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
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
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
Jadczyk, A., Modugno, M. (1994): Galilei general relativistic quantum mechanics. Report Dept. Appl. Math, Univ. of Florence, pp. 215
Abraham, R., Marsden, J.E. (1978): Foundations of Mechanics. 2nd ed., Benjamin-Cummings
Janyška, J. (1995): Remarks on symplectic and contact 2-forms in relativistic theories. Bollettino U.M.L 7, 9-B, 587–616
Janyška, J. (1995): Natural quantum Lagrangians in Galilei general relativistic quantum Lagrangians. Rendiconti di Matematica, S. VII, Vol. 15, Roma, 457–468
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
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
Vitolo, R. (1996): Spherical symmetry in classical and quantum Galilei general relativity. Annales de l’Institut Henri Poincaré, 64,(2), 177–203
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
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
Vitolo, R. (1998): Quantising the rigid body. In: Proceedings of the VII Conference on Differential Geometry and Applications, Brno 1998, 653–664
Vitolo, R. ( 1999): Quantum structures in Galilei general relativity. Ann. Inst.’ H. Poinc. 70,(3), 239–257
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
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
Sailer, D., Vitolo, R. (2000): Symmetries in covariant classical mechanics. J. Math. Phys. 41(10), 6824–6842
Trautman, A. (1963): Sur la théorie Newtonienne de la gravitation. C. R. Acad. Sc. Paris t. 257, 617–620
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
Dombrowski, H.D., Horneffer, K. (1964): Die Differentialgeometrie des Galileischen Relativitätsprinzips. Math. Z. 86, 291–311
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
Duval, C. (1993): On Galilean isometries. Clas. Quant. Grav. 10, 2217–2221
Duval, C., Burdet, G., Künzle, H.P., Perrin, M. (1985): Bargmann structures and Newton-Cartan theory. Phys. Rev. D 31(8), 1841–1853
Duval, C., Gibbons, G., Horvaty, P. (1991): Celestial mechanics, conformai structures, and gravitational waves. Phys. Rev. D 43(12), 3907–3921
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
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
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
Kuchař, K. (1980): Gravitation, geometry and nonrelativistic quantum theory. Phys. Rev. D 22(6), 1285–1299
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
Künzle, H.P. (1974): Galilei and Lorentz invariance of classical particle interaction. Symposia Mathematica 14, 53–84
Künzle, H.P. (1976): Covariant Newtonian limit of Lorentz space-times. G.R.G. 7(5), 445–457
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
Künzle, H.R, Duval, C. (1984): Dirac field on Newtonian space-time. Ann. Inst. H. Poinc. 41(4), 363–384
Le Bellac, M., Levy-Leblond, J.M. (1973): Galilean electromagnetism. Nuovo Cim. B 14(2), 217–233
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
Mangiarotti, L. (1979): Mechanics on a Galilean manifold. Riv. Mat. Univ. Parma 5(4), 1–14
Schmutzer, E., Plebanski, E. (1977): Quantum mechanics in non inertial frames of reference. Fortschritte der Physik 25, 37–82
Tulczyjew W.M. (1981): Classical and quantum mechanics of particles in external gauge fields. Rend. Sem. Mat. Univ. Torino 39, 111–124
Tulczyjew, W.M. (1985): An intrinsic formulation of nonrelativistic analytical mechanics and wawe mechanics. J. Geom. Phys. 2,(3), 93–105
Kolář, I., Michor, P., Slovák, J. (1993): Natural operations in differential geometry. Springer-Verlag, Berlin
Libermann, P., Marie, Ch.M. (1987): Symplectic geometry and analytical mechanics. Reidel Publ., Dordrecht
Woodhouse N. (1992): Geometric quantization. Second Edn, Clarendon Press, Oxford
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
Janyska, J. (1998): Natural Lagrangians for quantum structures over 4-dimensional spaces. Rendiconti di Matematica, S. VII, Vol. 18, Roma, 623–648
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
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
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
García, PL. (1972): Connections and 1-jet fibre bundle. Rendic. Sem. Mat. Univ. Padova 47, 227–242
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
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
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
Cattaneo, V. (1970): Invariance Relativiste, Symetries Internes et Extensions d’Algébre de Lie. Thesis Université Catholique de Louvain
de Leon, M., Marrero, J.C., Padron, E. (1997): On the geometric quantization of Jacobi manifolds. J. Math. Phys. 38,(12), 6185–6213
Fanchi, J.R. (1993): Review of invariant time formulations of relativistic quantum theories. Found. Phys. 23, 487–548
Fanchi, J.R. (1994): Evaluating the validity of parametrized relativistic wave equations. Found. Phys. 24, 543–562
Fernández, M., Ibañez, R., de Leon, M. (1996): Poisson cohomology and canonical cohomology of Poisson manifolds. Archivium Mathematicum (Brno) 32, 29–56
Gotay, M.J. (1986): Constraints, reduction and quantization. J. Math. Phys. 27(8), 2051–2066
Horwitz, L.P. (1992): On the definition and evolution of states in relativistic classical and quantum mechanics. Foun. Phys. 22, 421–448
Horwitz, L.P, Rotbart, FC. (1981): Non relativistic limit of relativistic quantum mechanics. Phys. Rev. D 24, 2127–2131
Kyprianidis, A. (1987): Scalar time parametrization of relativistic quantum mechanics: The covariant Schr odinger formalism. Phys. Rep 155, 1–27
Marmo, G., Morandi, G., Simoni. A. (1988): Quasi-invariance and central extensions. Phys. Rev. D 37, p. 2196–2206
Marsden, J.E., Ratiu, T. (1995): Introduction to mechanics and symmetry. Texts in Appl. Math. 17, Springer, New York
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
Peres, A. (1995): Relativistic quantum measurements, in Fundamental problems of quantum theory, Ann. N. Y. Acad. Sci., 755
Piron, C., Reuse, F. (1979): On classical and quantum relativistic dynamics. Found. Phys. 9, 865–882
Simms, D.J. (1968): Lie groups and quantum mechanics, in Lect. Notes Math., Vol. 52, Springer, Berlin Heidelberg New York
Simms, D.J., Woodhouse, N. (1977): Lectures on Geometric Quantization. Lect. Notes Phys. 53, Springer, Berlin Heidelberg New York
Sniaticki, J. (1980): Geometric quantization and quantum mechanics. Springer-Verlag, New York
Tuynman, G.M., Wigerinck, W.A.J.J. (1987): Central extensions and physics. J. Geom. Phys. 4,(3), 207–258
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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
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