Foamy structure of spacetime

Abstract

We examine spectrum of the physical volume operator within the non-standard loop quantum cosmology. The spectrum is discrete with equally distant levels defining a quantum of the volume. The discreteness may imply a foamy structure of spacetime at semi-classical level which may be detected in astro-cosmo observations. Comment: 5 pages, no figures, improved version

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arXiv:0907.4647v3 [gr-qc] 26 Oct 2009
Foamy structure of spacetime
Przemys? law Ma? lkiewicz1, ∗and W? lodzimierz Piechocki1, †
1Theoretical Physics Department, Institute for Nuclear Studies,
Ho˙ za 69, 00-681 Warszawa, Poland
(Dated: October 26, 2009)
We examine spectrum of the physical volume operator within the non-standard
loop quantum cosmology. The spectrum is discrete with equally distant levels defin-
ing a quantum of the volume. The discreteness may imply a foamy structure of
spacetime at semi-classical level which may be detected in astro-cosmo observations.
PACS numbers: 98.80.Qc,04.60.Pp
1. Introduction. Various forms of discrete-
ness of spacetime underly many approaches
in fundamental physics.
few: noncommutative geometry [1], causal
sets approach [2], gravitational Wilson loops
[3], Regge calculus [4], path integral over ge-
ometries [5], spin foam model [6], and cate-
gories [7]. The discreteness may translate at
the semi-classical level into a foamy structure
of space. Such expected property of space-
time creates large activity in observational
astrophysics and cosmology (see, e.g. Lorentz
and CPT violation [8], dispersion of cosmic
photons [10], electrons [9] and neutrinos [11],
birefringence effects [12]).
The goal of our paper is presentation of
the physics of geometry at short distances.
We study the spectrum of the volume oper-
ator. We find that the spectrum is bounded
from below and discrete. The minimum dis-
tance between the levels of the spectrum de-
fines a quantum of the volume. We suggest
that there may exist elementary quanta of en-
ergy connected with the multiplicity of the
quantum of the volume, i.e. with a foamy like
structure of space. They may have a form of
elementary objects like photon, electron, pro-
ton, dark matter particle, etc.
Our results suggest that the foamy struc-
ture of space is likely to be a real property
of the Universe so its identification via astro-
cosmo observations has sound motivation and
is important for the fundamental physics.
Our results are obtained within the non-
Just to name a
standard loop quantum cosmology (LQC) de-
veloped recently [13, 14, 15, 16, 17]. In this
method one first solves classical constraints
to identify the physical phase space and finds
an algebra of observables, then one imposes
quantum rules. Standard LQC [20, 21] means
first quantization of the kinematics, then im-
position of constraints in the form of opera-
tors acting on the kinematical Hilbert space.
Both methods should ‘commute’, i.e. give the
same results. In the case of quantization of
the Maxwell electrodynamics such treatment
of constraints leads to equivalent results [22].
Thus, another aim of our paper is of method-
ological nature: testing the equivalence of
both methods in the case of gravitational in-
teraction.
This Letter is meant to address a wide
physical community. It popularizes and in-
terprets the results of a ‘technical’ paper [17]
directed to experts in quantum cosmology,
and submitted for publication elsewhere.
A direct way of testing the singularity as-
pects of a given cosmological model is by an
examination of the energy density of matter
as a function of time [14]. However, the ge-
ometry of space, as a function of time, is sen-
sitive to these aspects too. We carry out the
corresponding discussion in [17].
2. Modified Hamiltonian. In what follows, for
simplicity of exposition, we restrict ourselves
to the quantum flat Friedmann-Robertson-
Walker (FRW) model with massless scalar
field.
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The classical dynamics may be defined by
the FRW Hamiltonian
?
H = N−
3
8πGγ2β2v +p2
φ
2v
?
≈ 0, (1)
which is known to be a dynamical constraint
[14]; N denotes the lapse function and γ
is the so-called Barbaro-Immirzi parameter;
(β,v,φ,pφ) are the kinematical phase space
variables; v1/3∼ a, where a is the scale fac-
tor; β ∼ ˙ a/a so it corresponds to the Hubble
parameter; pφis the momentum of the mass-
less scalar field φ.
The Hamiltonian modified by the so-called
holonomy functions specific to LQC, corre-
sponding to (1), is found to be [15]
?
8πGγ2
H(λ)= N−
3
sin2(λβ)
λ2
v +p2
φ
2v
?
≈ 0.
(2)
The parameter λ is a free parameter of the
non-standard LQC method parameterizing
holonomies. It is clear that in the limit λ → 0
the Hamiltonian (2) turns into (1). In what
follows we consider (2) in the gauge
N−1:=
3
8πGγ2v
?
κγ|pφ|+v|sin(λβ)|
λ
?
, (3)
where κ2≡ 4πG/3. Consequently, (2) leads
to the dynamical constraint
H(λ):= κγ|pφ| − v|sin(λβ)|
λ
≈ 0. (4)
The FRW constraint in the gauge corre-
sponding to (3), reads
H := κγ|pφ| − v |β| ≈ 0.
Since sin(·) is bounded from above, there
exists ǫ ∈ R, due to (4), such that v > ǫ > 0.
Thus, there exists ε ∈ R such that the scale
factor a > ε > 0. As sin(·) is a periodic
function, the variable β which occurs in (4)
is bounded. Thus, the Hubble parameter is
bounded, which means that there is no Big-
Bang. The variables β and v which satisfy
(5) do not have such properties so there is
(5)
Big-Bang. Thus, the modification of the clas-
sical Hamiltonian turns Big-Bang into Big-
Bounce. In what follows we show that quan-
tization of the bouncing dynamics inevitably
leads to discrete spectrum of the volume op-
erator.
It turns out that the physical phase space
may be parameterized by the elementary ob-
servables [15],
O1:= pφ,O2:= φ−sgn(pφ)
3κ
arth?cos(λβ)?,
(6)
i.e.
bracket with (4) on the constraint surface
H(λ)≈ 0.
The classical dynamics has been solved an-
alytically [15] and the explicit form of the so-
lution for the variable v, which is of interest
in the present paper, is given by
functions having vanishing Poisson
v(φ) = κγλ|O1| cosh3κ(φ − O2).
The variable φ changes monotonically with
an evolution so it has been chosen to be an
evolution parameter of the system [15].
3. Volume Operator. The variable v has the
interpretation of a volume of some piece of
space [19]. To define quantum operator cor-
responding to v, we use the classical observ-
ables (6)
(7)
v = |w|, w := κγλ O1 cosh3κ(φ−O2). (8)
Thus, quantization of v reduces to the quan-
tization problem of w. Quantization of the
latter may be done in a standard way as fol-
lows [17]
ˆ wf(x) := κγλ1
2
?? O1cosh3κ(φ −? O2)
+ cosh3κ(φ −? O2)? O1
where f ∈ L2(R), and where φ is a scalar field
used both at classical and quantum levels as
an evolution parameter [14].
For the elementary observables O1and O2
we use the Schr¨ odinger representation
?f(x), (9)
O1−→? O1f(x) := −i?∂xf(x),
O2−→? O2f(x) := ? xf(x) := xf(x).(10)

