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arXiv:gr-qc/0512139v2 17 Mar 2006
Naturalness in an emergent analogue spacetime
Stefano Liberati∗
International School for Advanced Studies and INFN,
Via Beirut 2-4, 34014 Trieste, Italy,
Matt Visser†and Silke Weinfurtner‡
School of Mathematics, Statistics, and Computer Science,
Victoria University of Wellington, PO Box 600, Wellington, New Zealand
(Dated: 23 December 2005; revised 11 February 2006; LATEX-ed February 7, 2008)
Effective field theories (EFTs) have been widely used as a framework in order to place constraints
on the Planck suppressed Lorentz violations predicted by various models of quantum gravity. There
are however technical problems in the EFT framework when it comes to ensuring that small Lorentz
violations remain small — this is the essence of the “naturalness” problem. Herein we present
an “emergent” space-time model, based on the “analogue gravity” programme, by investigating a
specific condensed-matter system that is in principle capable of simulating the salient features of an
EFT framework with Lorentz violations. Specifically, we consider the class of two-component BECs
subject to laser-induced transitions between the components, and we show that this model is an
example for Lorentz invariance violation due to ultraviolet physics. Furthermore our model explicitly
avoids the “naturalness problem”, and makes specific suggestions regarding how to construct a
physically reasonable quantum gravity phenomenology.
Introduction: The purpose of quantum gravity phe-
nomenology (QGP) is to analyze the physical conse-
quences arising from various models of quantum gravity
(QG). One hope for obtaining an experimental grasp on
QG is the generic prediction arising in many (but not
all) quantum gravity models that ultraviolet physics at
or near the Planck scale, MPlanck= 1.2 × 1019GeV/c2,
(or in some models the string scale), typically induces vi-
olations of Lorentz invariance (LI) at lower scales [1, 2].
Interestingly most investigations, even if they arise from
quite different fundamental physics, seem to converge
on the prediction that the breakdown of Lorentz invari-
ance (LI) can generically become manifest in the form
of modified dispersion relations exhibiting extra energy-
dependent or momentum-dependent terms, apart from
the usual quadratic one occurring in the Lorentz invari-
ant dispersion relation. In particular one most often con-
siders Lorentz invariance violations (LIV) in the boost
subgroup, leading to an expansion of the dispersion rela-
tion in momentum-dependent terms,
ω2= ω2
0+(1 + η2) c2k2+η4
?
¯ h
MLIV
?2
k4+... , (1)
where the coefficients ηnare dimensionless (and possibly
dependent on the particle species considered), and we
have restricted our expansion to CPT invariant terms
(otherwise one would also get odd powers in k). The
particular inertial frame for these dispersion relations is
generally specified to be the frame set by Cosmological
Microwave Background (CMB), and MLIVis the scale of
Lorentz symmetry breaking which furthermore is gener-
ally assumed to be of the order of MPlanck.
Although several alternative scenarios have been con-
sidered in the literature in order to justify modified kine-
matics of the kind of Eq. (1), so far the most commonly
explored avenue is an effective field theory (EFT) ap-
proach. In the present article we wish to focus on the
class of non-renormalizable EFTs with Lorentz violations
associated to dispersion relations like Eq. (1). Relaxing
our CPT invariance condition this class would include
the model developed in [3], and subsequently studied by
several authors, where an extension of QED including
only mass dimension five Lorentz-violating operators was
considered. (That ansatz leads to order k3LI and CPT
violating terms in the dispersion relation.) Very accu-
rate constraints have been obtained for this model using
a combination of experiments and observations (mainly
in high energy astrophysics). See e.g. [2].
