×

Momentum maps for mixed states in quantum and classical mechanics. (English) Zbl 1448.81070

Summary: 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 [J. von Neumann, Annals of Math. (2) 33, 587–642 (1932; JFM 58.1270.04)] and Koopman wavefunctions [B. Kostant, “Line bundles and the prequantized Schrödinger equation,” Colloquium on Group Theoretical Methods in Physics, Centre de Physique Théorique, Marseille, June 05 (1972) ] are shown to be special cases of this construction.

MSC:

81P16 Quantum state spaces, operational and probabilistic concepts
37K06 General theory of infinite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, conservation laws
53D20 Momentum maps; symplectic reduction
70H05 Hamilton’s equations
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
22E70 Applications of Lie groups to the sciences; explicit representations
81S10 Geometry and quantization, symplectic methods
53D50 Geometric quantization

Citations:

JFM 58.1270.04
PDFBibTeX XMLCite
Full Text: DOI arXiv

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. · Zbl 0875.58002
[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, 1265-1284 (1991) · 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. · Zbl 1113.81001
[7] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. A, 392, 45-57 (1984) · Zbl 1113.81306 · 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. · Zbl 1472.81012
[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. · Zbl 1331.81088
[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. · Zbl 1371.81034
[12] M. Born; R. Oppenheimer, Zur quantentheorie der molekeln, Ann. Physik, 389, 457-484 (1927) · JFM 53.0845.04
[13] D. C. Brody; L. P. Hughston, Geometric quantum mechanics, J Geom. Phys., 38, 19-53 (2001) · Zbl 1067.81081 · 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. · Zbl 1364.81001
[15] P. R. Chernoff; 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, 35-53 (1977) · Zbl 0336.70001
[16] A. Clebsch, Uber die Integration der hydrodynamischen Gleichungen, J. Reine Angew. Math., 56, 1-10 (1859) · ERAM 056.1468cj · doi:10.1515/crll.1859.56.1
[17] M. de Gosson, Quantum harmonic analysis of the density matrix, Quanta, 7, 74-110 (2018) · Zbl 1446.81006 · 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.
[19] P. A. M. Dirac, On the analogy between classical and quantum mechanics, Rev. Mod. Phys., 17, 195-199 (1945) · Zbl 0060.45102 · 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; C. Tronci, Geometry of nonadiabatic quantum hydrodynamics, Acta Appl. Math., 162, 63-103 (2019) · Zbl 1421.35274 · 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. · Zbl 1219.76005
[23] F. Gay-Balmaz and C. Tronci, Vlasov moment flows and geodesics on the Jacobi group, J. Math. Phys., 53 (2012), 123502, 36 pp. · Zbl 1287.37045
[24] F. Gay-Balmaz; C. Tronci; C. Vizman, Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups, J. Geom. Mech., 5, 39-84 (2013) · Zbl 1349.37077 · doi:10.3934/jgm.2013.5.39
[25] F. Gay-Balmaz; C. Vizman, Dual pairs in fluid dynamics, Ann. Global Anal. Geom., 41, 1-24 (2012) · Zbl 1234.53027 · doi:10.1007/s10455-011-9267-z
[26] J. W. Gray, Some global properties of contact structures, Ann. Math., 69, 421-450 (1959) · Zbl 0092.39301 · doi:10.2307/1970192
[27] V. Guillemin and S. Sternberg, The moment map and collective motion, Ann. Phys., 127 1980, 220-253. · Zbl 0453.58015
[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, 383-400 (1980) · Zbl 0451.58018
[29] B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267. Springer, New York, 2013. · Zbl 1273.81001
[30] D. D. Holm; B. Kuperschmidt, Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity, Phys. D, 6, 347-363 (1983) · Zbl 1194.76285 · doi:10.1016/0167-2789(83)90017-9
