American Institute of Mathematical Sciences

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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

Cesare Tronci

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:
[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

J. Anandan, A geometric approach to quantum mechanics, Found. Phys., 21 (1991), 1265-1284.  doi: 10.1007/BF00732829.  Google Scholar

[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.  Google Scholar

[7]

M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. A, 392 (1984), 45-57.  doi: 10.1098/rspa.1984.0023.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Born and R. Oppenheimer, Zur quantentheorie der molekeln, Ann. Physik, 389 (1927), 457-484.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

A. Clebsch, Uber die Integration der hydrodynamischen Gleichungen, J. Reine Angew. Math., 56 (1859), 1-10.  doi: 10.1515/crll.1859.56.1.  Google Scholar

[17]

M. de Gosson, Quantum harmonic analysis of the density matrix, Quanta, 7 (2018), 74-110.  doi: 10.12743/quanta.v7i1.74.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

P. A. M. Dirac, The Lagrangian in quantum mechanics, Feynman's Thesis: A New Approach to Quantum Theory, World Scientific, (2005), 111–119. Google Scholar

[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.  Google Scholar

[22]

F. Gay-Balmaz and C. Tronci, Madelung transform and probability currents in hybrid classical-quantum dynamics, arXiv: 1907.06624. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. W. Gray, Some global properties of contact structures, Ann. Math., 69 (1959), 421-450.  doi: 10.2307/1970192.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[34]

G. Hunter, Conditional probability amplitudes in wave mechanics, Int. J. Quant. Chem., 9 (1975), 237-242.   Google Scholar

[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.  Google Scholar

[36]

A. F. Izmaylov and I. Franco, Entanglement in the Born-Oppenheimer approximation, J. Chem. Theor. Comp., 13 (2016), 20-28.   Google Scholar

[37]

T. W. B. Kibble, Geometrization of quantum mechanics, Comm. Math. Phys., 65 (1979), 189-201.  doi: 10.1007/BF01225149.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

Y. L. Klimontovich, On the method of "second quantization" in phase space, Sov. Phys. JTEP, 6 (1958), 753-760.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[44]

R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138 (1986), 193-291.  doi: 10.1016/0370-1573(86)90103-1.  Google Scholar

[45]

E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1927), 322-326.   Google Scholar

[46]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 2013. Google Scholar

[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.  Google Scholar

[48] D. Marx and J. Hutter, Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods, Cambridge University Press, 2009.   Google Scholar
[49]

R. Montgomery, Heisenberg and isoholonomic inequalities, Symplectic Geometry and Mathematical Physics, Progr. Math., Birkhäuser Boston, Boston, MA, 99 (1991), 303-325.   Google Scholar

[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). Google Scholar

[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.  Google Scholar

[52]

W. Pauli, General Principles of Quantum Mechanics, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[56]

Y. M. Shirokov, Quantum and classical mechanics in the phase space representation, Sov. J. Part. Nucl., 10 (1979), 1-18.   Google Scholar

[57]

E. C. G. Sudarshan, Interaction between classical and quantum systems and the measurement of quantum observables, Pramana-J. Phys., 6 (1976), 117. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[60] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955.   Google Scholar
[61]

J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. Math., 33 (1932). 587–642. doi: 10.2307/1968537.  Google Scholar

[62]

A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557.  doi: 10.4310/jdg/1214437787.  Google Scholar

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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

J. Anandan, A geometric approach to quantum mechanics, Found. Phys., 21 (1991), 1265-1284.  doi: 10.1007/BF00732829.  Google Scholar

[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.  Google Scholar

[7]

M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. A, 392 (1984), 45-57.  doi: 10.1098/rspa.1984.0023.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Born and R. Oppenheimer, Zur quantentheorie der molekeln, Ann. Physik, 389 (1927), 457-484.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

A. Clebsch, Uber die Integration der hydrodynamischen Gleichungen, J. Reine Angew. Math., 56 (1859), 1-10.  doi: 10.1515/crll.1859.56.1.  Google Scholar

[17]

M. de Gosson, Quantum harmonic analysis of the density matrix, Quanta, 7 (2018), 74-110.  doi: 10.12743/quanta.v7i1.74.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

P. A. M. Dirac, The Lagrangian in quantum mechanics, Feynman's Thesis: A New Approach to Quantum Theory, World Scientific, (2005), 111–119. Google Scholar

[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.  Google Scholar

[22]

F. Gay-Balmaz and C. Tronci, Madelung transform and probability currents in hybrid classical-quantum dynamics, arXiv: 1907.06624. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. W. Gray, Some global properties of contact structures, Ann. Math., 69 (1959), 421-450.  doi: 10.2307/1970192.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[34]

G. Hunter, Conditional probability amplitudes in wave mechanics, Int. J. Quant. Chem., 9 (1975), 237-242.   Google Scholar

[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.  Google Scholar

[36]

A. F. Izmaylov and I. Franco, Entanglement in the Born-Oppenheimer approximation, J. Chem. Theor. Comp., 13 (2016), 20-28.   Google Scholar

[37]

T. W. B. Kibble, Geometrization of quantum mechanics, Comm. Math. Phys., 65 (1979), 189-201.  doi: 10.1007/BF01225149.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

Y. L. Klimontovich, On the method of "second quantization" in phase space, Sov. Phys. JTEP, 6 (1958), 753-760.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[44]

R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138 (1986), 193-291.  doi: 10.1016/0370-1573(86)90103-1.  Google Scholar

[45]

E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1927), 322-326.   Google Scholar

[46]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 2013. Google Scholar

[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.  Google Scholar

[48] D. Marx and J. Hutter, Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods, Cambridge University Press, 2009.   Google Scholar
[49]

R. Montgomery, Heisenberg and isoholonomic inequalities, Symplectic Geometry and Mathematical Physics, Progr. Math., Birkhäuser Boston, Boston, MA, 99 (1991), 303-325.   Google Scholar

[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). Google Scholar

[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.  Google Scholar

[52]

W. Pauli, General Principles of Quantum Mechanics, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[56]

Y. M. Shirokov, Quantum and classical mechanics in the phase space representation, Sov. J. Part. Nucl., 10 (1979), 1-18.   Google Scholar

[57]

E. C. G. Sudarshan, Interaction between classical and quantum systems and the measurement of quantum observables, Pramana-J. Phys., 6 (1976), 117. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[60] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955.   Google Scholar
[61]

J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. Math., 33 (1932). 587–642. doi: 10.2307/1968537.  Google Scholar

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