi-
nate transformations that leave the canonical noncommutative algebra invariant.
They form a subgroup of the unimodular transformations of a classical spacetime.
The version of General Relativity based on volume-preserving diffeomorphism
is known as the unimodular theory of gravitation7. Unimodular gravity here ap-
pears as a direct consequence of spacetime noncommutativity defined by a constant
antisymmetric θµν . One way to formulate gravity on a noncommutative spacetime
has been presented in refs.6. Our approach might not be unique, but if the non-
commutative model is reasonable, it must have a limit in which one recovers the
commutative unimodular gravity theory in the limit in which θµν goes to zero. For
small θµν we thus expect
SNC =
d4xR(gµν) +O(θ), (13)
where R(gµν) is the usual Ricci scalar Once matter is included, one finds the fol-
lowing equations of motion:
Rµν −
gµνR = −8πG(T µν −
gµνT λλ) +O(θ). (14)
These equations do not involve a cosmological constant and the contribution of
vacuum fluctuations automatically cancel on the right-hand side of eq.(14). As done
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
6 Xavier Calmet
in e.g. ref.13 we can use the Bianchi for R and the equations of motion for T =
−8πGT λλ and find:
Dµ(R+ T ) = 0 (15)
which can be integrated easily and give R + T = −Λ, where Λ is an integration
constant. It can then be shown that the differential equations (14) imply
Rµν − 1
gµνR− Λgµν = −8πGT µν −O(θ), (16)
i.e. Einstein’s equations20 of General Relativity with a cosmological constant Λ
that appears as an integration constant and is thus uncorrelated to any of the
parameters of the action (13). As we have shown, one needs to impose energy
conservation and the Bianchi identities to derive eq.(16) from eq.(14). Because any
solution of Einstein’s equations with a cosmological constant can, at least over any
topologically R4 open subset of spacetime, be written in a coordinate system with
g = −1, the physical content of unimodular gravity is identical at the classical level
to that of Einstein’s gravity with some cosmological constant13.
4. Cosmological implications of spacetime quantization
We now come to the link between a fundamental length and cosmology and rephrase
the arguments developed in refs.14,15,16,17,18 within the framework of a fundamen-
tal length. It has been shown that the quantization of an unimodular gravity action
proposed by Henneaux and Teitelboim12, which is an extension of the action de-
fined in eq. (13), leads to an uncertainty relation between the fluctuations of the
volume V and those of the cosmological constant Λ: δV δΛ ∼ 1 using natural units,
i.e. ~ = lp = c = mp = 1. Now if spacetime is quantized, as it is the case for
noncommuting coordinates, we expect the number of cells of spacetime to fluctu-
ate according to a Poisson distribution, δN ∼
N , where N is the number of
cells. This is however obviously an assumption which could only be justified by a
complete understanding of noncommutative quantum gravity. It is then natural to
assume that the volume fluctuates with the number of spacetime cells δV = δN .
One finds δV ∼
V and thus Λ ∼ V − 12 , i.e., we obtain an effective cosmological
constant which varies with the four-volume as obtained in a different context in
refs.19,14,15,16. In deriving this result, we have assumed as in refs.14,15 that the
fluctuation are around zero as explained below. A minimal length thus leads leads
to a vacuum energy density ρ
ρ ∼ 1√
. (17)
Here we assume that the scale for the quantization of spacetime is the Planck
scale. A crucial assumption made in refs.14,15,16 as well is that the value of cosmo-
logical constant fluctuates around zero. This was made plausible by Baum21 and
Hawking22 using an Euclidean formulation of quantum gravity.
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
Xavier Calmet 7
Now the question is really to decide what we mean by the four-volume V . If
this is the four-volume related to the Hubble radius RH as in refs.
14,15,16 then this
model predicts ρ ∼ (10−3eV )4 which is the right order for today’s energy density, it
is however not obvious what is the equation of state for this effective cosmological
constant. The choice V = R4H might be ruled out because of the equation of state
of such a dark energy model as shown in ref.23 in the context of holographic dark
energy which leads to similar phenomenology. However if we assume that the four-
volume is related to the future event horizon as suggested by M. Li24, again in
the context of holographic dark energy, then we get an equation of state which is
compatible with the data w = −0.903 + 1.04z which is precisely the equation of
state for the holographic dark energy obtained in ref.24. Details will appear in a
forthcoming publication.
