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Quantum gravity phenomenology in
an emergent spacetime: concepts,
constraints and speculations
Fourth Meeting on
Constrained Dynamics and Quantum Gravity
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Cala Gonone (Sardinia, Italy). September 12-16, 2005
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Stefano Liberati
SISSA/INFN
Trieste
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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: astro-ph/0505267, Annals of Phys. Special Issue Jan 2006
Lorentz violation: first evidence of QG?
Old “dogma” we cannot access any quantum gravity effect…
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
Real LIV with a preferred frame
See e.g. D. Mattingly, “Modern tests
of Lorentz invariance,”
Liv. Rev. Rel. [arXiv:gr-qc/0502097].
Apparent LIV with an extended SR
(i.e. possibly a new special relativity with two
invariant scales: c and lp)
QFT+LV
Renormalizable, or higher
dimension operators
Spacetime foam leading to
stochastic Lorentz violations
Non-commutative spacetime
EFT, non-renormalizable ops,
Extended Standard Model
(all op. of mass dimension> 4)
Renormalizible ops. (lab constraints)
Mainly astrophysical constraints
E.g. QED, dim 3,4 operators
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.
Amelino-Camelia, et al.
Nature 393, 763 (1998)
 Anomalous threshold reactions (usually forbidden, e.g. gamma decay, Vacuum
Cherekov)
 m 2c 2
pn2 
m2
p n2
n m 2 M n2
 Shift of standard
reactions
(e.g.
gamma
absorption
or
GZK)



p

E  cthresholds
p 
1


crit
n2
n2 

p2 M
p2 limits”
M(e.g.
synchrotron
 Reactions affected by
“speeds
radiation):
2
2
2

LIT.synchrotron
frequency:
Jacobson, SL, critical
D. Mattingly
Nature 424, 1019 (2003)

 cLI 
3 eB
2 m
2
  (1 v2 )1/ 2
1/ 2
m 2

E
 

2


E 2

M

QG 
There is now a maximum achievable synchrotron frequency max for ALL electrons!
Hence one gets a constraints by asking max≥ (max)observed

 Also: LV induced decays not characterized
by a threshold (e.g. decay of a particle
 variation of LV couplings (as the lab moves with respect
from one helicity to the other or photon splitting), Sidereal
to a preferred frame or directions), Cosmological variation of couplings…. (see e.g. Mattingly review)
Novelties in threshold reactions: why
 Asymmetric configurations:
Pair production can happen with
asymmetric distribution
of the final momenta
2E0
E f 
p 2
if
2E0
p 2
p
Upper thresholds:
The range of available energies of
the incoming particles for which
the reactions happens is changed.
Lower threshold can be shifted and
upper thresholds can be introduced
2
p p s
0
p p s
Sufficient condition for
asymmetric Threshold.
If LI holds there is never an
upper threshold
However the presence of
different coefficients for
different particles allows Ei
to intersect two or more
times Ef switching on and
off the reaction!
Applications: the GZK cut-off
Since the sixties it is well-known that the universe is opaque to protons (and other nuclei) on
cosmological distances via the interactions
In this way, the initial proton energy is degraded with an attenuation length of about 50 Mpc.
Since plausible astrophysical sources for UHE particles (like AGNs) are located at distances
larger than 50-100 Mpc, one expects the so-called Greisen-Zatsepin-Kuzmin (GZK) cutoff in
the cosmic ray flux at the energy given by
• HiRes collaboration claim that they see the
expected event reduction
• A recent reevaluation of AGASA data seems
to confirm the violation of the GZK cutoff.
• Several explanations proposed (e.g. Z-burst,
Wimpzillas), remarkably LIV appears as one one
of the less exotic…
• Everybody is waiting for the Auger experiment
to give a definitive answers…
Possible constraints from GZK
Constraint from photon-pion production
p+ CMB  p+0 if GZK confirmed
The range of p, for n = 3 dispersion
modifications where the GZK cutoff is
between 21019 eV and 71019 eV:
constraints of order 10-11 on both p,
Constraint from absence of proton vacuum
Cherenkov p+  p+ 
p and  are in multiples of 10-10
(Jacobson, SL, Mattingly: PRD 2003)
In 2004 Gagnon and Moore performed analysis taking into account the
partonic structure founding same orders of magnitude for the constraints
Applications: QED with LIV at O(E/M)
Let’s consider all the Lorentz-violating dimension 5 terms (n=3
LIV in dispersion relation) that are quadratic in fields, gauge &
rotation invariant, not reducible to lower order terms (MyersPospelov, 2003). For E»m
Warning: All these
LIV terms also
violate CPT
electron helicities have
independent LIV coefficients
u= unit timelike 4-vector that fix the
preferred system of reference
Moreover electron and positron have
inverted and opposite positive and negatives
helicities LIV coefficients (JLMS, 2003).
photon helicities have
opposite LIV coefficients
Positive
helicity
Negative
helicity
Electron
+
-
Positron
--
-+
The EM spectrum of the Crab nebula
synchrotron
Inverse Compton
From Aharonian and Atoyan, astro-ph/9803091
Crab nebula (and other SNR) well
explained by synchrotron self-Compton
(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 synchrotron ambient
photons
We shall assume SSC correct and use Crab observation to constrain LV.
Crab alone provides three of the best constraints. We use:
 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: 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)
Using the Crab nebula we infer:
-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
(Collins et al. 2004).
This need not be the case if a symmetry or other mechanism protects the lower dimensions operators
from violations of Lorentz symmetry.
Idea: SUSY protect dim<5 operators but SUSY is broken…
Can we get some hint of how things might work using some toy model?