In spite of the remarkable success of this framework
as a “test theory”, it is interesting to note that there
are still significant open issues concerning its theoreti-
cal foundations.This is often referred to as the nat-
uralness problem and can be expressed in the follow-
ing way.Looking back at our ansatz (1) we can see
that the lowest-order correction, proportional to η2, is
not explicitly Planck suppressed. This implies that such
term would always be dominant with respect to the
higher order ones and grossly incompatible with obser-
vations (given that we have very good constraints on
the universality of the speed of light for different ele-
mentary particles). Following the observational leads it
has been therefore often assumed either that some sym-
metry (other than Lorentz invariance) enforces the η2
coefficients to be exactly zero, or that the presence of
some other characteristic EFT mass scale µ ≪ MPlanck
(e.g., some particle physics mass scale) associated with
the Lorentz symmetry breaking might enter in the low-
est order dimensionless coefficient η2 — which will be
then generically suppressed by appropriate ratios of this
characteristic mass to the Planck mass: η2∝ (µ/MPl)σ
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2
where σ ≥ 1 is some positive power (often taken as one or
two). If this is the case then one has two distinct regimes:
For low momenta p/(MPlc) ≪ (µ/MPl)σthe lower-order
(quadratic in the momentum) deviations in (1) will dom-
inate over the higher-order ones, while at high energies
p/(MPlc) ≫ (µ/MPl)σthe higher order terms will be
dominant.
The naturalness problem arises because such a scenario
is not well justified within an EFT framework; in other
words there is no natural suppression of the low-order
modifications in these models. In fact we implicitly as-
sumed that there are no extra Planck suppressions hidden
in the dimensionless coefficients ηnwith n > 2. EFT can-
not justify why only the dimensionless coefficients of the
n ≤ 2 terms should be suppressed by powers of the small
ratio µ/MPl. Even worse, renormalization group argu-
ments seem to imply that a similar mass ratio, µ/MPl
would implicitly be present also in all the dimensionless
n > 2 coefficients — hence suppressing them even fur-
ther, to the point of complete undetectability. Further-
more it is easy to show [4] that, without some protecting
symmetry, it is generic that radiative corrections due to
particle interactions in an EFT with only Lorentz vio-
lations of order n > 2 in (1) for the free particles, will
generate n = 2 Lorentz violating terms in the dispersion
relation, which will then be dominant. Observational ev-
idence [1, 5] suggests that for a variety of standard model
particles |η2|<
imply that the higher order terms are at least as sup-
pressed as this, and hence beyond observational reach.
In order to contribute to this debate, we have cho-
sen a rather unconventional path: We have investigated
a condensed matter analog model (AM) of an emergent
spacetime [6], that reproduces the salient features of the
the non-renormalizable EFT with LIV adopted in quan-
tum gravity phenomenology studies.
looked for a condensed matter system characterized by 1)
relativistic kinematics for the low-energy quasi-particles
2) presence of Lorentz violation at high energies due to
the underling microscopic structure 3) coexistence on the
same background of more than one quasi-particle species
(in order to be able to see the presence of deviations from
LI at order k2).
Standard Bose–Einstein condensates (BEC) are in this
sense interesting systems as they fulfill the first two of the
above requirements (see e.g. [7, 8]): their excitations can
be described at low energy as relativistic phonons prop-
agating on a geometrical background, and at higher or-
der their dispersion relations show modifications of order
k4(this is the so called Bogoliubov dispersion relation,
which has been experimentally confirmed [9]). Unfortu-
nately in a single BEC system there is only one species of
phonon, and hence it is impossible to address the ques-
tion of naturalness. For this reason we chose to investi-
gate the energy dependent behavior of quasi-particles in
a 2-component Bose–Einstein condensate. For a specific
∼10−21. Naturalness in EFT would then
In particular we
choice of parameters such a system allows both a mass-
less and a massive quasi-particle (see [10]), which share
the same relativistic causal structure in the low energy
limit. It is then natural to investigate the expected vio-
lation of Lorentz invariance as the high energy regime of
the theory is probed. (For a detailed discussion, see [11].)