5. Conclusions
We have argued that an unification of quantum mechanics and general relativity
implies that there is a fundamental length in Nature in the sense that no opera-
tional procedure would be able to measure distances shorter than the Planck length.
Further we give an explicit realization of an old proposal by Anderson and Finkel-
stein who had argued that a fundamental length in nature would imply unimodular
gravity. Finally, using hand waving arguments we show that a minimal length might
be related to the cosmological constant, which if this scenario is realized, is time
dependent and thus only effectively a constant. Much more work remains to be
done to establish this connection. It would be interesting to related the time de-
pendence of the cosmological constant to that of other parameters of the standard
model such as the fine-structure constant. Indeed as argued in refs.25 if one of the
parameters of the standard model, such as a gauge coupling, a mass term or any
other cosmological parameter, is time dependent, it is quite natural to expect that
the remaining parameters of the theory will be time dependent as well.
Acknowledgments
I would like to thank Professor Xiao-Gang He and the Physics Department of the
National Taiwan University for their hospitality during my stay at NTU. I am
grateful to Professors Xiao-Gang He and Pauchy W-Y. Hwang for their invitation
to present this work at the CosPA 2006 meeting. This work was supported in part
by the IISN and the Belgian science policy office (IAP V/27).
References
1. C. A. Mead, Phys. Rev. 135, B849 (1964).
2. X. Calmet, M. Graesser and S. D. H. Hsu, Phys. Rev. Lett. 93, 211101 (2004); Int. J.
Mod. Phys. D 14, 2195 (2005); X. Calmet, arXiv:hep-th/0701073.
3. K. S. Thorne, Nonspherical gravitational collapse: A short review, in J. R . Klauder,
Magic Without Magic, San Francisco 1972, 231–258.
http://arxiv.org/abs/hep-th/0701073
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
8 Xavier Calmet
4. J. L. Anderson and D. Finkelstein, Am. J. Phys. 39, 901 (1971), see also D. R. Finkel-
stein, A. A. Galiautdinov and J. E. Baugh, J. Math. Phys. 42, 340 (2001).
5. X. Calmet, Phys. Rev. D 71, 085012 (2005).
6. X. Calmet and A. Kobakhidze, Phys. Rev. D 72, 045010 (2005); Phys. Rev. D 74,
047702 (2006); X. Calmet, arXiv:hep-th/0510165, to appear in Europhys.Lett.
7. A. Einstein, Siz. Preuss. Acad. Scis., (1919); “Do Gravitational Fields Play an essential
Role in the Structure of Elementary Particle of Matter,” in The Principle of Relativity:
A Collection of Original Memoirs on the Special and General Theory of Relativity by
A. Einstein et al (Dover, New York, 1923), pp. 191–198, English translation.
8. J. J. van der Bij, H. van Dam and Y. J. Ng, Physica 116A, 307 (1982).
9. M. Henneaux and C. Teitelboim, Phys. Lett. B 143, 415 (1984).
10. F. Wilczek, Phys. Rept. 104, 143 (1984).
11. W. Buchmuller and N. Dragon, Phys. Lett. B 207, 292 (1988).
12. M. Henneaux and C. Teitelboim, Phys. Lett. B 222, 195 (1989).
13. W. G. Unruh, Phys. Rev. D 40, 1048 (1989).
14. Y. J. Ng and H. van Dam, Int. J. Mod. Phys. D 10, 49 (2001).
15. Y. J. Ng, Mod. Phys. Lett. A 18, 1073 (2003).
16. M. Ahmed, S. Dodelson, P. B. Greene and R. Sorkin, Phys. Rev. D 69, 103523 (2004).
17. R .D. Sorkin, in Relativity and Gravitation: Classical and Quantum (Proceedings of
the SILARG VII Conference, held Cocoyoc, Mexico, December, 1990), edited by J.C.
