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Covariant Jacobi brackets for test particles

Departamento de Física Teórica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain

Dipartimento di Fisica “E. Pancini” dell’Universitá “Federico II” di Napoli, Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy

Sezione INFN di Napoli, Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy

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F. Di Cosmo

Dipartimento di Fisica “E. Pancini” dell’Universitá “Federico II” di Napoli, Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy

Sezione INFN di Napoli, Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy

E-mail Address: dicosmo@na.infn.it

Corresponding author

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A. Ibort

Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain

ICMAT, Instituto de Ciencias Matemáticas (CSIC – UAM – UC3M – UCM), Nicolás Cabrera, 13–15, Campus de Cantoblanco, UAM, 28049 Madrid, Spain

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G. Marmo

Dipartimento di Fisica “E. Pancini” dell’Universitá “Federico II” di Napoli, Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy

Sezione INFN di Napoli, Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy

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https://doi.org/10.1142/S021773231750122XCited by:8 (Source: Crossref)

Abstract

We show that the space of observables of test particles has a natural Jacobi structure which is manifestly invariant under the action of the Poincaré group. Poisson algebras may be obtained by imposing further requirements. A generalization of Peierls procedure is used to extend this Jacobi bracket to the space of time-like geodesics on Minkowski spacetime.

Keywords:

PACS: 02.40.-k, 03.30.+p, 11.10.Ef

References

1. P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949). Crossref, Web of Science, ADS, Google Scholar
2. E. P. Wigner, Phys. Rev. 77, 711 (1950). Crossref, Web of Science, ADS, Google Scholar
3. J. Grabowski and G. Marmo, Binary operations in classical and quantum mechanics, in Classical and Quantum Integrability, Banach Center Publications, Vol. 59 (Polish Academy of Science, 2003), pp. 163–172. Google Scholar
4. E. Ercolessi, G. Marmo, G. Morandi, La Rivista del Nuovo Cimento 33, 410 (2010). ADS, Google Scholar
5. A. A. Kirillov, Russ. Math. Surv. 31, 55 (1976). Crossref, Google Scholar
6. R. E. Peierls, Proc. R. Soc. Lond. A 214, 1117 (1952). Crossref, Web of Science, Google Scholar
7. M. Asorey, F. M. Ciaglia, F. Di Cosmo and A. Ibort, Mod. Phys. Lett. A 32, 19 (2017). Google Scholar
8. F. Lizzi, G. Marmo, G. Sparano and A. M. Vinogradov, J. Geom. Phys. 14, 211 (1994). Crossref, Web of Science, ADS, Google Scholar
9. F. M. Ciaglia, F. Di Cosmo, M. Laudato and G. Marmo, Int. J. Geom. Methods Mod. Phys. 14, 1740003 (2017). Link, Web of Science, Google Scholar
10. G. Esposito, G. Marmo and E. C. G. Sudarshan, From Classical to Quantum Mechnics (Cambridge Univ. Press, 2004). Crossref, Google Scholar
11. V. I. Arnol’d and A. B. Givental’, Symplectic geometry, in Dynamical Systems IV, Encyclopaedia of Mathematical Sciences, Vol. 4 (Springer-Verlag, 1990). Crossref, Google Scholar
12. J. Grabowski and G. Marmo, J. Phys. A: Math. Gen. 34, 10975 (2001). Crossref, ADS, Google Scholar
13. P. Aniello, F. M. Ciaglia, F. Di Cosmo, G. Marmo and J. M. Pérez-Pardo, Ann. Phys. 373, 532 (2016). Crossref, Web of Science, ADS, Google Scholar
14. F. Di Cosmo and A. Zampini, Int. J. Geom. Methods Mod. Phys. 14, 1740005 (2017). Link, Web of Science, Google Scholar
15. G. Marmo, G. Morandi and N. Mukunda, J. Geom. Mech. 1, 317 (2009). Crossref, Web of Science, Google Scholar
16. B. A. Dubrovin, M. Giordano, G. Marmo and A. Simoni, Int. J. Mod. Phys. A 8, 3747 (1993). Link, Web of Science, ADS, Google Scholar
17. G. Morandi, C. Ferrario, G. L. Vecchio, G. Marmo and C. Rubano, Phys. Rep. 188, 147 (1990). Crossref, Web of Science, ADS, Google Scholar
18. A. P. Balachandran, G. Marmo, N. Mukunda, J. Nilsson, E. C. G. Sudarshan and F. Zaccaria, Phys. Rev. D 27, 10 (1983). Google Scholar
19. A. Bautista, A. Ibort and J. Lafuente, The contact structure in the space of light rays, in A Mathematical Tribute to Professor José Maria Montesinos Amilibia (Universidad Complutense de Madrid, 2016), pp. 133–159, arXiv:1510.08139. Google Scholar
20. R. D. Sorkin, Int. J. Geom. Methods Mod. Phys. 14, 1740007 (2017). Link, Web of Science, ADS, Google Scholar
21. A. Ibort, M. de León and G. Marmo, J. Phys. A: Math. Gen. 30, 8 (1997). Google Scholar
22. M. Asorey, A. P. Balachandran, F. Lizzi and G. Marmo, JHEP 2017, 136 (2017). Crossref, Web of Science, Google Scholar