Momentum maps for mixed states in quantum and classical mechanics

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December 2019, 11(4): 639-656. doi: 10.3934/jgm.2019032

Momentum maps for mixed states in quantum and classical mechanics

Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK, Mathematical Sciences Research Institute, Berkeley, CA 94720, USA

For Darryl Holm, on the occasion of his 70th birthday

Received July 2018 Revised July 2019 Published November 2019

Fund Project: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at MSRI, during the Fall 2018 semester. In addition, the author was supported by the Leverhulme Trust Research Grant No. 2014-112, and by the London Mathematical Society Grant No. 31633 (Applied Geometric Mechanics).

This paper presents the momentum map structures which emerge in the dynamics of mixed states. Both quantum and classical mechanics are shown to possess analogous momentum map pairs associated to left and right group actions. In the quantum setting, the right leg of the pair identifies the Berry curvature, while its left leg is shown to lead to different realizations of the density operator, which are of interest in quantum molecular dynamics. Finally, the paper shows how alternative representations of both the density matrix and the classical density are equivariant momentum maps generating new Clebsch representations for both quantum and classical dynamics. Uhlmann's density matrix [58] and Koopman wavefunctions [41] are shown to be special cases of this construction.

Keywords: Momentum map, quantum mechanics, mixed states.

Mathematics Subject Classification: 70Hxx, 82Cxx, 37K05, 81Rxx.

Citation: Cesare Tronci. Momentum maps for mixed states in quantum and classical mechanics. Journal of Geometric Mechanics, 2019, 11 (4) : 639-656. doi: 10.3934/jgm.2019032

References:

