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Hermann, Robert

Quantum mechanics and geometric analysis on manifolds. (English) Zbl 0506.58017

Int. J. Theor. Phys. 21, 803-828 (1982).
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MSC:

53D50 Geometric quantization
53B50 Applications of local differential geometry to the sciences
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
81P99 Foundations, quantum information and its processing, quantum axioms, and philosophy
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Keywords:

differential operator; co-Riemannian metric; conformal gauge symmetry; classical configuration space; hydrodynamical interpretation of quantum mechanics

Citations:

Zbl 0483.93007
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Full Text: DOI

References:

[1] Abraham, R., and Marsden, J. (1978).Foundations of Mechanics, 2nd ed. Addison-Wesley, Reading, Massachusetts. · Zbl 0393.70001
[2] Grossman, A., Loupias, G., and Stein, E. M. (1968). ”An Algebra of Pseudodifferential Operators and Quantum Mechanics in Phase Spaces,”Annales de l’Institut Fourier, Grenoble,18, 343–368. · Zbl 0176.45102
[3] Hermann, R. (1965). ”Remarks on the Geometric Nature of Quantum Phase Space,”Journal of Mathematical Physics,6, 1768–1771. · doi:10.1063/1.1704721
[4] Hermann, R. (1966).Lie Groups for Physicists. W. A. Benjamin, New York. · Zbl 0135.06901
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[6] Hermann, R. (1970a).Lie Algebras and Qunatum Mechanics. W. A. Benjamin, New York. · Zbl 0206.26901
[7] Hermann, R. (1970b).Vector Bundles in Mathematical Physics, Parts I and II. W. A. Benjamin, New York. · Zbl 0213.23603
[8] Hermann, R. (1972a).Lectures on Mathematical Physics, Vol. II. W. A. Benjamin, Reading, Massachusetts. · Zbl 0228.00002
[9] Hermann, R. (1972b). ”Currents in Classical Field Theories,”Journal of Mathematical Physics,13, 97. · doi:10.1063/1.1665858
[10] Hermann, R. (1973a).Geometry, Physics and Systems, Marcel Dekker, New York. · Zbl 0285.58001
[11] Hermann, R. (1973b).Topics in the Mathematics of Quantum Mechanics, Vol. VI of Interdisciplinary Mathematics. Math Sci Press, Brookline, Massachusetts. · Zbl 0287.00004
[12] Hermann, R. (1973c). ”Geodesics of Singular Riemannian Metrics,”Bulletin of the American Mathematical Society,79, 780–782. · Zbl 0269.53020 · doi:10.1090/S0002-9904-1973-13313-0
[13] Hermann, R. (1973d).Topics in General Relativity, Vol. V of Interdisciplinary Mathematics. Math Sci Press, Brookline, Massachusetts. · Zbl 0286.00002
[14] Hermann, R. (1974).Physical Aspects of Lie Group Theory. University of Montreal Press, Montreal. · Zbl 0331.22001
[15] Hermann, R. (1975).Gauge Fields and Cartan-Ehresmann Connections, Part A, Vol. X of Interdisciplinary Mathematics. Math Sci Press, Brookline, Massachusetts. · Zbl 0314.53039
[16] Hermann, R. (1976).Geometric Structure of Systems-Control Theory and Physics, Part B, Vol. XI of Interdisciplinary Mathematics. Math Sci Press, Brookline, Massachusetts. · Zbl 0472.49002
[17] Hermann, R. (1977a). ”Appendix on Quantum Mechanics,” inSymplectic Geometry and Fourier Analysis, by N. Wallach. Math Sci Press, Brookline, Massachusetts.
[18] Hermann, R. (1977b).Quantum and Fermion Differential Geometry, Part A, Math Sci Press, Brookline, Massachusetts. · Zbl 0367.53018
[19] Hermann, R. (1977c).Differential Geometry and the Calculus of Variations, 2nd ed. Math Sci Press, Brookline, Massachusetts. · Zbl 0371.49001
[20] Herman, R. (1978). ”Modern Differential Geometry in Elementary Particle Physics,”VII GIFT Conference on Theoretical Physics, Salamonca, Spain, 1977, Proceedings, A. Azcarraga, ed., Springer-Verlag, Berlin.
[21] Hermann, R. (1981). ”Infeld-Hull Factorization, Galois-Picard-Vessiot Theory for Differential Operators,”Journal of Mathematical Physics,22, 1163–1167. · Zbl 0467.34021 · doi:10.1063/1.525039
[22] Hermann, R. (1978)Yang-Mills, Kaluza-Klein and the Einstein Program. Math Sci Press, Brookline, Massachusetts. · Zbl 0392.53040
[23] Hermann, R. (1981). ”Geometric Structure of Filtering and Scattering Systems,”Journal of Mathematical Physics,22, 2203–2207. · Zbl 0483.93007 · doi:10.1063/1.524788
[24] Hermann, R. (1982). ”Differential Geometry and Lie Theory of Classical and Quantum Stochastic Systems,”Stochastics (to appear).
[25] Hermann, R., and Hurt, N. (1980).Quantum Statistical Mechanics and Lie Group Harmonic Analysis. Math Sci Press. Brookline, Massachusetts. · Zbl 0439.22011
[26] Hurt, N., and Hermann, R. (1979). ”Some Relations between System Theory and Statistical Mechanics,”Riceria de Automatica,10, 316–343.
[27] Lévy, P. (1924).Analyse Functionelle Concrète. Gauthier-Villars, Paris.
[28] Madelung, E. (1926). ”Quantum Theorie in hydrodynamischen For,”Zeitschift für Physik, 323–326. · JFM 52.0969.06
[29] Taylor, M. (1981).Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey. · Zbl 0453.47026
[30] Treves, F. (1980).Introduction to Pseudodifferential and Fourier Integral Operators. Vols. I and II. Plenum Press, New York.
[31] von Neumann, J. (1955).The Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, New Jersey. · Zbl 0064.21503
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