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Covariant Quantum Mechanics and Quantum Symmetries

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Recent Developments in General Relativity, Genoa 2000

Abstract

We sketch the basic ideas and results on the covariant formulation of quantum mechanics on a curved spacetime with absolute time equipped with given gravitational and electromagnetic fields. Moreover, we analyse the classical and quantum symmetries and show their relations.

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References

  1. Jadczyk, A., Modugno, M. (1992): An outline of a new geometric approach to Galilei general relativistic quantum mechanics. in Differential geometric methods in theoretical physics, ed. by C. N. Yang, M. L. Ge and X. W. Zhou, World Scientific, Singapore, pp. 543–556

    Google Scholar 

  2. Jadczyk, A., Modugno, M. (1993): A scheme for Galilei general relativistic quantum mechanics, in Proceedings of the 10th Italian Conference on general relativity and gravitational physics, Bardonecchia, 1–5 September, ed. by M Cerdonio, R. D’Auria, M. Francariglia, G. Magnano, World Scientific, New York

    Google Scholar 

  3. Jadczyk, A., Modugno, M. (1994): Galilei general relativistic quantum mechanics. Report Dept. Appl. Math, Univ. of Florence, pp. 215

    Google Scholar 

  4. Abraham, R., Marsden, J.E. (1978): Foundations of Mechanics. 2nd ed., Benjamin-Cummings

    Google Scholar 

  5. Janyška, J. (1995): Remarks on symplectic and contact 2-forms in relativistic theories. Bollettino U.M.L 7, 9-B, 587–616

    Google Scholar 

  6. Janyška, J. (1995): Natural quantum Lagrangians in Galilei general relativistic quantum Lagrangians. Rendiconti di Matematica, S. VII, Vol. 15, Roma, 457–468

    MATH  Google Scholar 

  7. Modugno, M., Vitolo, M. (1996): Quantum connection and Poincaré-Cartan form, in Gravitation, electromagnetism and geometrical structures, ed. by G. Ferrarese, Pitagora, Bologna, 237–279

    Google Scholar 

  8. Janyška, J., Modugno, M. (1996): Relations between linear connections on the tangent bundle and connections on the jet bundle of a fibred manifold. Arch. Math. (Brno), 32, 281–288; http://www.emis.de/journals

    MathSciNet  MATH  Google Scholar 

  9. Vitolo, R. (1996): Spherical symmetry in classical and quantum Galilei general relativity. Annales de l’Institut Henri Poincaré, 64,(2), 177–203

    MathSciNet  MATH  Google Scholar 

  10. Vitolo, R. (1996): Quantum structures in general relativistic theories. In Proceedings of the XII Italian Conference on General Relativity and Gravitational Physics, Roma, 1996; World Scientific, Singapore

    Google Scholar 

  11. Janyška, J., Modugno, M. (1999): On the graded Lie algebra of quantisable forms, in Differential Geometry and Applications, ed. by I. Kolář, O. Kowalski, D. Krupka, J. Slovak, Proceedings of the 7th International Conference, Brno, 10–14 August 1998, Masaryk University, 601–620

    Google Scholar 

  12. Vitolo, R. (1998): Quantising the rigid body. In: Proceedings of the VII Conference on Differential Geometry and Applications, Brno 1998, 653–664

    Google Scholar 

  13. Vitolo, R. ( 1999): Quantum structures in Galilei general relativity. Ann. Inst.’ H. Poinc. 70,(3), 239–257

    MathSciNet  MATH  Google Scholar 

  14. Janyška, J. (2001): A remark on natural quantum Lagrangians and natural generalized Schrödinger operators in Galilei quantum mechanics, in Proceedings of the 20th Winter School of geometry and physics, Srni, January 15–22, 2000, Supplemento ai rendiconti del Circolo Matematico di Palermo, Serie II, Numero 66, pp. 117–128

