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Covariant Formulation of the Generalized Lorentz Invariance and Modified Dispersion Relations Alex E. Bernardini Departamento de Física – UFSCAR Financial Support FAPESP – Silafae Committee - CAB

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Seminar Basic concepts of Doubly Special Relativity and Generalized Lorentz Invariance; Covariant Formulation; Probes for the Neutrino End Point in the Beta Decay.

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Basic Principles Relativity of Inertial Frames; Equivalence Principle; >>> Theories where the Planck length (energy) is expected to play a fundamental role in a theory of Quantum Gravity (setting a physical scale) >>> Modification of special relativity in which the Planck energy, joins the Speed of light as an Invariant, in spite of a complete relativity of inertial frames and Agreement with Einstein’s theory at low energies. >>> A nonlinear modification of the action of the Lorentz Group on a momentum space, generated by adding Dilatation (or ?) to each boost.... some predecessors... (Amelino-Camelia, Magueijo, Smolin, …) Invariant (Planck) Energy Scale (E Planck ); Correspondence Principle: Energy<

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It is proposed a NONLINEAR modification of the ACTION of the Lorentz Group in momentum space which contains an OBSERVER INDEPENDENT Energy Scale It reduces to the usual LINEAR ACTION at low energies (E << ). For the new proposal, the concept of metric (a quadratic invariant) collapses at high energies, being replaced by the non-quadratic invariant Generalized LI with an Invariant Energy Scale

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How to construct the Generalized Nonlinear Actions? (J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, (2002))

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The operator D (not necessarily) represents a conformal transformation that preserves the algebra in spite of modifying the generators.

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Mapping (implicit form) parameterized by

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Mapping Look at ! Ex. 01 Invariant Energy Scale!

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Ex. 02 Varying Speed of Light - VSL

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Flavor Oscillation Problem For LV Systems with an Invariant Energy Scale, there is a general addition rule for composite systems Ex. 03

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Deformed Dispersion Relations Just illustrative ! By following this rule…

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Modifications to the Beta Decay “End-Point”.

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...we will turn back to this point later !!!

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Covariant Formulation of Modified Lorentz Actions...let us turn back to our proposition !!!

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The structure of the algebra is preserved if

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... calculations...

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(J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, (2002)) The previous results can be specialized by setting

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(J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, (2002)) The previous results can be specialized by setting

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Solution for generalized (n µ ) (Dilatation Transformation) -> Covariant representation of the spacetime coordinates. Conclusions I (Preliminary!) (A. E. Bernardini, R. da Rocha, Phys. Rev. D75, (2007)) (A. E. Bernardini, R. da Rocha, EuroPhys. Lett. 81, (2008)) We give a covariant formulation for the generalized Lorentz invariance.

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How to describe Very Special Relativity in the framework of Lorentz-Algebra Preserving Systems? -> We have to find the right transformation ! How to describe Very Special Relativity in the framework of Lorentz-Algebra Preserving Systems? The operator represents conformal transformation that preserves the algebra in spite of modifying the generators. The operator D (not necessarily) represents a conformal transformation that preserves the algebra in spite of modifying the generators. Just to remind Ex. 04

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Why is it important? Or Where can it be observed? Modifications to the predictions for the end-point of the Beta Decay

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The equation of the motion the motion Phenomenology

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LV extensions of the SM

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More general solutions -> Other Dynamical Equations -> New Dispersion Relations, etc. Solution for generalized (n µ ) (Dilatation Transformation) -> Covariant representation of the spacetime coordinates. Measurable modifications for the predictions of the end-point of the neutrino beta-decay: VSR or LV SM extensions. Final Conclusions (A. E. Bernardini & R. da Rocha, Phys. Rev. D75, (2007)) (A. E. Bernardini & R. da Rocha, EuroPhys. Lett. 81, (2008)) (A. E. Bernardini, Phys. Rev. D75, (2007)) (A. E. Bernardini & O. Bertolami, Phys. Rev. D75, (2008))

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-> We propose the Ansatz

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The equation of motion for a “Dirac” particle in the framework of Lorentz Violating Systems - Phenomenological Perspectives - AND Alex E. Bernardini Department of Cosmic Rays and Chronology IFGW Unicamp BRASIL Financial Support

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Very Special Relativity - VSR 2

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The Problem: The Solution (!?!?) The equation of motion

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What does it change? 1) From the point of view of the generator algebra: New relations between the standard generators - The same algebra 2) From the phenomenological point of view:

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