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Quantum gravity phenomenology via
Lorentz violations: concepts,
constraints and lessons from
analogue models
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Stefano Liberati
SISSA/INFN
Trieste
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Analogue gravity work made in collaboration with Matt Visser and Silke Weinfurtner
QG phenomenology work made in collaboration with Ted Jacobson and David Mattingly
The Quantum Gravity problem…
 Why we need a QG theory?


Philosophy of unification QM-GR (reductionism in physics)
Lack of predictions by current theories
(e.g. GR singularities, time machines, DE?)
 To eventually understand QG, we will need to


observe phenomena that depend on QG
extract reliable predictions from candidate theories & compare with
observations
Old “dogma” we cannot access any quantum gravity effect…
Lorentz violation: first evidence of QG?
In recent years several ideas about sub-Planckian consequences of QG physics have been
explored:
e.g. Extra dimensions effects on gravity at sub millimeter scales, TeV BH at LHC, violations of
spacetime symmetries…
Idea: LI linked to scale-free spacetime -> unbounded boosts expose ultra-short distances…
Suggestions for Lorentz violation come from:

need to cut off UV divergences of QFT & BH entropy
 transplanckian problem in BH evaporation end Inflation
 tentative calculations in various QG scenarios, e.g.
 semiclassical spin-network calculations in Loop QG
 string theory tensor VEVs
 spacetime foam
 non-commutative geometry
 some brane-world backgrounds
Very different approaches but common prediction of
modified dispersion relations for elementary particles
QG phenomenology
via modified dispersion relations
Almost all of the above cited framework do lead to modified dispersion
relations that can be cast in this form
E 2  p 2  m 2  ( p, M)
M  spacetime structure scale, generally assumed  M Planck  1019 GeV

If we presume that any Lorentz violation is associated with quantum
and we violate only boost symmetry
 gravity
(no violation of rotational symmetry)
Note: (1,2) are dimensionless coefficients which must be necessarily
small -> standard conjecture
1(/M)+1, 2(/M) with  1 and with <<M
where  is some particle physics mass scale
This assures that at low p the LIV are small and that at high p the highest
order ones pn with n 3 are dominant…
Theoretical Frameworks for LV
Apparent LIV with an extended SR
Real LIV with a preferred frame
QFT+LV
Renormalizable, or higher
dimension operators
Extended Standard Model
Renormalizible ops.
E.g. QED, dim 3,4 operators
(i.e. possibly a new special relativity with two
invariant scales: c and lp)
Spacetime foam leading to
stochastic Lorentz violations
Non-commutative spacetime
EFT, non-renormalizable ops,
(all op. of mass dimension> 4)
E.g. QED, dim 5 operators
Astrophysical tests of Lorentz violation

Cumulative effects: times of flight & birefringence:
Time of fight: time delay in arrival of different colors
Birefringence: linear polarization direction is rotated through
a frequency dependent angle due to different
phase velocities of photons polarizations
 Anomalous threshold reactions (usually forbidden,
Cherekov):
e.g. gamma decay, Vacuum
 m 2c 2
 m 2
p n2
pn2 (e.g.
2
n2
standard
reactions
gamma
absorption
or
GZK):
2
2 thresholds
2
n m


p

M

E  c p 
1


crit
2
n2
2
n2 

p
M
p
M


 Reactions affected by “speeds limits” (e.g. synchrotron radiation):
 Shift of


LI
LI synchrotron critical frequency:  c 
3 eB
2 m
2
  (1 v2 )1/ 2
1/ 2
m 2

E
 

2


E 2

M

QG 
The key point is that for negative ,  is now a bounded function of E! There is now
a maximum achievable synchrotron frequency max for ALL electrons!
 asking max≥ (max)observed
So one gets a constraints from

