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On the classical and quantum evolution of Lagrangian half-forms in phase space. (English) Zbl 1049.53055

Here the author has proved that one can recover Maslov’s asymptotic formula for the solution of the Schrödinger’s equation via transportation of a Lagrangian half-form by the flow associated with the classical Hamiltonian. Next, he considers the case when the Hamiltonian flow is replaced by the flow associated with the Bohmian (the Hamiltonian with an extra term) and this leads to the conclusion that the use of Lagrangian half-forms is equivalent to a quantum mechanics on phase space as the local expressions of the half-forms on a quantized Lagrangian submanifold in phase space correspond just to the wave functions of quantum mechanics

MSC:

53D12 Lagrangian submanifolds; Maslov index
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
53D50 Geometric quantization
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
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References:

[1] V.I. Arnold , Mathematical Methods of Classical Mechanics , 2 nd ed., Graduate Texts in Mathematics , Springer , Berlin , New York , 1989 . MR 997295
[2] S.E. Cappell , R. Lee and E.Y. Miller , On the Maslov index , Comm. Pure Appl. Math. 47 ( 1994 ) 121 - 180 . MR 1263126 | Zbl 0805.58022 · Zbl 0805.58022 · doi:10.1002/cpa.3160470202
[3] D. Bohm and B.J. Hiley , The Undivided Universe: An Ontological Interpretation of Quantum Theory , Routledge , London and New York , 1993 . MR 1326828 | Zbl 0990.81503 · Zbl 0990.81503
[4] P. Dazord , Invariants homotopiques attachés aux fibrés symplectiques , Ann. Inst. Fourier , Grenoble 29 ( 2 ) ( 1979 ) 25 - 78 . Numdam | MR 539693 | Zbl 0378.58011 · Zbl 0378.58011 · doi:10.5802/aif.743
[5] M. Demazure , Classe de Maslov II , Exposé No. 10, in: Séminaire sur le Fibré Cotangent , Orsay 1975-1976 .
[6] G.B. Folland , Harmonic Analysis in Phase Space , Annals of Mathematics Studies , Princeton University Press , Princeton, NJ , 1989 . MR 983366 | Zbl 0682.43001 · Zbl 0682.43001
[7] M. De Gosson , La définition de l’indice de Maslov sans hypothèse de transversalité , C. R. Acad. Sci. Paris Série I 309 ( 1990 ) 279 - 281 . Zbl 0705.22012 · Zbl 0705.22012
[8] M. De Gosson , La relation entre Sp\infty , revêtement universel du groupe symplectique Sp et Sp x Z , C. R. Acad. Sci. Paris 310 ( 1990 ) 245 - 248 . MR 1042855 | Zbl 0732.22001 · Zbl 0732.22001
[9] M. De Gosson , Maslov indices on the metaplectic group Mp (n), Ann. Inst. Fourier , Grenoble 40 ( 3 ) ( 1990 ) 537 - 555 . Numdam | MR 1091832 | Zbl 0705.22013 · Zbl 0705.22013 · doi:10.5802/aif.1223
[10] M. De Gosson , The structure of q-symplectic geometry , J. Math. Pures Appl. 71 ( 1992 ) 429 - 453 . MR 1191584 | Zbl 0829.58015 · Zbl 0829.58015
[11] M. De Gosson , Cocycles de Demazure-Kashiwara et géométrie métaplectique , J. Geom. Phys. 9 ( 1992 ) 255 - 280 . MR 1171138 | Zbl 0776.53022 · Zbl 0776.53022 · doi:10.1016/0393-0440(92)90031-U
[12] M. De Gosson , On the Leray-Maslov quantization of Lagrangian manifolds , J. Geom. Phys. 13 ( 1994 ) 158 - 168 . MR 1260596 | Zbl 0795.58022 · Zbl 0795.58022 · doi:10.1016/0393-0440(94)90025-6
[13] M. De Gosson , Maslov Classes, Metaplectic Representation and Lagrangian Quantization , Research Notes in Mathematics , Vol. 95 , Akademie-Verlag , Berlin , 1996 . Zbl 0872.58031 · Zbl 0872.58031
[14] M. De Gosson , On half-form quantization of Lagrangian manifolds , Bull. Sci. Math. 1997 . · Zbl 0878.58023
[15] V. Guillemin and S. Sternberg , Geometric Asymptotics, Math. Surveys Monographs , Vol. 14 , Amer. Math. Soc. , Providence , RI , 1977 . MR 516965
[16] V. Guillemin and S. Sternberg , Symplectic Techniques in Physics , Cambridge University Press , Cambridge, MA , 1984 . MR 770935 | Zbl 0576.58012 · Zbl 0576.58012
[17] P.R. Holland , The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics , Cambridge University Press , Cambridge, MA , 1993 . MR 1341368 | Zbl 0854.00009 · Zbl 0854.00009 · doi:10.1017/CBO9780511622687
[18] J. Leray , Lagrangian Analysis , MIT Press , Cambridge, MA , London , 1981 ; Analyse Lagamgienne RCP 25 , Strasbourg 1978 ; Collège de France , 1976-1977 . MR 644633
[19] J. Leray , The meaning of Maslov’s asymptotic method the need of Planck’s constant in mathematics , Bull. Amer. Math. Soc. , Symposium on the Mathematical Heritage of Henri Poincaré, 1980 . Article | Zbl 0532.35068 · Zbl 0532.35068
[20] G. Lion and M. Vergne , The Weil Representation, Maslov Index and Theta Series , Progress in Math. , Birkhäuser , Boston , 1980 . MR 573448 | Zbl 0444.22005 · Zbl 0444.22005
[21] V.P. Maslov , Théorie des Perturbations et Méthodes Asymptotiques , Dunod , Paris , 1972 ; Perturbation Theory and Asymptotic Methods , Moscow , MGU , 1965 (in Russian).
[22] V.P. Maslov and M.V. Fedoriuk , Semi-Classical Approximations in Quantum Mechanics , Reidel , Boston , 1981 .
[23] A.S. Mischenko , V.E. Shatalov and B. Yu. Sternin , Lagrangian Manifolds and the Maslov Canonical Operator , Springer , Berlin , 1990 . Zbl 0727.58001 · Zbl 0727.58001
[24] J.M. Souriau , Indice de Maslov des variétés Lagrangiennes orientables , C. R. Acad. Sci. Paris Série A 276 ( 1973 ) 1025 - 1026 . MR 319227 | Zbl 0254.58007 · Zbl 0254.58007
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