Sound waves in 2-component BECs: The basis for our
model is an ultra-cold dilute atomic gas of N bosons
in two single-particle states |A? and |B?. For example
one could consider a two-component condensate of87Rb
atoms in different hyperfine levels (see e.g. [12].) The
two states have slightly different energies, which permits
us, from a theoretical point of view, to keep mA?= mB,
even if in experimentally realizable situations mA≈ mB.
There are three atom-atom coupling constants, UAA,
UBB, and UAB, and for our purposes it is essential to
include an additional coupling λ that drives transitions
between the two single-particle states. For temperatures
at or below the critical BEC temperature, almost all
atoms occupy the respective ground states |A?,|B? and it
is hence meaningful to adopt the mean-field description
for these modes. Ignoring back reaction effects of the
quantum fluctuations one then obtains a pair of coupled
Gross–Pitaevskii equations (GPE)
i¯ h∂tΨi =
?
+Uii|Ψi|2+ Uij|Ψj|2
−
¯ h2
2mi∇2+ Vi− µi
?
Ψi+ λΨj,(2)
where (i,j) → (A,B) or (i,j) → (B,A) and Ψi is the
classical wave function of the condensate ?Ψ?. Now con-
sider small perturbations (sound waves) in the conden-
sate cloud. The excitation spectrum is obtained by lin-
earizing around some background, and after a straight-
forward analysis leads to a differential equation for the
(rescaled) perturbations in the phases,˜θ = Ξ−1/2¯θ =
Ξ−1/2[θA1,θB1]T, where Ξ is a 2 × 2 matrix constructed
from the atomic couplings Uij[11].
Hydrodynamic limit: If one ignores the effect of the
quantum potential then one obtains
∂2
t˜θ = −∂t
?
I ? v0· ∇˜θ
??C2
?
− ∇ ·
?
? v0I˙˜θ
?
+∇ ·
0−? v0I ? v0
?∇˜θ
?
+ Ω2˜θ,(3)
where C2
structed from the parameters appearing in the GPE, and
? v0 is the common flow velocity of the background con-
densates. If [C2
eigenstates of the system results in a pair of independent
“curved spacetime” Klein–Gordon equations
0and Ω2are 2 × 2 symmetric matrices con-
0, Ω2] = 0 then decomposition onto the
1
?−gI/II∂a
??−gI/II(gI/II)ab∂b˜θI/II
?
+ ω2
I/II˜θI/II= 0,
(4)
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3
where the “acoustic metrics” are given by
(gI/II)ab∝
?
−
?
c2
I/II− v2
−? v0
0
?
| −? v0
| Id×d
T
?
,(5)
and where the overall conformal factor depends on the
spatial dimension d. The metric components depend only
on the background velocity ? v0, the background densities
ρ0i, and the speeds of sound c2
These are given by the eigenvalues of the matrix C2
I/IIfor the two eigenmodes.
0:
c2
I/II=tr[C2
0] ±?tr[C2
0]2− 4det[C2
2
0]
. (6)
The speed of sound in the AM takes on the role of the
speed of light. The matrix Ω2can be shown to have
zero determinant, and so the eigenfrequencies of the two
phonon modes are: ω2
of the modes are then defined as m2
and thus the AM corresponds to one massless particle
mI = 0 and one massive particle. They both “experi-
ence” the same space-time if the sound speeds are equal,
which requires tr[C2
c0= tr[C2
0]/2. Hence we now have an AM representing
both massive and massless particles, propagating on the
same background at low energies. Let us now extend the
analysis to high-energies and explore the structure of the
corresponding LIV.
QGP beyond the hydrodynamic limit: Starting from
the GPE, we now linearize around a uniform condensate
and set the background velocity to zero, ? v0 =?0, but
retain the quantum pressure term. The equation for the
phase perturbations in momentum space is now [11]
I= 0 and ω2
II= tr[Ω2]. The masses
I/II= ¯ h2ω2
I/II/c4
I/II
0]2− 4det[C2
0] = 0, i.e. cI = cII =
ω2˜θ =
??
Ξ + X k2[D k2+ Λ]
?