D’Olivo, E. Nahmad-Achar, M. Rosenbaum, M.P. Ryan, L.F. Urrutia and F. Zertuche,
(World Scientific, Singapore, 1991, pp. 150-173.
18. R. D. Sorkin, Int. J. Th. Phys. 36: 2759–2781 (1997).
19. W. Chen and Y. S. Wu, Phys. Rev. D 41, 695 (1990) [Erratum-ibid. D 45, 4728
(1992)].
20. A. Einstein, “The Foundation Of The General Theory Of Relativity,” Annalen Phys.
49, 769 (1916).
21. E. Baum, Phys. Lett. B 133, 185 (1983).
22. S. W. Hawking, Phys. Lett. B 134, 403 (1984).
23. S. D. H. Hsu, Phys. Lett. B 594, 13 (2004).
24. M. Li, Phys. Lett. B 603, 1 (2004).
25. X. Calmet and H. Fritzsch, Eur. Phys. J. C 24, 639 (2002); Phys. Lett. B 540, 173
(2002); arXiv:hep-ph/0211421; Europhys. Lett. 76, 1064 (2006).
http://arxiv.org/abs/hep-th/0510165
http://arxiv.org/abs/hep-ph/0211421

  We show that an unification of quantum mechanics and general relativity
implies that there is a fundamental length in Nature in the sense that no
operational procedure would be able to measure distances shorter than the
Planck length. Furthermore we give an explicit realization of an old proposal
by Anderson and Finkelstein who argued that a fundamental length in nature
implies unimodular gravity. Finally, using hand waving arguments we show that a
minimal length might be related to the cosmological constant which, if this
scenario is realized, is time dependent.

Introduction
The idea that a unification of quantum mechanics and general relativity implies
the notion of a fundamental length is not new1. However, it has only recently
been established that no operational procedure could exclude the discreteness of
space-time on distances shorter than the Planck length2. This makes the case for
a fundamental length of the order of the Planck length much stronger. It seems
reasonable to think that any quantum description of general relativity will have to
include the fact that measurement of distance shorter than the Planck length are
forbidden. It is notoriously difficult to build a quantum theory of gravity. Besides
technical difficulties the lack of experimental guidance, the Planck length being
so miniscule lP ∼ 10−33cm, is flagrant. In this work we shall however argue that
a fundamental length in nature, even if it is as small as the Planck scale may
have dramatic impacts on our universe. In particular, we will argue that it may be
related to the vacuum energy, i.e. dark energy and thus account for roughly 70% of
the energy of the universe.
We shall first present our motivation for a minimal length which follows from
quantum mechanics, general relativity and causality. We will then argue that a fun-
http://arxiv.org/abs/0704.1360v1
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
2 Xavier Calmet
damental length in nature may lead to unimodular gravity. If this is the case, the
cosmological constant is an integration parameter and is thus arbitrary. Finally, we
shall consider argument based on spacetime quantization to argue that the cosmo-
logical constant might not be actually constant but might be time dependent.
2. Minimal Length from Quantum Mechanics and General
Relativity
We first review the results obtained in ref.2. We show that quantum mechanics
and classical general relativity considered simultaneously imply the existence of a
minimal length, i.e. no operational procedure exists which can measure a distance
less than this fundamental length. The key ingredients used to reach this conclusion
are the uncertainty principle from quantum mechanics, and gravitational collapse
from classical general relativity.
A dynamical condition for gravitational collapse is given by the hoop
conjecture3: if an amount of energy E is confined at any instant to a ball of size
R, where R < E, then that region will eventually evolve into a black hole. We use
natural units where ~, c and Newton’s constant (or lP ) are unity. We also neglect
numerical factors of order one.
From the hoop conjecture and the uncertainty principle, we immediately deduce
the existence of a minimum ball of size lP . Consider a particle of energy E which
is not already a black hole. Its size r must satisfy
r ∼> max [ 1/E , E ] , (1)
where λC ∼ 1/E is its Compton wavelength and E arises from the hoop conjecture.
Minimization with respect to E results in r of order unity in Planck units or r ∼ lP .
