Skip to main content

Note about string theory with deformed dispersion relations

Abstract

The goal of this short paper is to find Lagrangian for bosonic string with deformed dispersion relation proposed by J. Magueijo and L. Smolin in 2004. We also show that in the preferred case \(f=g\) this Lagrangian reduces into Nambu-Gotto form of relativistic string without modification of the dispersion relation.

This is a preview of subscription content, access via your institution.

Data Availability

No datasets were generated or analysed during the current study

Notes

  1. See also [4,5,6, 16, 17].

  2. We introduced the vector \(K^M\) in order to deal with covariant prescription that could simplify calculations and hence we can set \(K^M\) to be equal \(\delta ^M_0\) in the end. In other words, \(K^M\) is not related to any possible Killing vectors of the background metric.

  3. More generally we could consider situation when \(f=Dg\) where D is constant. Then (17) implies that B is equal to zero and (16) reduces to Nambu-Gotto action with prefactor \(\sqrt{D}\) that could be eliminated by appropriate rescaling of \(x^\mu \) coordinates.

References

  1. Amelino-Camelia, G.: Testable scenario for relativity with minimum length. Phys. Lett. B 510, 255–263 (2001). https://doi.org/10.1016/S0370-2693(01)00506-8. [arXiv:hep-th/0012238 [hep-th]]

    ADS  Article  MATH  Google Scholar 

  2. Amelino-Camelia, G., Piran, T.: Planck scale deformation of Lorentz symmetry as a solution to the UHECR and the TeV gamma paradoxes. Phys. Rev. D 64, 036005 (2001). https://doi.org/10.1103/PhysRevD.64.036005. [arXiv:astro-ph/0008107 [astro-ph]]

    ADS  Article  Google Scholar 

  3. Biermann, P., Sigl, G.: Introduction to cosmic rays. Lect. Notes Phys. 576, 1–26 (2001). [arXiv:astro-ph/0202425 [astro-ph]]

    ADS  Article  Google Scholar 

  4. Deriglazov, A.A.: Doubly special relativity in position space starting from the conformal group. Phys. Lett. B 603, 124–129 (2004). https://doi.org/10.1016/j.physletb.2004.10.024. [arXiv:hep-th/0409232 [hep-th]]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. Ghosh, S.: A Lagrangian for DSR Particle and the Role of Noncommutativity. Phys. Rev. D 74, 084019 (2006). https://doi.org/10.1103/PhysRevD.74.084019. [arXiv:hep-th/0608206 [hep-th]]

    ADS  MathSciNet  Article  Google Scholar 

  6. Ghosh, S.: DSR relativistic particle in a Lagrangian formulation and non-commutative spacetime: A gauge independent analysis. Phys. Lett. B 648, 262–265 (2007). https://doi.org/10.1016/j.physletb.2007.03.016. [arXiv:hep-th/0602009 [hep-th]]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. Girelli, F., Konopka, T., Kowalski-Glikman, J., Livine, E.R.: The Free particle in deformed special relativity. Phys. Rev. D 73, 045009 (2006). https://doi.org/10.1103/PhysRevD.73.045009. [arXiv:hep-th/0512107 [hep-th]]

    ADS  MathSciNet  Article  Google Scholar 

  8. Girelli, F., Liberati, S., Sindoni, L.: Planck-scale modified dispersion relations and Finsler geometry. Phys. Rev. D 75, 064015 (2007). https://doi.org/10.1103/PhysRevD.75.064015. [arXiv:gr-qc/0611024 [gr-qc]]

    ADS  MathSciNet  Article  Google Scholar 

  9. Kowalski-Glikman, J.: Doubly special relativity: Facts and prospects. [arXiv:gr-qc/0603022 [gr-qc]]

  10. Kowalski-Glikman, J.: Observer independent quantum of mass. Phys. Lett. A 286, 391–394 (2001). https://doi.org/10.1016/S0375-9601(01)00465-0. [arXiv:hep-th/0102098 [hep-th]]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. Kowalski-Glikman, J.: Introduction to doubly special relativity. Lect. Notes Phys. 669, 131–159 (2005). https://doi.org/10.1007/11377306_5. [arXiv:hep-th/0405273 [hep-th]]

    ADS  Article  Google Scholar 

  12. Magueijo, J., Smolin, L.: Lorentz invariance with an invariant energy scale. Phys. Rev. Lett. 88, 190403 (2002). https://doi.org/10.1103/PhysRevLett.88.190403. [arXiv:hep-th/0112090 [hep-th]]

    ADS  Article  Google Scholar 

  13. Magueijo, J., Smolin, L.: Generalized Lorentz invariance with an invariant energy scale. Phys. Rev. D 67, 044017 (2003). https://doi.org/10.1103/PhysRevD.67.044017. [arXiv:gr-qc/0207085 [gr-qc]]

    ADS  MathSciNet  Article  Google Scholar 

  14. Magueijo, J., Smolin, L.: String theories with deformed energy momentum relations, and a possible nontachyonic bosonic string. Phys. Rev. D 71, 026010 (2005). https://doi.org/10.1103/PhysRevD.71.026010. [arXiv:hep-th/0401087 [hep-th]]

    ADS  MathSciNet  Article  Google Scholar 

  15. Mignemi, S.: Transformations of coordinates and Hamiltonian formalism in deformed special relativity. Phys. Rev. D 68, 065029 (2003). https://doi.org/10.1103/PhysRevD.68.065029. [arXiv:gr-qc/0304029 [gr-qc]]

    ADS  MathSciNet  Article  Google Scholar 

  16. Pramanik, S., Ghosh, S.: GUP-based and Snyder Non-Commutative Algebras, Relativistic Particle models and Deformed Symmetries and Interaction: A Unified Approach. Int. J. Mod. Phys. A 28(27), 1350131 (2013). https://doi.org/10.1142/S0217751X13501315. [arXiv:1301.4042 [hep-th]]

    ADS  Article  MATH  Google Scholar 

  17. Pramanik, S., Ghosh, S., Pal, P.: Electrodynamics of a generalized charged particle in doubly special relativity framework. Annals Phys. 346, 113–128 (2014). https://doi.org/10.1016/j.aop.2014.04.009. [arXiv:1212.6881 [hep-th]]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. Takeda, M., Hayashida, N., Honda, K., Inoue, N., Kadota, K., Kakimoto, F., Kamata, K., Kawaguchi, S., Kawasaki, Y., Kawasumi, N., et al.: Small-scale anisotropy of cosmic rays above \(10^19ev\) observed with the akeno giant air shower array. Astrophys. J. 522, 225–237 (1999). https://doi.org/10.1086/307646. [arXiv:astro-ph/9902239 [astro-ph]]

    ADS  Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the grant “Integrable Deformations” (GA20-04800S) from the Czech Science Foundation (GACR).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. Klusoň.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Klusoň, J. Note about string theory with deformed dispersion relations. Gen Relativ Gravit 54, 61 (2022). https://doi.org/10.1007/s10714-022-02945-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10714-022-02945-0

Keywords

  • String Theory
  • Canonical Formalism
  • Modified Dispersion Relation