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[2]	A. Abedi, N. T. Maitra and E. K. U. Gross, Correlated electron-nuclear dynamics: Exact factorization of the molecular wavefunction, The J. Chem. Phys., 137 (2012), 22A530.
[3]	R. Abraham, J. E. Marsden and T. M. Ratiu, Tensor Analysis and Applications, Second edition, Applied Mathematical Sciences, 75. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.
[4]	F. Agostini, A. Abedi, Y. Suzuki, S. K. Min, N. T. Maitra and E. K. U. Gross, The exact forces on classical nuclei in non-adiabatic charge transfer, J. Chem. Phys., 142 (2015), 084303.
[5]	J. Anandan, A geometric approach to quantum mechanics, Found. Phys., 21 (1991), 1265-1284. doi: 10.1007/BF00732829.
[6]	M. Baer, Beyond Born-Oppenheimer: Conical Intersections and Electronic Nonadiabatic Coupling Terms, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. doi: 10.1002/0471780081.
[7]	M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. A, 392 (1984), 45-57. doi: 10.1098/rspa.1984.0023.
[8]	D. Bondar, R. Cabrera, R. R. Lompay, M. Y. Ivanov and H. Rabitz, Operational dynamic modeling transcending quantum and classical mechanics, Phys. Rev. Lett., 109 (2012), 190403.
[9]	D. Bondar, F. Gay-Balmaz and C. Tronci, Koopman wavefunctions and classical-quantum correlation dynamics, Proc. R. Soc. A, 475 (2019), 20180879, 18 pp. doi: 10.1098/rspa.2018.0879.
[10]	E. Bonet Luz and C. Tronci, Geometry and symmetry of quantum and classical-quantum variational principles, J. Math. Phys., 56 (2015), 082104, 19 pp. doi: 10.1063/1.4929567.
[11]	E. Bonet Luz and C. Tronci, Hamiltonian approach to Ehrenfest expectation values and Gaussian quantum states, Proc. R. Soc. A, 472 (2016), 20150777, 15 pp. doi: 10.1098/rspa.2015.0777.
[12]	M. Born and R. Oppenheimer, Zur quantentheorie der molekeln, Ann. Physik, 389 (1927), 457-484.
[13]	D. C. Brody and L. P. Hughston, Geometric quantum mechanics, J Geom. Phys., 38 (2001), 19-53. doi: 10.1016/S0393-0440(00)00052-8.
[14]	J. F. Cariñena, A. Ibort, G. Marmo and G. Morandi, Geometry from Dynamics, Classical and Quantum, Springer, Dordrecht, 2015. doi: 10.1007/978-94-017-9220-2.
[15]	P. R. Chernoff and J. E. Marsden, Some remarks on Hamiltonian systems and quantum mechanics, Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, Univ. Western Ontario Ser. Philos. Sci., Reidel, Dordrecht, 6 (1977), 35-53.
[16]	A. Clebsch, Uber die Integration der hydrodynamischen Gleichungen, J. Reine Angew. Math., 56 (1859), 1-10. doi: 10.1515/crll.1859.56.1.
[17]	M. de Gosson, Quantum harmonic analysis of the density matrix, Quanta, 7 (2018), 74-110. doi: 10.12743/quanta.v7i1.74.
[18]	M. de Gosson, Symplectic Geometry and Quantum Mechanics, Operator Theory: Advances and Applications, 166. Advances in Partial Differential Equations (Basel). Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7575-2.
[19]	P. A. M. Dirac, On the analogy between classical and quantum mechanics, Rev. Mod. Phys., 17 (1945), 195-199. doi: 10.1103/RevModPhys.17.195.
[20]	P. A. M. Dirac, The Lagrangian in quantum mechanics, Feynman's Thesis: A New Approach to Quantum Theory, World Scientific, (2005), 111–119.
[21]	M. S. Foskett, D. D. Holm and C. Tronci, Geometry of nonadiabatic quantum hydrodynamics, Acta Appl. Math., 162 (2019), 63-103. doi: 10.1007/s10440-019-00257-1.
[22]	F. Gay-Balmaz and C. Tronci, Madelung transform and probability currents in hybrid classical-quantum dynamics, arXiv: 1907.06624.
[23]	F. Gay-Balmaz and C. Tronci, Vlasov moment flows and geodesics on the Jacobi group, J. Math. Phys., 53 (2012), 123502, 36 pp. doi: 10.1063/1.4763467.
[24]	F. Gay-Balmaz, C. Tronci and C. Vizman, Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups, J. Geom. Mech., 5 (2013), 39-84. doi: 10.3934/jgm.2013.5.39.
[25]	F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Global Anal. Geom., 41 (2012), 1-24. doi: 10.1007/s10455-011-9267-z.
[26]	J. W. Gray, Some global properties of contact structures, Ann. Math., 69 (1959), 421-450. doi: 10.2307/1970192.
[27]	V. Guillemin and S. Sternberg, The moment map and collective motion, Ann. Phys., 127 1980, 220–253. doi: 10.1016/0003-4916(80)90155-4.
[28]	C. Günther, Presymplectic manifolds and the quantization of relativistic particle systems, Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Math., Springer, Berlin, 836 (1980), 383-400.
[29]	B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267. Springer, New York, 2013. doi: 10.1007/978-1-4614-7116-5.
[30]	D. D. Holm and B. Kuperschmidt, Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity, Phys. D, 6 (1983), 347-363. doi: 10.1016/0167-2789(83)90017-9.