If the particle is a black hole, then its radius grows with mass: r ∼ E ∼ 1/λC . This
relationship suggests that an experiment designed (in the absence of gravity) to
measure a short distance l << lP will (in the presence of gravity) only be sensitive
to distances 1/l.
Let us give a concrete model of minimum length. Let the position operator
x̂ have discrete eigenvalues {xi}, with the separation between eigenvalues either of
order lP or smaller. For regularly distributed eigenvalues with a constant separation,
this would be equivalent to a spatial lattice. We do not mean to imply that nature
implements minimum length in this particular fashion - most likely, the physical
mechanism is more complicated, and may involve, for example, spacetime foam or
strings. However, our concrete formulation lends itself to detailed analysis. We show
below that this formulation cannot be excluded by any gedanken experiment, which
is strong evidence for the existence of a minimum length.
Quantization of position does not by itself imply quantization of momentum.
Conversely, a continuous spectrum of momentum does not imply a continuous spec-
trum of position. In a formulation of quantum mechanics on a regular spatial lattice,
with spacing a and size L, the momentum operator has eigenvalues which are spaced
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
Xavier Calmet 3
by 1/L. In the infinite volume limit the momentum operator can have continuous
eigenvalues even if the spatial lattice spacing is kept fixed. This means that the
displacement operator
x̂(t)− x̂(0) = p̂(0) t
does not necessarily have discrete eigenvalues (the right hand side of (2) assumes
free evolution; we use the Heisenberg picture throughout). Since the time evolu-
tion operator is unitary the eigenvalues of x̂(t) are the same as x̂(0). Importantly
though, the spectrum of x̂(0) (or x̂(t)) is completely unrelated to the spectrum of
the p̂(0), even though they are related by (2). A measurement of arbitrarily small
displacement (2) does not exclude our model of minimum length. To exclude it, one
would have to measure a position eigenvalue x and a nearby eigenvalue x′, with
|x− x′| << lP .
Many minimum length arguments are obviated by the simple observation of the
minimum ball. However, the existence of a minimum ball does not by itself preclude
the localization of a macroscopic object to very high precision. Hence, one might
attempt to measure the spectrum of x̂(0) through a time of flight experiment in
which wavepackets of primitive probes are bounced off of well-localised macroscopic
objects. Disregarding gravitational effects, the discrete spectrum of x̂(0) is in princi-
ple obtainable this way. But, detecting the discreteness of x̂(0) requires wavelengths
comparable to the eigenvalue spacing. For eigenvalue spacing comparable or smaller
than lP , gravitational effects cannot be ignored, because the process produces min-
imal balls (black holes) of size lP or larger. This suggests a direct measurement of
the position spectrum to accuracy better than lP is not possible. The failure here
is due to the use of probes with very short wavelength.
A different class of instrument, the interferometer, is capable of measuring dis-
tances much smaller than the size of any of its sub-components. Nevertheless, the
uncertainty principle and gravitational collapse prevent an arbitrarily accurate mea-
surement of eigenvalue spacing. First, the limit from quantum mechanics. Consider
the Heisenberg operators for position x̂(t) and momentum p̂(t) and recall the stan-
dard inequality
(∆A)2(∆B)2 ≥ − 1
(〈[Â, B̂]〉)2 . (3)
Suppose that the position of a free mass is measured at time t = 0 and again at a
later time. The position operator at a later time t is
x̂(t) = x̂(0) + p̂(0)
. (4)
We assume a free particle Hamiltonian here for simplicity, but the argument can be
generalized2. The commutator between the position operators at t = 0 and t is
[x̂(0), x̂(t)] = i
, (5)
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
4 Xavier Calmet
so using (3) we have
|∆x(0)||∆x(t)| ≥ t
. (6)
We see that at least one of the uncertainties ∆x(0) or ∆x(t) must be larger than
of order
t/M . As a measurement of the discreteness of x̂(0) requires two position
measurements, it is limited by the greater of ∆x(0) or ∆x(t), that is, by
t/M ,
∆x ≡ max [∆x(0),∆x(t)] ≥
, (7)
where t is the time over which the measurement occurs andM the mass of the object
whose position is measured. In order to push ∆x below lP , we take M to be large. In
order to avoid gravitational collapse, the size R of our measuring device must also
grow such that R > M . However, by causality R cannot exceed t. Any component of
the device a distance greater than t away cannot affect the measurement, hence we
should not consider it part of the device. These considerations can be summarized
in the inequalities
t > R > M . (8)
Combined with (7), they require ∆x > 1 in Planck units, or
∆x > lP . (9)
Notice that the considerations leading to (7), (8) and (9) were in no way specific
to an interferometer, and hence are device independent. In summary, no device
subject to quantum mechanics, gravity and causality can exclude the quantization
of position on distances less than the Planck length.
