* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Annalen der Physik
Quantum group wikipedia , lookup
Coherent states wikipedia , lookup
Quantum machine learning wikipedia , lookup
Bell's theorem wikipedia , lookup
Double-slit experiment wikipedia , lookup
Matter wave wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Density matrix wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Path integral formulation wikipedia , lookup
Quantum state wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum field theory wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
EPR paradox wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Renormalization group wikipedia , lookup
Renormalization wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Quantum key distribution wikipedia , lookup
Wave–particle duality wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Scalar field theory wikipedia , lookup
Hidden variable theory wikipedia , lookup
Canonical quantum gravity wikipedia , lookup
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 1015 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 FF * 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 Ht 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. .