The EM spectrum of the Crab nebula
From
Aharonian and Atoyan,
astro-ph/9803091
Crab alone provides three of the best constraints.
We use:
synchrotron
Inverse Compton
Crab nebula (and other SNR) well explained by synchrotron self-Compton model.
SSC Model:
1. Electrons are accelerated to very high energies at pulsar
2. High energy electrons emit synchrotron radiation
3. High energy electrons undergo inverse Compton with ambient photons
We shall assume SSC correct and use Crab observation to constrain LV.
Observations from Crab
 Gamma rays up to 50 TeV reach us from Crab: no photon
annihilation up to 50 TeV.
 By energy conservation during the IC process we can infer that
electrons of at least 50 TeV propagate in the nebula: no vacuum
Cherenkov up to 50 TeV
 The synchrotron emission extends up to 100 MeV
(corresponding to ~1500 teV electrons if LI is preserved):
LIV for electrons (with negative ) should allow an Emax100 MeV.
B at most 0.6 mG
Constraints for EFT
with O(E/M) LIV
T. Jacobson, SL, D. Mattingly: PRD 66, 081302 (2002); PRD 67, 124011-12 (2003)
T. Jacobson, SL, D. Mattingly: Nature 424, 1019 (2003)
T. Jacobson, SL, D. Mattingly, F. Stecker: PRL 93 (2004) 021101
T. Jacobson, SL, D. Mattingly: Annals of Phys. Special Issue Jan 2006
TOF: ||0(100) from MeV emission GRB
Birefringence: ||10-4 from UV light of radio
galaxies (Gleiser and Kozameh, 2002)
-decay: for ||10-4 implies || 0.2
from 50 TeV gamma rays from Crab nebula
Inverse Compton Cherenkov: at least one of 
10-2 from inferred presence of 50 TeV electrons
Synchrotron: at least one of  -10-8
Synch-Cherenkov: for any particle with  satisfying
synchrotron bound the energy should not be so
high to radiate vacuum Cherenkov
An open problem: un-naturalness of small LV.
Renormalization group arguments might suggest that lower powers of momentum in
will be suppressed by lower powers of M so that n≥3 terms will be further suppressed
w.r.t. n≤2 ones. I.e. one could have that
˜1
E  p  m  
2
2
2
2
M2
˜2
Mp  
1

M
˜3
p  
2

M2
˜4
p  
3

M3
˜n
p  ... 
4

M n1
pn
Alternatively one can see that even if one postulates classically a dispersion relation with
only terms (n)pnMn-2 with n3 and (n)O(1) then radiative (loop) corrections involving this
term will generate terms of the form (n)p2+(n)p M which are unacceptable observationally.
This need not be the case if a symmetry or other mechanism protects the lower dimensions operators
from violations of Lorentz symmetry.
Ideas: SUSY protect dim<5 operators or brane-world scenarios or analog models…
Analogue gravity as a test field
Idea: Study in condensed matter systems typical phenomena
of semiclassical gravity.
At least three possible motivations for this study:
I.
Experimental issues: try to gain an indirect observational confirmations of
SG predictions (e.g. laboratory BH evaporation).
II. Theoretical issues for GR: use the analogy to guess behavior in real
semiclassical gravity
III. Theoretical issues for CM: use analogy to have new interpretation of known
phenomena or prediction of new ones.
LIV in BEC analogue gravity
The quasiparticles propagate as on a curved spacetime with an energy dependent
metric. This metric is the standard acoustic spacetimes metric when the quantum
potential term is negligible, and shows at high frequencies a violation of the
Lorentz “acoustic” invariance. In particular adopting the eikonal approximation
ˆ     t,x  
ˆ t,x
t,x
1 t,x  Re exp i 
n1 t,x   Re n exp i 
So the dispersion relation for the
BEC quasi-particles is