3. Minimal Length and Unimodular Gravity
General relativity is a scaleless theory:
SGR =
−gR(g) (10)
varying this action with respect to the metric gµν leads to the well-known Einstein
equations. The action (10) is invariant under general coordinate transformations
and this may seem at odd with the notation of a minimal or fundamental length
in nature. This may suggest that a quantum mechanical description of general
relativity will fix the measure of Einstein-Hilbert action
−g to some constant
linked to the fundamental length. In that case one is led to unimodular gravity:
SGR =
d4xR(g) (11)
with the constraint
−g = constant which implies that only variation of the metric
which respect this contraint may be considered. This is basically the argument made
by Anderson and Finkelstein 4 in favor of a unimodular theory of gravity.
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
Xavier Calmet 5
There may be different ways to implement a minimal length in a theory, but
we shall concentrate on one approach based on a noncommutative spacetime which
indeed leads to a unimodular theory of gravity. Positing a noncommutative relation
between e.g. x and y implies ∆x∆y ≥ |θxy| ∼ l2, with [x̂, ŷ] = iθxy and where l
is the minimal length introduced in the theory. This also implies that a spacetime
volume is quantized ∆V ≥ l4.
One of the motivations to consider a noncommutative spacetime is that the non-
commutative relations for the coordinates imply the existence of a minimal which
can be thought of being proportional to the square root of the vacuum expecta-
tion value of θµν i.e. lmin ∼
θ. If this length is fundamental it should not de-
pend on the observer. Assuming the invariance of this fundamental length, one can
show that there is a class of spacetime symmetries called noncommutative Lorentz
transformations5 which preserve this length. It has recently been shown6, that there
are also general coordinate transformations ξµ(x̂) that leave the canonical noncom-
mutative algebra invariant and thus conserve the minimal length:
[x̂µ, x̂ν ] = iθµν , (12)
where θµν is constant and antisymmetric. They are of the form: ξµ(x̂) = θµν∂νf(x̂),
where f(x̂) is an arbitrary field. The Jacobian of these restricted coordinate trans-
formations is equal to one. This implies that the four-volume element is invariant:
d4x′ = d4x. These noncommutative transformations correspond to volume preserv-
ing diffeomorphisms which preserve the noncommutative algebra. A canonical non-
commutative spacetime thus restricts general coordinate transformations to volume
preserving coordinate transformations. These transformations are the only coordi-
nate transformations that leave the canonical noncommutative algebra invariant.
They form a subgroup of the unimodular transformations of a classical spacetime.
The version of General Relativity based on volume-preserving diffeomorphism
is known as the unimodular theory of gravitation7. Unimodular gravity here ap-
pears as a direct consequence of spacetime noncommutativity defined by a constant
antisymmetric θµν . One way to formulate gravity on a noncommutative spacetime
has been presented in refs.6. Our approach might not be unique, but if the non-
commutative model is reasonable, it must have a limit in which one recovers the
commutative unimodular gravity theory in the limit in which θµν goes to zero. For
small θµν we thus expect
SNC =
d4xR(gµν) +O(θ), (13)
where R(gµν) is the usual Ricci scalar Once matter is included, one finds the fol-
lowing equations of motion:
Rµν −
gµνR = −8πG(T µν −
gµνT λλ) +O(θ). (14)
These equations do not involve a cosmological constant and the contribution of
vacuum fluctuations automatically cancel on the right-hand side of eq.(14). As done
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
6 Xavier Calmet
in e.g. ref.13 we can use the Bianchi for R and the equations of motion for T =
−8πGT λλ and find:
Dµ(R+ T ) = 0 (15)
which can be integrated easily and give R + T = −Λ, where Λ is an integration
constant. It can then be shown that the differential equations (14) imply
Rµν − 1
gµνR− Λgµν = −8πGT µν −O(θ), (16)
i.e. Einstein’s equations20 of General Relativity with a cosmological constant Λ
that appears as an integration constant and is thus uncorrelated to any of the
parameters of the action (13). As we have shown, one needs to impose energy
conservation and the Bianchi identities to derive eq.(16) from eq.(14). Because any
solution of Einstein’s equations with a cosmological constant can, at least over any
topologically R4 open subset of spacetime, be written in a coordinate system with
g = −1, the physical content of unimodular gravity is identical at the classical level
to that of Einstein’s gravity with some cosmological constant13.