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In the representation (10) an explicit form
of the operator ˆ w is
ˆ wf(x) = iκγλ?
2
?2cosh3κ(φ − x)
−3κsinh3κ(φ − x)?f(x). (11)
4. Eigenvalue problem. It turns out that the
solution to the eigenvalue problem
d
dx
ˆ wfa(x) = afa(x),a ∈ R, (12)
reads [17]
fa(x) :=
?
3κ
πexp?i
2a
3κ2γλ?arctane3κ(φ−x)?
cosh
1
23κ(φ − x)
.
(13)
The condition ?fb|fa? = 0 leads to
a − b = 6κ2γλ?m = 8πGγλ?m,
where m ∈ Z. Thus, the set
Fb:= { fa| a = b + 8πGγλ?m},
where b ∈ R, is orthonormal. Each subspace
Fb⊂ L2(R) spans a pre-Hilbert space. The
completion of each span Fb in the norm of
L2(R) defines an infinite dimensional separa-
ble Hilbert space Hb. Since
?fb| ˆ wfa? − ?ˆ wfb|fa? = (a − b)?fb|fa?, (16)
the operator ˆ w is symmetric on Fb for any
b ∈ R. In fact, it is a self-adjoint operator on
the span of Fb(see, [17] for a proof).
5. Spectrum. Due to the the relation (8) and
the spectral theorem on self-adjoint operators
[26, 27], we may carry out quantization of the
volume function on each Fbas follows
v = |w|
A common feature of all Fb is the exis-
tence of the minimum gap △ := 8πGγ?λ in
the spectrum, which defines a quantum of the
volume. In the limit λ → 0, corresponding to
the classical FRW model without the loop ge-
ometry modification, there is no quantum of
the volume.
(14)
(15)
−→ˆ vfa:= |a|fa. (17)
It results from (14) that for b = 0 and
m = 0 the minimum eigenvalue of ˆ v equals
zero. It is a special case that corresponds
to the classical situation when v = 0, which
due to (4) means that pφ = 0 so there is
no classical dynamics (for more details see
[15]). Thus, we have a direct correspondence
between classical and quantum levels corre-
sponding to this very special state. It is clear
that all other states describe bouncing dy-
namics.
6. Free parameter. There exists a fundamen-
tal problem underlying LQC (see, [13] and
references therein), which is the unknown nu-
merical value of the parameter λ [28].
Determination of λ by standard LQC
means [18, 19]: (a) considering eigenvalue
problem for the area operator,?
?
to discrete eigenvalues, {0,?,...}, of kine-
matical?
leads to λ = 3√3/2.
One postulates in standard LQC that a
surface cannot be squeezed to the zero value
due to the existence in the Universe of the
quantum of area.
Physical justification for the assumption
on the existence of quantum of area, offered
by standard LQC, seems to be doubtful be-
cause: (d)?
?
spectrum of LQG was used to replace con-
tinuous spectrum of standard LQC, which is
the spectral discretization by hand; and (f)
standard LQC is not a cosmological sector of
LQG, but a quantization method inspired by
LQG [29].
This is why we propose to treat λ as a free
parameter yet to be determined.
7. Conclusions. As the Universe expands
a discrete spectrum of the volume operator
favors a foamy structure which turns into a
Ar =?
|p|, in
kinematical phase space of standard LQC:
Ar |µ? =
continuous since µ ∈ R; (b) making reference
Ar of LQG, where ? := 2√3πγl2
and (c) assuming that ar(λ) ≡ ?, which
4πγl2
3
p
|µ||µ? =: ar(µ)|µ? so ar(µ) is
p;
Ar has been examined in kinemat-
ical Hilbert space of LQG, i.e. spectrum of
Ar ignores the algebra of constraints of LQG
so it has poor physical meaning; (e) discrete