    Google Scholar 

  15. Modugno, M., Tejero Prieto, C., Vitolo, R. (2000): Comparison between geometric quantisation and covariant quantum mechanic, in Proceedings Lie Theory and Its Applications in Physics — Lie III, 11–14 July 1999, Clausthal, Germany, ed. by H.-D. Doebner, V.K. Dobrev, J. Hilgert, World Scientific, Singapore, 155–175

    Google Scholar 

  16. Sailer, D., Vitolo, R. (2000): Symmetries in covariant classical mechanics. J. Math. Phys. 41(10), 6824–6842

    Article  MathSciNet  ADS  Google Scholar 

  17. Trautman, A. (1963): Sur la théorie Newtonienne de la gravitation. C. R. Acad. Sc. Paris t. 257, 617–620

    MathSciNet  MATH  Google Scholar 

  18. Trautman, A. (1966): Comparison of Newtonian and relativistic theories of space-time, in Perspectives in geometry and relativity, N. 42, Indiana University press, 413–425

    MathSciNet  Google Scholar 

  19. Dombrowski, H.D., Horneffer, K. (1964): Die Differentialgeometrie des Galileischen Relativitätsprinzips. Math. Z. 86, 291–311

    Article  MathSciNet  MATH  Google Scholar 

  20. Duval, C. (1985): The Dirac & Levy-Leblond equations and geometric quantization, in Diff. Geom. Meth. in Math. Phys., Proceedings of the 14th International Conference held in Salamanca, Spain, June 24–29, ed. by P.L. García, A. Pérez-Rendón, L.N.M. 1251, Springer-Verlag, Berlin, pp. 205–221

    Google Scholar 

  21. Duval, C. (1993): On Galilean isometries. Clas. Quant. Grav. 10, 2217–2221

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Duval, C., Burdet, G., Künzle, H.P., Perrin, M. (1985): Bargmann structures and Newton-Cartan theory. Phys. Rev. D 31(8), 1841–1853

    Article  MathSciNet  ADS  Google Scholar 

  23. Duval, C., Gibbons, G., Horvaty, P. (1991): Celestial mechanics, conformai structures, and gravitational waves. Phys. Rev. D 43(12), 3907–3921

    Article  MathSciNet  ADS  Google Scholar 

  24. Duval, C., Künzle, H.P. (1984): Minimal gravitational coupling in the Newtonian theory and the covariant Schr odinger equation. G.R.G. 16(4), 333–347

    Article  Google Scholar 

  25. Ehlers, J. (1989): The Newtonian limit of general relativity, in Fisica matematica classica e relatività, Rapporti e Compatiilitá, Elba 9–13 giugno 1989, pp. 95–106

    Google Scholar 

  26. Havas, P. (1964): Four-dimensional formulation of Newtonian mechanics and their relation to the special and general theory of relativity. Rev. Modern Phys. 36, 938–965

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Kuchař, K. (1980): Gravitation, geometry and nonrelativistic quantum theory. Phys. Rev. D 22(6), 1285–1299

    Article  MathSciNet  ADS  Google Scholar 

  28. Künzle, H.R (1972): Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics. Ann. Inst. H. Poinc. 17(4), 337–362

    Google Scholar 

  29. Künzle, H.P. (1974): Galilei and Lorentz invariance of classical particle interaction. Symposia Mathematica 14, 53–84

    Google Scholar 

  30. Künzle, H.P. (1976): Covariant Newtonian limit of Lorentz space-times. G.R.G. 7(5), 445–457

    MATH  Google Scholar 

  31. Künzle, H.P. (1984): General covariance and minimal gravitational coupling in Newtonian spacetime, in Geometrodynamics Proceedings, ed by A. Prastaro, Tecnoprint, Bologna, pp. 37–48

    Google Scholar 

  32. Künzle, H.R, Duval, C. (1984): Dirac field on Newtonian space-time. Ann. Inst. H. Poinc. 41(4), 363–384

    MATH  Google Scholar 

  33. Le Bellac, M., Levy-Leblond, J.M. (1973): Galilean electromagnetism. Nuovo Cim. B 14(2), 217–233

    Google Scholar 

  34. Levy-Leblond, J.M. (1971): Galilei group and Galilean invariance, in Group theory and its applications, ed. by E. M. Loebl, Vol. 2, Academic, New York, pp. 221–299

