Fins nsler r geom ometry ry and nd deSitter mo momentum space - - PowerPoint PPT Presentation

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Fins nsler r geom ometry ry and nd deSitter mo momentum space - - PowerPoint PPT Presentation

Jun 10, 2023 430 likes •747 views

Fins nsler r geom ometry ry and nd deSitter mo momentum space Loret Niccol arXiv:1407.8143 With: Giovanni Amelino-Camelia, Leonardo Barcaroli, Giulia Gubitosi and Stefano Liberati Gali lilean Relativity Gali lilean Relativity

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Fins nsler r geom

  • metry

ry and nd deSitter mo momentum space

arXiv:1407.8143

Loret Niccolò

With: Giovanni Amelino-Camelia, Leonardo Barcaroli, Giulia Gubitosi and Stefano Liberati

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Gali lilean Relativity Gali lilean Relativity

Invariant (Casimir) Transformation (Boost)

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Special Relat ativity Special Relat ativity

Invariant (Casimir) Transformation (Boost)

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??? Relat ativity ??? Relat ativity

Invariant (Casimir) Transformation (Boost)

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??? Relat ativity ??? Relat ativity

???

Invariant (Casimir) Transformation (Boost)

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Deformed Spe pecial l Relat ativity Deformed Spe pecial l Relat ativity

κ-Poincaré algebra in bicrossproduct basis

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Deformed Spe pecial l Relat ativity Deformed Spe pecial l Relat ativity

κ-Poincaré algebra in bicrossproduct basis Casimir Symmetry genera rators representation

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Curved momentum-space Curved momentum-space

Modified symmetries Non-trivial properties of momentum-space geometry

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Curved momentum-space Curved momentum-space

In the κ-Poincaré (bicro rosspro roduct basis) case (Modified) Dispersion Relation obtained through Modified symmetries Non-trivial properties of momentum-space geometry

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Finsl sler Geometry Finsl sler Geometry

Finsler Norm

  • Positive function in the tangent bundle
  • Homogeneus of degree one in ẋ

Velocity-dependent genera ralization of Riemannian metric

Rund 1959

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Finsler Geometry of a a par article with MDR Finsler Geometry of a a par article with MDR

Girelli, Liberati, Sindoni, PRD 2007

Action of a fr free relativistic particle

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Finsler Geometry of a a par article with MDR Finsler Geometry of a a par article with MDR

Girelli, Liberati, Sindoni, PRD 2007

Action of a fr free relativistic particle By using Hamilton's equation We find

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Finsler Geometry of a a par article with MDR Finsler Geometry of a a par article with MDR

Girelli, Liberati, Sindoni, PRD 2007

Action of a fr free relativistic particle By using Hamilton's equation We find

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Finsl sler Geometry an and Finsl sler Geometry an and κ κ-Poincar aré sy symmetries

  • Poincar

aré sy symmetries

We apply this procedure re to the case

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Finsl sler Geometry an and Finsl sler Geometry an and κ κ-Poincar aré sy symmetries

  • Poincar

aré sy symmetries

We apply this procedure re to the case

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Finsl sler Geometry an and Finsl sler Geometry an and κ κ-Poincar aré sy symmetries

  • Poincar

aré sy symmetries

We apply this procedure re to the case

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Finsler spacetime metric Finsler spacetime metric

In terms of momenta

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Finsler spacetime metric Finsler spacetime metric

In terms of momenta Despite its horri rible aspect this metri ric defines some simple re relati tions:

  • Its invers

rse is related to the part rticle's MDR

  • In terms of g momenta

are simply related to velocities

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On the invar ariance of the Lag agran angian On the invar ariance of the Lag agran angian

Edge terms

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On the invar ariance of the Lag agran angian On the invar ariance of the Lag agran angian = 0

Edge terms

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On the invar ariance of the Lag agran angian On the invar ariance of the Lag agran angian = 0

Edge terms Invari riant Lagrangian

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Relat ative Locality and Rainbow metrics Relat ative Locality and Rainbow metrics

Loret, arXiv:1404.5093

Invariant t “rainbow” line element This would allow us to satisfy one of the key requirments of rainbow metri rics which is This invariant Lagrangian suggest us to check whether

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Geod

  • desic equations

Geod

  • desic equations

In terms of ζ(ẋ):

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Geod

  • desic equations

Geod

  • desic equations

In terms of ζ(ẋ): From the homogeneity of F(ẋ) one can show th that the metric g(ẋ) satisfies

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Geod

  • desic equations

Geod

  • desic equations

In terms of ζ(ẋ): In terms of g(ẋ): Where From the homogeneity of F(ẋ) one can show th that the metric g(ẋ) satisfies

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Wordli lines and symmetries Wordli lines and symmetries

In the k-Poincaré ré case We choose a parametrization

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Finsler Killing equation Finsler Killing equation

We look for solutions and charges

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Finsler Killing equation Finsler Killing equation

We look for solutions and charges

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Finsler Killing equation Finsler Killing equation

We look for solutions and charges

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Summar ary and outlook Summar ary and outlook

  • Finsler

r generalization of Riemannian geometr try can be used to describe spacetime geometry seen by a particle with given (modified) dispersion re relation

  • This provides a consistent framework to derive physical properties
  • f the particle: propagation, symmetries
  • Equivalent to a ‘ra

rainbow’ metri ric associated to classical particles with κ-Poincaré inspired symmetries

  • Can it be used to tre

reat t more complicated cases, when gravity is introduced?

  • How to intro

roduce interactions?