4. Cosmological implications of spacetime quantization
We now come to the link between a fundamental length and cosmology and rephrase
the arguments developed in refs.14,15,16,17,18 within the framework of a fundamen-
tal length. It has been shown that the quantization of an unimodular gravity action
proposed by Henneaux and Teitelboim12, which is an extension of the action de-
fined in eq. (13), leads to an uncertainty relation between the fluctuations of the
volume V and those of the cosmological constant Λ: δV δΛ ∼ 1 using natural units,
i.e. ~ = lp = c = mp = 1. Now if spacetime is quantized, as it is the case for
noncommuting coordinates, we expect the number of cells of spacetime to fluctu-
ate according to a Poisson distribution, δN ∼
N , where N is the number of
cells. This is however obviously an assumption which could only be justified by a
complete understanding of noncommutative quantum gravity. It is then natural to
assume that the volume fluctuates with the number of spacetime cells δV = δN .
One finds δV ∼
V and thus Λ ∼ V − 12 , i.e., we obtain an effective cosmological
constant which varies with the four-volume as obtained in a different context in
refs.19,14,15,16. In deriving this result, we have assumed as in refs.14,15 that the
fluctuation are around zero as explained below. A minimal length thus leads leads
to a vacuum energy density ρ
ρ ∼ 1√
. (17)
Here we assume that the scale for the quantization of spacetime is the Planck
scale. A crucial assumption made in refs.14,15,16 as well is that the value of cosmo-
logical constant fluctuates around zero. This was made plausible by Baum21 and
Hawking22 using an Euclidean formulation of quantum gravity.
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
Xavier Calmet 7
Now the question is really to decide what we mean by the four-volume V . If
this is the four-volume related to the Hubble radius RH as in refs.
14,15,16 then this
model predicts ρ ∼ (10−3eV )4 which is the right order for today’s energy density, it
is however not obvious what is the equation of state for this effective cosmological
constant. The choice V = R4H might be ruled out because of the equation of state
of such a dark energy model as shown in ref.23 in the context of holographic dark
energy which leads to similar phenomenology. However if we assume that the four-
volume is related to the future event horizon as suggested by M. Li24, again in
the context of holographic dark energy, then we get an equation of state which is
compatible with the data w = −0.903 + 1.04z which is precisely the equation of
state for the holographic dark energy obtained in ref.24. Details will appear in a
forthcoming publication.
5. Conclusions
We have argued that an unification of quantum mechanics and general relativity
implies that there is a fundamental length in Nature in the sense that no opera-
tional procedure would be able to measure distances shorter than the Planck length.
Further we give an explicit realization of an old proposal by Anderson and Finkel-
stein who had argued that a fundamental length in nature would imply unimodular
gravity. Finally, using hand waving arguments we show that a minimal length might
be related to the cosmological constant, which if this scenario is realized, is time
dependent and thus only effectively a constant. Much more work remains to be
done to establish this connection. It would be interesting to related the time de-
pendence of the cosmological constant to that of other parameters of the standard
model such as the fine-structure constant. Indeed as argued in refs.25 if one of the
parameters of the standard model, such as a gauge coupling, a mass term or any
other cosmological parameter, is time dependent, it is quite natural to expect that
the remaining parameters of the theory will be time dependent as well.