    Google Scholar 

  35. Mangiarotti, L. (1979): Mechanics on a Galilean manifold. Riv. Mat. Univ. Parma 5(4), 1–14

    MathSciNet  Google Scholar 

  36. Schmutzer, E., Plebanski, E. (1977): Quantum mechanics in non inertial frames of reference. Fortschritte der Physik 25, 37–82

    Article  MathSciNet  ADS  Google Scholar 

  37. Tulczyjew W.M. (1981): Classical and quantum mechanics of particles in external gauge fields. Rend. Sem. Mat. Univ. Torino 39, 111–124

    MathSciNet  MATH  Google Scholar 

  38. Tulczyjew, W.M. (1985): An intrinsic formulation of nonrelativistic analytical mechanics and wawe mechanics. J. Geom. Phys. 2,(3), 93–105

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. Kolář, I., Michor, P., Slovák, J. (1993): Natural operations in differential geometry. Springer-Verlag, Berlin

    MATH  Google Scholar 

  40. Libermann, P., Marie, Ch.M. (1987): Symplectic geometry and analytical mechanics. Reidel Publ., Dordrecht

    Book  MATH  Google Scholar 

  41. Woodhouse N. (1992): Geometric quantization. Second Edn, Clarendon Press, Oxford

    MATH  Google Scholar 

  42. Janyška, J., Modugno, M. (1996): Classical particle phase space in general relativity, in Proc. Conf. Diff. Geom. Appl., Brno 28 August-1 September 1995, Masaryk University, 1996, ed. by J. anyška, I. Kolar, J. Slovak, pp. 573–602; http://www.emis.de/proceedings

  43. Janyska, J. (1998): Natural Lagrangians for quantum structures over 4-dimensional spaces. Rendiconti di Matematica, S. VII, Vol. 18, Roma, 623–648

    MathSciNet  MATH  Google Scholar 

  44. Janyška, J., Modugno, M. (2000): Quantisable functions in general relativity, in Opérateurs différentiels et Physique Mathématique, ed. by J. Vaillant, J. Carvalho e Silva, Textos Mat. Ser. B, 24, 161–181

    Google Scholar 

  45. Janyška, J., Modugno, M. ( 1997): On quantum vector fields in general relativistic quantum mechanics. General Mathematics 5, Proceeding of the 3rd International Workshop on Differential Geometry and its Applications, Sibiu (Romania) 1997, 199–217

    Google Scholar 

  46. Jadczyk, A., Janyška, J., Modugno, M. (1998): Galilei general relativistic quantum mechanics revisited, in Geometria, física-matemática e outros ensaios, ed. by A.S. Alves, F.J. Craveiro de Carvalho, J.A. Pereira da Silva, Departamento de Matematica, Universidade de Coimbra, Coimbra, pp. 253–313

    Google Scholar 

  47. García, PL. (1972): Connections and 1-jet fibre bundle. Rendic. Sem. Mat. Univ. Padova 47, 227–242

    MATH  Google Scholar 

  48. Canarutto, D., Jadczyk, A., Modugno, M. (1995): Quantum mechanics of a spin particle in a curved spacetime with absolute time. Rep. on Math. Phys. 36, 1, 95–140

    MathSciNet  Google Scholar 

The following additional references are useful for a comparison with the current literature

  • Albert, C. (1989): Le théorème de reduction de Marsden-Weinstein en géométrie cosymplectique et de contact. J. Geom. Phys. 6(4), 627–649

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Balachandran, A.P., Gromm, H., Sorkin, R.D. (1987): Quantum symmetries from quantum phases. Fermions from Bosons, a Z2 Anomaly and Galileian Invariance. Nucl. Phys. B 281, 573–583