Dilute BEC analogue gravity
ˆ     t,x   
ˆ t,x  where
t,x
ˆ   = wave function of the BEC (c - number)
 t,x   t,x
 t,x  nt,x   N /V
2
ˆ t,x  q - number, non - condensate amplitude, excitations/fluctuations

The propagations of quantum excitations in a BEC system simulates that of a
scalar field on a curved spacetime with an energy dependent metric. In particular
adopting the eikonal approximation the dispersion relation for the BEC quasiparticles is
2

i
v o ki

2 
 ck   k 
2m 
2
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)=
with  healing length=1/(8a)1/2
»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 pressing issues in quantum gravity phenomenology
In fact in order to “see” a modification of the coefficient of k2 in
the dispersion relation one needs at least two particles
So what we need is an analogue model which has at least two kind of
quasiparticles which “feel” the same effective geometry at low
energies and show LIV in the dispersion relations at high energies
Fortunately we have such a model:
a system of two coupled BEC
2-BEC: teaser
This system reproduces a QFT in a spacetime with two scalar fields,
one massive the other massless.
As such this is an ideal system for reproducing the salient features of
the situations studied in quantum gravity phenomenology…
Separately each BEC as a dispersion relation of the form
2k2+k4/K2 K1/
When one switched the laser coupling  on, at low frequencies
one can tune the system so to have a common speed of light and gets
12k12, 12k12+m2 (cs=1) [Note that the mass is generated via the coupling]
What happens if one looks at the behaviour at high energies?
Will the Lorentz violation remain at order k4 or a naturalness problem will arise?
See Silke Weinfurtner’s talk later…
The future?
 Definitively rule out n=3 LV, O(E/M), EFT including chirality
effects
 Strengthen the n=3 bounds. E.g. via possible role of positrons in Crab nebula emission.
 naturalness problem: better understanding via an analog model?
 Constraint on n=4 (favored if CPT also for QG):
 From Auger, Euso, OWL
 No GZK protons Cherenkov: ≤O(10-5)
m 2 ~  p 4 / M 2  p ~ mM 1/ 4
 If GZK cutoff seen: ≈≥O(-10-2)
 From Amanda, IceCube, Euso, OWL
p ~ 100 TeV (neutrino),
3 10
eV (proton),
PeV (electron)
 Neutrinos: 100 TeV neutrinos give order unity constraint
by absence
of vacuum100
Cherenkov
but rate of
1520
energy loss too low. Recent calculations shows one need 10 10 eV UHE cosmological neutrinos.
Possibly to be seen via EUSO and/or OWL satellites
18
 From AGILE or GLAST we shall hardly get constraints in n=4.

For GRB we need anyway
 better measures of energy, timing, polarization from distant -ray sources. O(1) constraint on || requires
polarization detection of at 100 MeV
 AGILE/GLAST could see TOF n=3 LIV, unfortunately no polarization
Probably new advances will require better understanding of the astrophysics
Better understanding of the naturalness problem could tell us something important about EFT in
an emergent spacetime…