[31] D. D. Holm; 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, 203-235 (2005) · doi:10.1007/0-8176-4419-9_8
[32] D. D. Holm; C. Tronci, Geodesic Vlasov equations and their integrable moment closures, J. Geom. Mech., 1, 181-208 (2009) · Zbl 1190.82033 · doi:10.3934/jgm.2009.1.181
[33] G. Hunter, Conditional probability amplitudes in wave mechanics, Int. J. Quant. Chem., 9, 237-242 (1975)
[34] 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. · Zbl 1132.53043
[35] A. F. Izmaylov; I. Franco, Entanglement in the Born-Oppenheimer approximation, J. Chem. Theor. Comp., 13, 20-28 (2016)
[36] T. W. B. Kibble, Geometrization of quantum mechanics, Comm. Math. Phys., 65, 189-201 (1979) · Zbl 0412.58006 · doi:10.1007/BF01225149
[37] A. A. Kirillov, Geometric quantization, Dynamical Systems Ⅳ, Encyclopaedia Math. Sci., Springer, 4, 139-176 (2001) · doi:10.1007/978-3-662-06791-8_2
[38] U. Klein, From Koopman-von Neumann theory to quantum theory, Quantum Stud. Math. Found., 5, 219-227 (2018) · Zbl 1400.81095 · doi:10.1007/s40509-017-0113-2
[39] Y. L. Klimontovich, On the method of “second quantization” in phase space, Sov. Phys. JTEP, 6, 753-760 (1958) · Zbl 0083.45101
[40] B. O. Koopman, Hamiltonian systems and transformation in Hilbert space, Proc. Nat. Acad. Sci., 17 (1931), 315. · JFM 57.1010.02
[41] 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.
[42] 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. · JFM 06.0092.01
[43] R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138, 193-291 (1986) · doi:10.1016/0370-1573(86)90103-1
[44] E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40, 322-326 (1927) · JFM 52.0969.06
[45] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 2013. · Zbl 0933.70003
[46] J. E. Marsden; A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D, 7, 305-323 (1983) · Zbl 0576.58008 · doi:10.1016/0167-2789(83)90134-3
[47] R. Montgomery, Heisenberg and isoholonomic inequalities, Symplectic Geometry and Mathematical Physics, Progr. Math., Birkhäuser Boston, Boston, MA, 99, 303-325 (1991) · Zbl 0744.58028
[48] P. J. Morrison, Hamiltonian field description of two-dimensional vortex fluids and guiding center plasmas, Princeton University Plasma Physics Laboratory Report, PPPL-1788, (1981).
[49] T. Ohsawa and C. Tronci, Geometry and dynamics of Gaussian wave packets and their Wigner transforms, J. Math. Phys., 58 (2017), 092105, 19 pp. · Zbl 1372.81104
[50] W. Pauli, General Principles of Quantum Mechanics, Springer-Verlag, Berlin-New York, 1980.
[51] 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.
[52] A. Sawicki; A. Huckleberry; M. Kuś, Symplectic geometry of entanglement, Comm. Math. Phys., 305, 441-468 (2011) · Zbl 1223.81070 · doi:10.1007/s00220-011-1259-0
[53] A. Sawicki, M. Oszmaniec and M. Kuś, Convexity of momentum map, Morse index, and quantum entanglement, Rev. Math. Phys., 26 (2014), 14500044, 39 pp. · Zbl 1290.81011
[54] Y. M. Shirokov, Quantum and classical mechanics in the phase space representation, Sov. J. Part. Nucl., 10, 1-18 (1979)
[55] E. C. G. Sudarshan, Interaction between classical and quantum systems and the measurement of quantum observables, Pramana-J. Phys., 6 (1976), 117.
[56] A. Uhlmann, Parallel transport and “quantum holonomy” along density operators, Rep. Math. Phys., 24, 229-240 (1986) · Zbl 0644.46058 · doi:10.1016/0034-4877(86)90055-8
[57] L. Van Hove, On Certain Unitary Representations of an Infinite Group of Transformations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. · Zbl 0989.81051
[58] J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. Math., 33 (1932). 587-642. · Zbl 0005.12203
[59] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18, 523-557 (1983) · Zbl 0524.58011 · doi:10.4310/jdg/1214437787
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.