    Article  ADS  Google Scholar 

  • Cattaneo, V. (1970): Invariance Relativiste, Symetries Internes et Extensions d’Algébre de Lie. Thesis Université Catholique de Louvain

    Google Scholar 

  • de Leon, M., Marrero, J.C., Padron, E. (1997): On the geometric quantization of Jacobi manifolds. J. Math. Phys. 38,(12), 6185–6213

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Fanchi, J.R. (1993): Review of invariant time formulations of relativistic quantum theories. Found. Phys. 23, 487–548

    Article  MathSciNet  ADS  Google Scholar 

  • Fanchi, J.R. (1994): Evaluating the validity of parametrized relativistic wave equations. Found. Phys. 24, 543–562

    Article  MathSciNet  ADS  Google Scholar 

  • Fernández, M., Ibañez, R., de Leon, M. (1996): Poisson cohomology and canonical cohomology of Poisson manifolds. Archivium Mathematicum (Brno) 32, 29–56

    MATH  Google Scholar 

  • Gotay, M.J. (1986): Constraints, reduction and quantization. J. Math. Phys. 27(8), 2051–2066

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Horwitz, L.P. (1992): On the definition and evolution of states in relativistic classical and quantum mechanics. Foun. Phys. 22, 421–448

    Article  MathSciNet  ADS  Google Scholar 

  • Horwitz, L.P, Rotbart, FC. (1981): Non relativistic limit of relativistic quantum mechanics. Phys. Rev. D 24, 2127–2131

    Article  MathSciNet  ADS  Google Scholar 

  • Kyprianidis, A. (1987): Scalar time parametrization of relativistic quantum mechanics: The covariant Schr odinger formalism. Phys. Rep 155, 1–27

    Article  MathSciNet  ADS  Google Scholar 

  • Marmo, G., Morandi, G., Simoni. A. (1988): Quasi-invariance and central extensions. Phys. Rev. D 37, p. 2196–2206

    Article  MathSciNet  ADS  Google Scholar 

  • Marsden, J.E., Ratiu, T. (1995): Introduction to mechanics and symmetry. Texts in Appl. Math. 17, Springer, New York

    Google Scholar 

  • Michel, L. (1965): Invariance in quantum mechanics and group extensions, in Group Theoretical Concepts and Methods in Elememtary Particle Physics, ed. by F. Gürsey, Gordon and Breach, New York

    Google Scholar 

  • Peres, A. (1995): Relativistic quantum measurements, in Fundamental problems of quantum theory, Ann. N. Y. Acad. Sci., 755

    Google Scholar 

  • Piron, C., Reuse, F. (1979): On classical and quantum relativistic dynamics. Found. Phys. 9, 865–882

    Article  MathSciNet  ADS  Google Scholar 

  • Simms, D.J. (1968): Lie groups and quantum mechanics, in Lect. Notes Math., Vol. 52, Springer, Berlin Heidelberg New York

    Google Scholar 

  • Simms, D.J., Woodhouse, N. (1977): Lectures on Geometric Quantization. Lect. Notes Phys. 53, Springer, Berlin Heidelberg New York

    Google Scholar 

  • Sniaticki, J. (1980): Geometric quantization and quantum mechanics. Springer-Verlag, New York

    Book  Google Scholar 

  • Tuynman, G.M., Wigerinck, W.A.J.J. (1987): Central extensions and physics. J. Geom. Phys. 4,(3), 207–258

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Janyška, J., Modugno, M., Saller, D. (2002). Covariant Quantum Mechanics and Quantum Symmetries. In: Cianci, R., Collina, R., Francaviglia, M., Fré, P. (eds) Recent Developments in General Relativity, Genoa 2000. Springer, Milano. https://doi.org/10.1007/978-88-470-2101-3_13

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  • DOI: https://doi.org/10.1007/978-88-470-2101-3_13

  • Publisher Name: Springer, Milano

  • Print ISBN: 978-88-470-0162-6

  • Online ISBN: 978-88-470-2101-3

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