;
t
ki   i 
2


  vioki  c sk 2   k 2 
2m 
This dispersion relation (already found by Bogoliubov in 1947) actually interpolates between
two different regimes depending on the value of the fluctuations wavelength |k| with
respect to the “acoustic Planck wavelength” C=h/(2mc)=
1/2
 with  healing length=1/(8a)
»C one gets the standard phonon dispersion relation ≈c|k|.
 For «C one gets instead the dispersion relation for an individual gas particle (breakdown
of the continuous medium approximation) ≈(h2k2)/(2m) .
 For
So we see that analogue gravity via BEC reproduces that kind of LIV
that people has conjectured in quantum gravity phenomenology…
The need for a more complex
model: 2-BEC
Unfortunately the 1-BEC system just discussed is not enough complex to
discuss the most interesting issues in quantum gravity phenomenology
In fact
 Naturalness: in order to “see” a modification of the coefficient of
k2 in the dispersion relation one needs at least two particles
 Universality: the possibility or not of having different LIV
coefficients for different particles in the dispersion relation
requires at least two species of particles
So it would be nice to have an analogue model which has at least two
kind of quasiparticles which “feel” the same effective geometry at low
energies and show in the dispersion relations at high energies LIV of the
sort studied in QG phenomenology
An example of such a model is a 2-BEC system
2-BEC analogue system
Note: with the above notation the
scattering length and U are
proportional
We are interested in deviations from SR then:
• Homogeneous (position-independent) and time independent background
• BEC at rest (vA0=vB0=0)
• Equal background phases, A0=B0=0 (simplification)
As usual one analyzes the excitation spectrum by introducing the
Madelung representation
2-BEC wave equation
Mass Density Matrix
Coupling Matrix
Interaction Matrix
Quantum Potential Matrix
Linearized Quantum
Potential
2-BEC: dispersion relation
It is now convenient to introduce the new set of variables
So that the wave equation can be cast in the form
The dispersion relation can now be easily seen by considering the eikonal
approximation. In this case
and
“All” we need now is to make explicit
the content of this dispersion relation
2-BEC: masses
Let’s define
Then we can read the dispersion
relation as
That is
Masses
We can read the mass matrix by looking
at the constant factor left for H(0)
One gets
Note: in order to get quantities with dimension of
masses we need to define a speed of sound. This
requires some choices…
2-BEC: LIV
We now expect to get LIV with p4 (as in standard 1-BEC) but will we get also
p2 terms due to the particle interactions? If yes will we have the latter
dominant over the p4?
So we want to see if our dispersion relation takes the form
A simple way to determine the  coefficients consist in taking a suitable
number of derivatives in k2 of 2
and then look at the limit for k0.
Introducing the following 2x2 matrices
One gets:
Note: X=Y=0 and c2=tr[C02]/2 if one neglects the quantum potential
Note: there are LIV terms which do not disappear for X0
2-BEC LIV:
Case 1, universality and naturalness
Let’s see what happens if fine tune our system so that all the Lorentz
violations which are purely interaction-dependent vanish.
This is equivalent to cancel in our LIV coefficients all the terms which do
not depend on X and Y
This request implies the following
conditions to hold
These conditions assure that all the the violations of Lorentz
invariance which are not due to the UV physics of the analogue
spacetime are removed. These are also the same conditions one has to
impose in order to get “mono-metricity” in the hydrodynamical limit
Few computations show in this case that
2-BEC: the naturalness problem
We see that the coefficients of LIV are particle dependent (no universality) but
we might have a naturalness problem because 2
Does this mean that the k4 LIV is always more suppressed of the k2 one?
This would be the case if both 2 and 4 are found to be proportional to
some small ratio (/M)n given that the k4 term has a further 1/M2
suppression
A careful analysis shows that
As long we are in a regime where mII<<Meff (I.e. UAB<< UAA and/or UBB) this is
indeed a small ratio
But the 4 coefficients do not show any similar “Planck” suppression
This implies that at momenta k» the
quartic order LIV is dominant on the
quadratic one!
Note also that for mA=mB 4,I= 4,II=1/2
2-BEC LIV:
Case 2, mono-metricity at low momenta
Let’s see what happens instead if fine tune our
system so that all the Lorentz violations at order k2
vanish.
This request implies the
following condition to hold
Note that this condition alone is not enough for
removing in 4 all the QP-indipendent terms. In
particular it is not cancelled the term
This is not strictly speaking a QGP analogue but something new where
the LIV are not just a UV effect… but let’s see if it would be anyway
compatible with approximate SR at low energies.
The above condition is a quadratic in  . In order to get a simple
solution let’s specialize to the case mA= mB=m, A= B= 0, UAA=UBB=U0
In this limits the solutions read
2-BEC LIV: Case 2
The solution with 2 is pathological as in this case
mII=0 and tr[Y2]=2eff
The solution with 1=-0UAB is instead interesting.
In fact in this case
Note that in spite of the presence of the QP-independent term the LIV
coefficients are still O(1) and identical in the limit considered here. Again
a small mII requires very different UAB and U0.
The qp-indep term in d22/d(k2)2|k=0 is
Meanwhile the other qp-dep terms are at leading order
This tells us that the qp-independent term is always subdominat. One
might think that it would still be visible in the hydrodynamic regime (no
qp terms) but in order to see a term like this one would requires
frequencies beyond the hydro limit
Conclusions
• Potential violations of Lorentz invariance seems to be generic in QG
models
• These violations typically lead to modified dispersion relations for the
elementary particles suppressed by the QG scale
• This kind of Planck suppressed modifications have been put to the test
via laboratory and astrophysical observations
• EFT with LIV is the most explored avenue but has also some conceptual
problems
• Analogue models of gravity provide a natural environment where to
look for answers to these problems
Lessons from 2-BEC as an analogue QG system
• Interplay between interaction-dependent LIV and microstructure
(Quantum potential)-dependent LIV
• The system can be fine tuned in order to reproduce both a situations
with low energy (small) LIV and one with exact mono-metricity at low
energies
• Different quasi-particles have generically different coefficients of LIV if
there is more than one microphysical scale
2
• When both k and k4 deviations exists the latter are dominant. Because
of the different origin? (the latter are also there in a 1-BEC the former are
induced by the interactions)
The future?
• Is there an EFT interpretation for the Case1 and Case2
condensates conditions?
• What is going wrong with the naive EFT prediction?
• Could these systems tell us that LIV is not the only low energy
effect of QG? Can they inspire us to look for something else?
Once again analogue models seems able to teach us about
gravity and quantum gravity effects more than we expected to.
And I cherish more than anything else the Analogies, my most trustworthy
masters. They know all the secrets of Nature, and they ought least to be
neglected in Geometry.
Johannes Kepler