[31]	D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, The breadth of symplectic and Poisson Geometry, Progr. Math., Birkhäuser Boston, Boston, MA, 232 (2005), 203-235. doi: 10.1007/0-8176-4419-9_8.
[32]	D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, Oxford Texts in Applied and Engineering Mathematics, 12. Oxford University Press, Oxford, 2009.
[33]	D. D. Holm and C. Tronci, Geodesic Vlasov equations and their integrable moment closures, J. Geom. Mech., 1 (2009), 181-208. doi: 10.3934/jgm.2009.1.181.
[34]	G. Hunter, Conditional probability amplitudes in wave mechanics, Int. J. Quant. Chem., 9 (1975), 237-242.
[35]	R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a group of symplectomorphisms, Moscow Math. J., 6 (2006), 307–315, 407. doi: 10.17323/1609-4514-2006-6-2-307-315.
[36]	A. F. Izmaylov and I. Franco, Entanglement in the Born-Oppenheimer approximation, J. Chem. Theor. Comp., 13 (2016), 20-28.
[37]	T. W. B. Kibble, Geometrization of quantum mechanics, Comm. Math. Phys., 65 (1979), 189-201. doi: 10.1007/BF01225149.
[38]	A. A. Kirillov, Geometric quantization, Dynamical Systems Ⅳ, Encyclopaedia Math. Sci., Springer, 4 (2001), 139-176. doi: 10.1007/978-3-662-06791-8_2.
[39]	U. Klein, From Koopman-von Neumann theory to quantum theory, Quantum Stud. Math. Found., 5 (2018), 219-227. doi: 10.1007/s40509-017-0113-2.
[40]	Y. L. Klimontovich, On the method of "second quantization" in phase space, Sov. Phys. JTEP, 6 (1958), 753-760.
[41]	B. O. Koopman, Hamiltonian systems and transformation in Hilbert space, Proc. Nat. Acad. Sci., 17 (1931), 315. doi: 10.1073/pnas.17.5.315.
[42]	B. Kostant, Line bundles and the prequantized Schrödinger equation, Colloquium on Group Theoretical Methods in Physics, Centre de Physique Théorique, Marseille, (1972), Ⅳ.1–Ⅳ.22.
[43]	S. Lie, Begründung einer Invarianten-Theorie der Berührungs-Transformationen, Math. Ann., 8 (1875), 215–303, http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/lie_-_contact_transformations.pdf. doi: 10.1007/BF01443411.
[44]	R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138 (1986), 193-291. doi: 10.1016/0370-1573(86)90103-1.
[45]	E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1927), 322-326.
[46]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 2013.
[47]	J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D, 7 (1983), 305-323. doi: 10.1016/0167-2789(83)90134-3.
[48]	D. Marx and J. Hutter, Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods, Cambridge University Press, 2009.
[49]	R. Montgomery, Heisenberg and isoholonomic inequalities, Symplectic Geometry and Mathematical Physics, Progr. Math., Birkhäuser Boston, Boston, MA, 99 (1991), 303-325.
[50]	P. J. Morrison, Hamiltonian field description of two-dimensional vortex fluids and guiding center plasmas, Princeton University Plasma Physics Laboratory Report, PPPL-1788, (1981).
[51]	T. Ohsawa and C. Tronci, Geometry and dynamics of Gaussian wave packets and their Wigner transforms, J. Math. Phys., 58 (2017), 092105, 19 pp. doi: 10.1063/1.4995233.
[52]	W. Pauli, General Principles of Quantum Mechanics, Springer-Verlag, Berlin-New York, 1980.
[53]	I. Ramos-Prieto, A. R. Urzùa-Pineda, F. Soto-Eguibar and H. M. Moya-Cessa, KvN mechanics approach to the time-dependent frequency harmonic oscillator, Sci. Rep., 8 (2018), 8401.
[54]	A. Sawicki, A. Huckleberry and M. Kuś, Symplectic geometry of entanglement, Comm. Math. Phys., 305 (2011), 441-468. doi: 10.1007/s00220-011-1259-0.
[55]	A. Sawicki, M. Oszmaniec and M. Kuś, Convexity of momentum map, Morse index, and quantum entanglement, Rev. Math. Phys., 26 (2014), 14500044, 39 pp. doi: 10.1142/S0129055X14500044.
[56]	Y. M. Shirokov, Quantum and classical mechanics in the phase space representation, Sov. J. Part. Nucl., 10 (1979), 1-18.
[57]	E. C. G. Sudarshan, Interaction between classical and quantum systems and the measurement of quantum observables, Pramana-J. Phys., 6 (1976), 117.
[58]	A. Uhlmann, Parallel transport and "quantum holonomy" along density operators, Rep. Math. Phys., 24 (1986), 229-240. doi: 10.1016/0034-4877(86)90055-8.
[59]	L. Van Hove, On Certain Unitary Representations of an Infinite Group of Transformations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812838988.
[60]	J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955.
[61]	J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. Math., 33 (1932). 587–642. doi: 10.2307/1968537.
[62]	A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557. doi: 10.4310/jdg/1214437787.