Ξ + X k2?
˜θ
= H(k2)˜θ,(7)
where Ξ, X, D, and Λ are additional 2×2 symmetric ma-
trices constructed from the parameters appearing in the
GPE. The perturbation spectrum obeys the generalized
Fresnel equation:
det{ω2I − H(k2)} = 0, (8)
and the dispersion relations for the phonon modes are
ω2
I/II=tr[H(k2)] ±?tr[H(k2)]2− 4 det[H(k2)]
A Taylor-series expansion gives
2
.(9)
ω2
I/II= ω2
I/II
???
k→0+
dω2
I/II
dk2
?????
k→0
k2+1
2
d2ω2
d(k2)2
I/II
?????
k→0
?k2?2
(10)+O?(k2)3?,
I/II= ω2
so these two dispersion relations are in the desired form of
Eq (1). Note ω2
I/II(k2) only permits even powers
in k, as the dispersion relation is invariant under CPT.
This is by no means a surprising result, because the un-
derlying GPE (2) is also invariant under CPT.
It is now useful to define the symmetric matrices
C2= C2
scribe how the speed of sound is modified, and also to
define Y2= 2X1/2Ξ−1X1/2and Z2= 2X1/2DX1/2.
All three of ∆C2, Y2, and Z2are explicitly suppressed
by powers of the mass of the fundamental constituents
that condense to form the BEC. Note that all the rele-
vant matrices have been carefully symmetrized, and note
the important distinction between C2
fine c2=1
c2→ c2
fourth order coefficients in the dispersion relations (10)
are [11]:
0+ ∆C2and ∆C2= X1/2ΛX1/2which de-
0and C2. Now de-
2tr[C2], which approaches the speed of sound
0, in the hydrodynamic limit. The second and
dω2
I/II
dk2
?????
?????
k→0
= c2
?
1 ±
?2tr[Ω2C2
0] − tr[Ω2] tr[C2]
tr[C2]tr[Ω2]
??
(11)= c2(1 ± η2);
1
2
d2ω2
I/II
d(k2)2
k→0
=
?
tr[Z2] ± tr[Z2]
±2tr[Ω2C2
0] − tr[Ω2] tr[C2
tr[Ω2]
0]
tr[Y2]
±tr[C2]2− 4det[C2
tr[Ω2]
0]
∓tr[C2]2
tr[Ω2]η2
2
?
= 2η4(¯ h/MLIV)2. (12)
Lorentz violations from UV physics: In order to obtain
LIV purely due to ultraviolet physics, we demand exact
Lorentz invariance in the hydrodynamic limit. In other
words, we require all terms in the equations (11) and (12)
which might otherwise survive in the hydrodynamic limit
to be set to zero. The constraints we obtain are:
C1 : tr[C2
2tr[Ω2C2
0]2− 4det[C2
0] − tr[Ω2]tr[C2
0] = 0; (13)
C2 :
0] = 0. (14)
Beyond the hydrodynamic limit, but imposing C1 and
C2, the equations (11) and (12) simplify to:
dω2
I/II
dk2
?????
k→0
= c2
0+1 ± 1
2
tr[∆C2],(15)
d2ω2
I/II
d(k2)2
?????
k→0
=
tr[Z2] ± tr[Z2]
2
±tr[C2
0]
?
−tr[Y2] +tr[∆C2]
tr[Ω2]
?
.(16)
To enforce C1 and C2 the effective coupling between
the hyperfine states has to vanish, ˜UAB = 0.
can be done by imposing a particular transition rate
This
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4
λ = −2√ρA0ρB0 UAB. In addition, fix the hydrody-
namic speed of sound to be
c2
0=mBρA0UAA+ mAρB0UBB+ UAB(ρA0mA+ ρB0mB)
2mAmB
.