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continuous spacetime with time. The classi-
cal FRW model is commonly used in obser-
vational cosmology because it fits quite well
the data. Thus, the detection of any cosmic
events favoring the foamy spacetime would
give support to the quantum FRW model.
Our non-standard LQC method gives re-
sults concerning geometrical properties of
space on the physical phase space so they may
be verified by the data of observational cos-
mology.
There exist results concerning the spec-
trum of the volume operator obtained within
LQG (see, e.g. [30, 31]), but cannot be com-
pared easily with our results due to the lack
of a direct correspondence between LQG and
LQC methods (see Sec 6f).
Both standard and non-standard LQC
methods offer the resolution of the initial Big-
Bang singularity in the sense that the singu-
larity is replaced by the regular Big-Bounce
(BB) transition. However, the energy scale
specific to BB (the scale of unification of
gravity with quantum physics) has not been
determined satisfactory yet [14]. The prob-
lem reduces to the problem of determination
of the minimum length [13]. Can it be solved
by making use of the cosmic data? There ex-
ists speculation that the foamy structure of
spacetime may lead to the dependence of the
velocity of a photon on its energy. Such de-
pendance is weak, but may sum up to give a
measurable effect in the case of photons trav-
elling over cosmological distances across the
Universe [35]. Presently, available data sug-
gest that such dispersion effects do not occur
up to the energy scale 5 × 1017GeV [36] so
this type of effects may be present, but at
higher energies.
The quantum of volume may be used
as a measure of a size, λf, of a spacetime
foam. One may speculate that λf:= △1/3=
?8πGγ?λ?1/3.
that determine a size of spacetime ‘granu-
larity’ λf may fix the minimum length pa-
rameter λ of LQC. That would enable mak-
ing an estimate of the critical matter density
Thus, an astro-cosmo data
ρmax = 1/2(κγλ)2corresponding to the BB
[17].
The granularity of volume should lead to
the granularity of energy of physical fields.
We suggest, making use of the de Broglie re-
lation, that a specific particle representing a
quantum of energy may have a momentum pi
corresponding to its wavelength λisuch that
piλi= ?. The detection of an ultrahigh en-
ergy particle with specific pimay be used to
determine λi, and consequently set the up-
per limit for the fundamental length λf. The
set of parameters λi (for a set of particles)
may be treated further as multiplicities of λf
in which case the greatest common divisor of
all λiwould set the lowest upper limit for λf.
The standard and non-standard LQC
methods give comparable results as they pre-
dict the appearance of the Big-Bounce transi-
tion parameterized by a free parameter to be
determined [14]. Both methods seem to com-
mute so there exists an analogy to the case of
quantum electrodynamics [22]. However, our
method is fully controlled analytically as it
does not require any numerical work. It may
be also linked with the loop quantum gravity
(LQG) by finding relation with the reduced
phase space quantization [37].
Acknowledgments.
Iwo Bia? lynicki-Birula, Jan Derezi´ nski, Piotr
Dzier˙ zak, Anatol Odzijewicz, Wies? law Pusz,
and Jakub Rembieli´ nski for helpful discus-
sions.
We are grateful to
∗Electronic address: pmalk@fuw.edu.pl
†Electronic address: piech@fuw.edu.pl
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