show all references

References:

[1]	A. Abedi, N. T. Maitra and E. K. U. Gross, Exact factorization of the time-dependent electron-nuclear wave function, Phys. Rev. Lett., 105 (2010), 123002.
[2]	A. Abedi, N. T. Maitra and E. K. U. Gross, Correlated electron-nuclear dynamics: Exact factorization of the molecular wavefunction, The J. Chem. Phys., 137 (2012), 22A530.
[3]	R. Abraham, J. E. Marsden and T. M. Ratiu, Tensor Analysis and Applications, Second edition, Applied Mathematical Sciences, 75. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.
[4]	F. Agostini, A. Abedi, Y. Suzuki, S. K. Min, N. T. Maitra and E. K. U. Gross, The exact forces on classical nuclei in non-adiabatic charge transfer, J. Chem. Phys., 142 (2015), 084303.
[5]	J. Anandan, A geometric approach to quantum mechanics, Found. Phys., 21 (1991), 1265-1284. doi: 10.1007/BF00732829.
[6]	M. Baer, Beyond Born-Oppenheimer: Conical Intersections and Electronic Nonadiabatic Coupling Terms, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. doi: 10.1002/0471780081.
[7]	M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. A, 392 (1984), 45-57. doi: 10.1098/rspa.1984.0023.
[8]	D. Bondar, R. Cabrera, R. R. Lompay, M. Y. Ivanov and H. Rabitz, Operational dynamic modeling transcending quantum and classical mechanics, Phys. Rev. Lett., 109 (2012), 190403.
[9]	D. Bondar, F. Gay-Balmaz and C. Tronci, Koopman wavefunctions and classical-quantum correlation dynamics, Proc. R. Soc. A, 475 (2019), 20180879, 18 pp. doi: 10.1098/rspa.2018.0879.
[10]	E. Bonet Luz and C. Tronci, Geometry and symmetry of quantum and classical-quantum variational principles, J. Math. Phys., 56 (2015), 082104, 19 pp. doi: 10.1063/1.4929567.
[11]	E. Bonet Luz and C. Tronci, Hamiltonian approach to Ehrenfest expectation values and Gaussian quantum states, Proc. R. Soc. A, 472 (2016), 20150777, 15 pp. doi: 10.1098/rspa.2015.0777.
[12]	M. Born and R. Oppenheimer, Zur quantentheorie der molekeln, Ann. Physik, 389 (1927), 457-484.
[13]	D. C. Brody and L. P. Hughston, Geometric quantum mechanics, J Geom. Phys., 38 (2001), 19-53. doi: 10.1016/S0393-0440(00)00052-8.
[14]	J. F. Cariñena, A. Ibort, G. Marmo and G. Morandi, Geometry from Dynamics, Classical and Quantum, Springer, Dordrecht, 2015. doi: 10.1007/978-94-017-9220-2.
[15]	P. R. Chernoff and J. E. Marsden, Some remarks on Hamiltonian systems and quantum mechanics, Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, Univ. Western Ontario Ser. Philos. Sci., Reidel, Dordrecht, 6 (1977), 35-53.
[16]	A. Clebsch, Uber die Integration der hydrodynamischen Gleichungen, J. Reine Angew. Math., 56 (1859), 1-10. doi: 10.1515/crll.1859.56.1.
[17]	M. de Gosson, Quantum harmonic analysis of the density matrix, Quanta, 7 (2018), 74-110. doi: 10.12743/quanta.v7i1.74.
[18]	M. de Gosson, Symplectic Geometry and Quantum Mechanics, Operator Theory: Advances and Applications, 166. Advances in Partial Differential Equations (Basel). Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7575-2.
[19]	P. A. M. Dirac, On the analogy between classical and quantum mechanics, Rev. Mod. Phys., 17 (1945), 195-199. doi: 10.1103/RevModPhys.17.195.
[20]	P. A. M. Dirac, The Lagrangian in quantum mechanics, Feynman's Thesis: A New Approach to Quantum Theory, World Scientific, (2005), 111–119.
[21]	M. S. Foskett, D. D. Holm and C. Tronci, Geometry of nonadiabatic quantum hydrodynamics, Acta Appl. Math., 162 (2019), 63-103. doi: 10.1007/s10440-019-00257-1.
[22]	F. Gay-Balmaz and C. Tronci, Madelung transform and probability currents in hybrid classical-quantum dynamics, arXiv: 1907.06624.
[23]	F. Gay-Balmaz and C. Tronci, Vlasov moment flows and geodesics on the Jacobi group, J. Math. Phys., 53 (2012), 123502, 36 pp. doi: 10.1063/1.4763467.
[24]	F. Gay-Balmaz, C. Tronci and C. Vizman, Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups, J. Geom. Mech., 5 (2013), 39-84. doi: 10.3934/jgm.2013.5.39.
[25]	F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Global Anal. Geom., 41 (2012), 1-24. doi: 10.1007/s10455-011-9267-z.
[26]	J. W. Gray, Some global properties of contact structures, Ann. Math., 69 (1959), 421-450. doi: 10.2307/1970192.
[27]	V. Guillemin and S. Sternberg, The moment map and collective motion, Ann. Phys., 127 1980, 220–253. doi: 10.1016/0003-4916(80)90155-4.
[28]	C. Günther, Presymplectic manifolds and the quantization of relativistic particle systems, Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Math., Springer, Berlin, 836 (1980), 383-400.
[29]	B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267. Springer, New York, 2013. doi: 10.1007/978-1-4614-7116-5.
