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Photon propagation in
quantum gravity
A tool to test Lorentz Invariance
R. Gleiser, C. Kozameh, F. Parisi
Physics at the end of the XIX century
 Lord Kelvin reflecting on the status of physics.
“I am lucky to have seen all the major accomplishments in
Physics!

Maxwell´s theory to explain the ondulatory nature of light.
 Newton´s theory to explain celestial motion.
 Thermodynamics to explain the behavior of systems.
All that is left is to solve specific problems.”
Small problem: lack of invariance of Maxwell theory
under the Galilean group. It was thought the equations
were valid on a reference frame at rest with respect to
the ether. The Michelson & Morley experiment was
designed to measure the speed of the earth with respect
to the ether.
Einstein’s five papers of 1905
1. "On a heuristic viewpoint concerning the production and
transformation of light."
(light quantum/photoelectric effect) (17 March 1905)
Annalen der Physik, 17(1905), pp. 132-148.
2. "A New Determination of Molecular Dimensions"
(Doctoral dissertation) (30 April 1905).
Annalen der Physik, 19 (1906), pp. 289-305.
3. "On the motion of small particles suspended in liquids at
rest required by the molecular-kinetic theory of heat."
(Brownian motion paper) (11 May 1905)
Annalen der Physik, 17 (1905), pp. 549-560.
Einstein’s five papers of 1905
4. " On the electrodynamics of moving bodies"
(Special relativity) (30 June 1905)
Annalen der Physik, 17 (1905), pp. 891-921.
5. " Does the inertia of a body depend on its energy
content?"
(E=mc2) (27 September 1905).
Annalen der Physik, 18 (1905), pp. 639-641.
Einstein’s fourth miraculous paper
 Lorentz invariance is arguably the
most fundamental principle in
Physics.
 In 1905, Albert Einstein, inspired
by the Michelson and Morley's
experiment, presented his theory
of special relativity and redefined
our notion of space, time and
gravity.
Einstein’s fourth miraculous paper
 Today physicists are doing reruns of old experiments
with extraordinary precision testing the constancy of the
speed of light. Nature 427, 482 - 484 (2004)
 Recent claims coming from the two leading candidates
for a quantum theory of gravity challenge this basic
symmetry.
 Their predictions could be observed with present level of
technology.
 If true, there would have to be a mayor revision of our
understanding of the physical processes.
 In this talk we review these claims and show that
Lorentz invariance is preserved.
Possible Lorentz violating effects
1.
Superstrings. Very energetic photons interact with the
quantum structure of the space-time. Consequence:
their speed is lower than that of less energetic photons.
G. Amelio-Camelia, et al, Nature 393, 793 (1998).
cE  c(1  E
2.
E
)
Planck
Loop Quantum Gravity. Left and right helicity
photons travel at different speeds.
R. Gambini, J. Pullin, Phys. Rev. D 59, 124021 (1999).
k  k 1  4 P k
Possible Lorentz violating effects
Photons emitted simultaneously will separate as they
travel through space. Time delay seen by observer
t E  L ( E E )
c Planck
Physical processes
will mask this effect.
Unlikely to be observed
Another Lorentz violating effect.
3. For linearly polarized photons the polarization direction
depends on the energy of the photon.
R. Gleiser, C. Kozameh, Phys. Rev. D 59, 124021 (2001).
 k  2  P k 2 L
This effect has been used to set an upper bound   10
for visible light and   1015 for gamma rays.
5
Is Lorentz invariance violated?
●
None of these predicted effects have been observed. Is
Lorentz invariance really violated at the order of
approximation of these calculations?
●
Superstring model difficult to follow. Nevertheless the
dispersion relation arises from the equation
g 00 2  g ij k i k j  g 0i k j  0,
 2   ij k i k j  vi k j  0.
●
where vi is the speed of the receding string.
This equation says that any photon follows the same null
geodesic regardless of its energy.
Coupling radiation fields with gravity

The loop quantum gravity model has a more standard
approach. It is thus worth reviewing the relevant fields
involved in this interaction

Classical theory: Lagrangian and Hamiltonian formulation

Quantum theory: the quantum operators.

Semiclassical approximation:
- Effective interaction Hamiltonian.
- Field equations
Classical source-free Maxwell fields

Maxwell field:
F  d ( A)

Lagrangian density:
L  FF *

Euler-Lagrange equation:
d ( F *)  0

Local (t, xa) coords.
Ea  F0 a   t Aa ,
Ba  F *0 a   abc b Ac
 


E  t A, B    A.
Hamiltonian formulation

Conjugate momentum to Aa

Hamiltonian density
L
 
 E a .
  t Aa 
a
H  12 qab a b  12 q ab Ba Bb

Hamilton equations
H
b
 t Aa 

q

,
ab
a

H
a
 t  
  bcd  c q ab Bb
Aa
The quantum operators and states

ˆ ,E
ˆ
ˆab , A
The quantum operators are
(q
a
where the canonical pairs satisfy the c.c.r.

b
)


 
b
b 
ˆ

ˆ
Aa ( x, t ), ( x , t )  i a  ( x  x) .

The Hamiltonian density operator reads
1
ˆ
H  qˆ ab Eˆ a Eˆ b  qˆ ab Bˆ a Bˆb
2

The semiclassical states
 gˆ ab
SC    
 P
   ab  O
 

,

a
 E a   Eclass
The effective Hamiltonian
Taking expectation values of H with semiclassical states
H eff
 
 Hˆ  12 E a H ab E b  12 Ba H 1

H ab   ab   ( P ) k  k abc1 ..ck  c1 ... ck
k 1
Or, using vectorial notation
with

     1 
 H  BH B

H eff 


k
k
H  Id    P  k   .
1
2
k 1
ab
Bb
The field equations
The Hamilton equations of motion yield
 t Aa  qab ,
b
 t   bcd  c q Bb
a
ab
which means that either
 
A or E


 t A   HE ,

 1 
t E    H B .

satisfy the wave equation



 

 1 
2
 A  Ht E  H  H B   A
2
t

C. Kozameh, F. Parisi, Class. Quantum Grav. (2004).
Summary of results

Lorentz invariance is preserved by the interaction
between photons and quantum gravity states at a
semiclassical approximation.

Results extend to any gravitational state with rotational
invariance.

Any violation must appear at a higher order, i.e. when
we consider the back reaction to the metric.

Preliminary results indicate that this non lorentzian term
oscilates around a Lorentz preserving dispersion relation.

Details will be presented at the next commemoration
conference.
.