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

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

3. 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<<E (Planck) LV Systems ~ Special Relativity

4. 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

5. How to construct the Generalized Nonlinear Actions? (J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 19403 (2002))

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

8. Mapping (implicit form) parameterized by

9. Mapping Look at ! Ex. 01 Invariant Energy Scale!

11. Ex. 02 Varying Speed of Light - VSL

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

13. Deformed Dispersion Relations Just illustrative ! By following this rule…

14. Modifications to the Beta Decay “End-Point”.

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

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

17. The structure of the algebra is preserved if

19. ... calculations...

21. (J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 19403 (2002)) The previous results can be specialized by setting

23. (J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 19403 (2002)) The previous results can be specialized by setting

24. 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, 065014 (2007)) (A. E. Bernardini, R. da Rocha, EuroPhys. Lett. 81, 40010 (2008)) We give a covariant formulation for the generalized Lorentz invariance.

25. 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

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

27. The equation of the motion the motion Phenomenology

29. LV extensions of the SM

30. 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, 065014 (2007)) (A. E. Bernardini & R. da Rocha, EuroPhys. Lett. 81, 40010 (2008)) (A. E. Bernardini, Phys. Rev. D75, 097901 (2007)) (A. E. Bernardini & O. Bertolami, Phys. Rev. D75, 085032 (2008))

34. -> We propose the Ansatz

45. 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

46. Very Special Relativity - VSR 2

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

48. 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: