Photon propagation in quantum gravity

1
Photon propagation in quantum gravity

• A tool to test Lorentz Invariance
• R. Gleiser, C. Kozameh, F. Parisi

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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!
• Maxwells theory to explain the ondulatory
  nature of light.
• Newtons 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.

3
Einsteins five papers of 1905

• "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.
• "A New Determination of Molecular Dimensions"
  (Doctoral dissertation) (30 April 1905).
• Annalen der Physik, 19 (1906), pp. 289-305.
• "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.

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Einsteins five papers of 1905

• " On the electrodynamics of moving bodies"
• (Special relativity) (30 June 1905) Annalen der
  Physik, 17 (1905), pp. 891-921.
• " Does the inertia of a body depend on its energy
  content?"
• (Emc2) (27 September 1905).
• Annalen der Physik, 18 (1905), pp. 639-641.

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Einsteins 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.

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Einsteins 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.

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Possible Lorentz violating effects

• 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).
• Loop Quantum Gravity. Left and right helicity
  photons travel at different speeds.
• R. Gambini, J. Pullin, Phys. Rev. D 59, 124021
  (1999).

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Possible Lorentz violating effects

• Photons emitted simultaneously will separate as
  they travel through space. Time delay seen by
  observer
• Physical processes
• will mask this effect.
• Unlikely to be observed

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Another Lorentz violating effect.

• For linearly polarized photons the polarization
  direction depends on the energy of the photon.
• R. Gleiser, C. Kozameh, Phys. Rev. D 59, 124021
  (2001).
• This effect has been used to set an upper bound
  for visible light and for
  gamma rays.

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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
• where vi is the speed of the receding string.
• This equation says that any photon follows the
  same null geodesic regardless of its energy.

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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

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Classical source-free Maxwell fields

• Maxwell field
• Lagrangian density
• Euler-Lagrange equation
• Local (t, xa) coords.

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Hamiltonian formulation

• Conjugate momentum to Aa
• Hamiltonian density
• Hamilton equations

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The quantum operators and states

• The quantum operators are
• where the canonical pairs satisfy the c.c.r.
• The Hamiltonian density operator reads
• The semiclassical states

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The effective Hamiltonian

• Taking expectation values of H with semiclassical
  states
• Or, using vectorial notation
• with

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The field equations

• The Hamilton equations of motion yield
• which means that either satisfy the
  wave equation
• C. Kozameh, F. Parisi, Class. Quantum Grav.
  (2004).

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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.
• .