(17)
While one eigenfrequency always remains zero, ω0,I≡ 0,
for the second phonon mode we get
ω2
0,II=4UAB(ρA0mB+ ρB0mA)c2
0
¯ h2
.(18)
Thus the AM corresponds to one massless particle mI= 0
and one massive particle m2
in the acoustic Minkowski space-time in the hydrody-
namic limit. This mass is much smaller that any average
of the atomic masses if we set UAB≪ UAA+ UBB. (Al-
though not relevant to the aim of the present paper, we
stress that such a regime is potentially achievable in an
experimental setting.) For higher wave numbers we ob-
tain LIV in the form of equation (1), and the coefficients
η2and η4for the two modes are:
II= ¯ h2ω2
0,II/c4
0, propagating
η2,I/II ≈
η4,I/II ≈ 1;
where MLIV=√mAmBis defined as the scale of Lorentz
violations — which is our analogue Planck scale.
It is quite remarkable that the quadratic coefficients
(19) are exactly of the form postulated in several works on
non-renormalizable EFT with LIV (see e.g. [2]). They are
indeed the squared ratio of the particle mass to the scale
of Lorentz violation. Moreover we can see from (20) that
there is no further suppression — after having pulled out
a factor (¯ h/MLIV)2— for the quartic coefficients η4,I/II.
These coefficients are of order one and generically non-
universal, (though if desired they can be forced to be
universal by additional and specific fine tuning).
Discussion: The suppression of η2, combined with the
non-suppression of η4, is precisely the statement that
the “naturalness problem” does not arise in the current
model. We stress this is not a “tree level” result as the
dispersion relation was computed directly from the fun-
damental Hamiltonian and was not derived via any EFT
reasoning. Moreover avoidance of the naturalness prob-
lem is not directly related to the tuning of our system
to reproduce SR in the hydrodynamic limit. In fact our
conditions for recovering SR at low energies do not a
priori fix the the η2coefficient, as its strength after the
“fine tuning” could still be large (even of order one) if
the typical mass scale of the massive phonon is not well
below the atomic mass scale. Instead the smallness of η2
is directly related to the mass-generating mechanism.
The key question is now: Why does our model es-
cape the naive predictions of dominant lowest-dimension
Lorentz violations? (In fact in our model for any p ≫
?mI/II
MLIV
?2
=
?
quasiparticle mass
effective Planck scale
?2
; (19)
(20)
mII the k4LIV term dominates over the order k2one.)
We here propose a nice interpretation in terms of “emer-
gent symmetry”: Non-zero λ simultaneously produces a
non-zero mass for one of the phonons, and a correspond-
ing non-zero LIV at order k2. (Single BEC systems have
only k4LIV as described by the Bogoliubov dispersion
relation.) Let us now drive λ → 0, but keep the condi-
tions C1 and C2 valid at each stage. (This also requires
UAB → 0.)
describes two non-interacting phonons propagating on a
common background. (In fact η2→ 0 and cI= cII= c0.)
This system possesses a SO(2) symmetry. Non-zero laser
coupling λ softly breaks this SO(2), the mass degeneracy,
and low-energy Lorentz invariance. Such soft Lorentz vi-
olation is then characterized (as usual in EFT) by the
ratio of the scale of the symmetry breaking mII, and
that of the scale originating the LIV in first place MLIV.
We stress that the SO(2) symmetry is an “emergent
symmetry” as it is not preserved beyond the hydrody-
namic limit: the η4 coefficients are in general different
if mA?= mB, so SO(2) is generically broken at high en-
ergies. Nevertheless this is enough for the protection of
the lowest-order LIV operators. The lesson to be drawn
is that emergent symmetries are sufficient to minimize
the amount of Lorentz violation in the lowest-dimension
operators of the EFT.
One gets an EFT which at low energies
We acknowledge useful discussions and comments by
David Mattingly, Ted Jacobson, and Bei-Lok Hu.
∗Electronic address: liberati@sissa.it
†Electronic address: matt.visser@mcs.vuw.ac.nz
‡Electronic address: silke.weinfurtner@mcs.vuw.ac.nz
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