Acknowledgments
I would like to thank Professor Xiao-Gang He and the Physics Department of the
National Taiwan University for their hospitality during my stay at NTU. I am
grateful to Professors Xiao-Gang He and Pauchy W-Y. Hwang for their invitation
to present this work at the CosPA 2006 meeting. This work was supported in part
by the IISN and the Belgian science policy office (IAP V/27).
References
1. C. A. Mead, Phys. Rev. 135, B849 (1964).
2. X. Calmet, M. Graesser and S. D. H. Hsu, Phys. Rev. Lett. 93, 211101 (2004); Int. J.
Mod. Phys. D 14, 2195 (2005); X. Calmet, arXiv:hep-th/0701073.
3. K. S. Thorne, Nonspherical gravitational collapse: A short review, in J. R . Klauder,
Magic Without Magic, San Francisco 1972, 231–258.
http://arxiv.org/abs/hep-th/0701073
November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal-
8 Xavier Calmet
4. J. L. Anderson and D. Finkelstein, Am. J. Phys. 39, 901 (1971), see also D. R. Finkel-
stein, A. A. Galiautdinov and J. E. Baugh, J. Math. Phys. 42, 340 (2001).
5. X. Calmet, Phys. Rev. D 71, 085012 (2005).
6. X. Calmet and A. Kobakhidze, Phys. Rev. D 72, 045010 (2005); Phys. Rev. D 74,
047702 (2006); X. Calmet, arXiv:hep-th/0510165, to appear in Europhys.Lett.
7. A. Einstein, Siz. Preuss. Acad. Scis., (1919); “Do Gravitational Fields Play an essential
Role in the Structure of Elementary Particle of Matter,” in The Principle of Relativity:
A Collection of Original Memoirs on the Special and General Theory of Relativity by
A. Einstein et al (Dover, New York, 1923), pp. 191–198, English translation.
8. J. J. van der Bij, H. van Dam and Y. J. Ng, Physica 116A, 307 (1982).
9. M. Henneaux and C. Teitelboim, Phys. Lett. B 143, 415 (1984).
10. F. Wilczek, Phys. Rept. 104, 143 (1984).
11. W. Buchmuller and N. Dragon, Phys. Lett. B 207, 292 (1988).
12. M. Henneaux and C. Teitelboim, Phys. Lett. B 222, 195 (1989).
13. W. G. Unruh, Phys. Rev. D 40, 1048 (1989).
14. Y. J. Ng and H. van Dam, Int. J. Mod. Phys. D 10, 49 (2001).
15. Y. J. Ng, Mod. Phys. Lett. A 18, 1073 (2003).
16. M. Ahmed, S. Dodelson, P. B. Greene and R. Sorkin, Phys. Rev. D 69, 103523 (2004).
17. R .D. Sorkin, in Relativity and Gravitation: Classical and Quantum (Proceedings of
the SILARG VII Conference, held Cocoyoc, Mexico, December, 1990), edited by J.C.
D’Olivo, E. Nahmad-Achar, M. Rosenbaum, M.P. Ryan, L.F. Urrutia and F. Zertuche,
(World Scientific, Singapore, 1991, pp. 150-173.
18. R. D. Sorkin, Int. J. Th. Phys. 36: 2759–2781 (1997).
19. W. Chen and Y. S. Wu, Phys. Rev. D 41, 695 (1990) [Erratum-ibid. D 45, 4728
(1992)].
20. A. Einstein, “The Foundation Of The General Theory Of Relativity,” Annalen Phys.
49, 769 (1916).
21. E. Baum, Phys. Lett. B 133, 185 (1983).
22. S. W. Hawking, Phys. Lett. B 134, 403 (1984).
23. S. D. H. Hsu, Phys. Lett. B 594, 13 (2004).
24. M. Li, Phys. Lett. B 603, 1 (2004).
25. X. Calmet and H. Fritzsch, Eur. Phys. J. C 24, 639 (2002); Phys. Lett. B 540, 173
(2002); arXiv:hep-ph/0211421; Europhys. Lett. 76, 1064 (2006).
http://arxiv.org/abs/hep-th/0510165
http://arxiv.org/abs/hep-ph/0211421