[30]	D. D. Holm and B. Kuperschmidt, Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity, Phys. D, 6 (1983), 347-363. doi: 10.1016/0167-2789(83)90017-9.
[31]	D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, The breadth of symplectic and Poisson Geometry, Progr. Math., Birkhäuser Boston, Boston, MA, 232 (2005), 203-235. doi: 10.1007/0-8176-4419-9_8.
[32]	D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, Oxford Texts in Applied and Engineering Mathematics, 12. Oxford University Press, Oxford, 2009.
[33]	D. D. Holm and C. Tronci, Geodesic Vlasov equations and their integrable moment closures, J. Geom. Mech., 1 (2009), 181-208. doi: 10.3934/jgm.2009.1.181.
[34]	G. Hunter, Conditional probability amplitudes in wave mechanics, Int. J. Quant. Chem., 9 (1975), 237-242.
[35]	R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a group of symplectomorphisms, Moscow Math. J., 6 (2006), 307–315, 407. doi: 10.17323/1609-4514-2006-6-2-307-315.
[36]	A. F. Izmaylov and I. Franco, Entanglement in the Born-Oppenheimer approximation, J. Chem. Theor. Comp., 13 (2016), 20-28.
[37]	T. W. B. Kibble, Geometrization of quantum mechanics, Comm. Math. Phys., 65 (1979), 189-201. doi: 10.1007/BF01225149.
[38]	A. A. Kirillov, Geometric quantization, Dynamical Systems Ⅳ, Encyclopaedia Math. Sci., Springer, 4 (2001), 139-176. doi: 10.1007/978-3-662-06791-8_2.
[39]	U. Klein, From Koopman-von Neumann theory to quantum theory, Quantum Stud. Math. Found., 5 (2018), 219-227. doi: 10.1007/s40509-017-0113-2.
[40]	Y. L. Klimontovich, On the method of "second quantization" in phase space, Sov. Phys. JTEP, 6 (1958), 753-760.
[41]	B. O. Koopman, Hamiltonian systems and transformation in Hilbert space, Proc. Nat. Acad. Sci., 17 (1931), 315. doi: 10.1073/pnas.17.5.315.
[42]	B. Kostant, Line bundles and the prequantized Schrödinger equation, Colloquium on Group Theoretical Methods in Physics, Centre de Physique Théorique, Marseille, (1972), Ⅳ.1–Ⅳ.22.
[43]	S. Lie, Begründung einer Invarianten-Theorie der Berührungs-Transformationen, Math. Ann., 8 (1875), 215–303, http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/lie_-_contact_transformations.pdf. doi: 10.1007/BF01443411.
[44]	R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138 (1986), 193-291. doi: 10.1016/0370-1573(86)90103-1.
[45]	E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1927), 322-326.
[46]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 2013.
[47]	J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D, 7 (1983), 305-323. doi: 10.1016/0167-2789(83)90134-3.
[48]	D. Marx and J. Hutter, Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods, Cambridge University Press, 2009.
[49]	R. Montgomery, Heisenberg and isoholonomic inequalities, Symplectic Geometry and Mathematical Physics, Progr. Math., Birkhäuser Boston, Boston, MA, 99 (1991), 303-325.
[50]	P. J. Morrison, Hamiltonian field description of two-dimensional vortex fluids and guiding center plasmas, Princeton University Plasma Physics Laboratory Report, PPPL-1788, (1981).
[51]	T. Ohsawa and C. Tronci, Geometry and dynamics of Gaussian wave packets and their Wigner transforms, J. Math. Phys., 58 (2017), 092105, 19 pp. doi: 10.1063/1.4995233.
[52]	W. Pauli, General Principles of Quantum Mechanics, Springer-Verlag, Berlin-New York, 1980.
[53]	I. Ramos-Prieto, A. R. Urzùa-Pineda, F. Soto-Eguibar and H. M. Moya-Cessa, KvN mechanics approach to the time-dependent frequency harmonic oscillator, Sci. Rep., 8 (2018), 8401.
[54]	A. Sawicki, A. Huckleberry and M. Kuś, Symplectic geometry of entanglement, Comm. Math. Phys., 305 (2011), 441-468. doi: 10.1007/s00220-011-1259-0.
[55]	A. Sawicki, M. Oszmaniec and M. Kuś, Convexity of momentum map, Morse index, and quantum entanglement, Rev. Math. Phys., 26 (2014), 14500044, 39 pp. doi: 10.1142/S0129055X14500044.
[56]	Y. M. Shirokov, Quantum and classical mechanics in the phase space representation, Sov. J. Part. Nucl., 10 (1979), 1-18.
[57]	E. C. G. Sudarshan, Interaction between classical and quantum systems and the measurement of quantum observables, Pramana-J. Phys., 6 (1976), 117.
[58]	A. Uhlmann, Parallel transport and "quantum holonomy" along density operators, Rep. Math. Phys., 24 (1986), 229-240. doi: 10.1016/0034-4877(86)90055-8.
[59]	L. Van Hove, On Certain Unitary Representations of an Infinite Group of Transformations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812838988.
[60]	J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955.
[61]	J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. Math., 33 (1932). 587–642. doi: 10.2307/1968537.
[62]	A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557. doi: 10.4310/jdg/1214437787.

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