2015, 7(2): 169-202. doi: 10.3934/jgm.2015.7.169

On the relation between geometrical quantum mechanics and information geometry

1. 

Instituto de Matemática, Universidade Federal da Bahia, Av. Adhemar de Barros, S/N, Ondina, 40170-110 Salvador, BA, Brazil

Received  April 2012 Revised  April 2014 Published  June 2015

Let $(M,g)$ be a compact, connected and oriented Riemannian manifold with volume form $d$ ${vol}_g$. We denote by $\mathcal{D}$ the space of smooth probability density functions on $M\,,$ i.e. $\mathcal{D}:= \{\rho\in C^{\infty}(M,\mathbb{R})| \rho>0\,\,$and$\,\,\int_{M}\rho\cdot $d${vol}_{g}=1\}\,.$ We regard $\mathcal{D}$ as an infinite dimensional manifold.
    In this paper, we consider the almost Hermitian structure on $T\mathcal{D}$ associated, via Dombrowski's construction, to the Wasserstein metric $g^{\mathcal{D}}$ and a natural connection $\nabla^{\mathcal{D}}$ on $\mathcal{D}$. Using geometric mechanical methods, we show that the corresponding fundamental $2$-form on $T\mathcal{D}$ leads to the Schrödinger equation for a quantum particle living in $M$. Geometrically, we exhibit a map which pulls back the Fubini-Study symplectic form to the $2$-form on $T\mathcal{D}$. The integrability of the almost complex structure on $T\mathcal{D}$ is also discussed.
    These results echo other papers of the author where it is stressed that the Fisher metric and exponential connection are related (via Dombrowski's construction) to Kähler geometry and the quantum formalism in finite dimension.
Citation: Mathieu Molitor. On the relation between geometrical quantum mechanics and information geometry. Journal of Geometric Mechanics, 2015, 7 (2) : 169-202. doi: 10.3934/jgm.2015.7.169
References:
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R. Abraham and J. E. Marsden, Foundations of Mechanics,, $2^{nd}$ edition, (1978).

[2]

S. T. Ali, J.-P. Antoine, J.-P. Gazeau and U. A. Mueller, Coherent states and their generalizations: A mathematical overview,, Rev. Math. Phys., 7 (1995), 1013. doi: 10.1142/S0129055X95000396.

[3]

S.-I. Amari and H. Nagaoka, Methods of Information Geometry,, American Mathematical Society, (2000).

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V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics,, Springer, (2013).

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A. Ashtekar and T. A. Schilling, Geometrical formulation of quantum mechanics,, in On Einstein's Path (ed. A. Harvey), (1999), 23.

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N. Ay, J. Jost, H. V. Lê and L. Schwachhöfer, Information geometry and sufficient statistics,, Probability Theory and Related Fields, (2014). doi: 10.1007/s00440-014-0574-8.

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P. Goyal, From information geometry to quantum theory,, New J. Phys., 12 (2010). doi: 10.1088/1367-2630/12/2/023012.

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A. Grinbaum, Reconstruction of quantum theory,, British Journal for the Philosophy of Science, 58 (2007), 387. doi: 10.1093/bjps/axm028.

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L. Hardy, Quantum theory from five reasonable axioms,, preprint, ().

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A. Heslot, Une caractérisation des espaces projectifs complexes,, C. R. Acad. Sci. Paris Sér. I Math., 298 (1984), 95.

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J. Jost, Riemannian Geometry and Geometric Analysis, $3^{nd}$ edition,, Springer-Verlag, (2002). doi: 10.1007/978-3-662-04672-2.

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[39]

B. Khesin, J. Lenells, G. Misiołek and S. C. Preston, Geometry of diffeomorphism groups, complete integrability and geometric statistics,, Geom. Funct. Anal., 23 (2013), 334. doi: 10.1007/s00039-013-0210-2.

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

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J. R. Klauder and B.-S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics,, World Scientific, (1985). doi: 10.1142/0096.

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S. Kochen and E. P. Specker, Logical structures arising in quantum theory,, in The Theory of Models (Proceeding of the 1963 International Symposium at Berkeley) (eds. W. Addison, (1963), 177.

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A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis,, American Mathematical Society, (1997). doi: 10.1090/surv/053.

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S. Lang, Introduction to Differentiable Manifolds, $2^{nd}$ edition,, Springer-Verlag, (2002).

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J. Lott, Some geometric calculations on Wasserstein space,, Comm. Math. Phys., 277 (2008), 423. doi: 10.1007/s00220-007-0367-3.

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E. Madelung, Eine anschauliche deutung der gleichung von Schrödinger,, Naturwissenschaften, 14 (1926), 1004.

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E. Madelung, Quantentheorie in hydrodynamischer form,, Zeitschrift für Physik, 40 (1927), 322. doi: 10.1007/BF01400372.

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G. Marmo, E. J. Saletan, A. Simoni and B. Vitale, Dynamical Systems: A Differential Geometric Approach to Symmetry and Reduction,, John Wiley & Sons Ltd., (1985).

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J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, $2^{nd}$ edition,, Springer-Verlag, (1999). doi: 10.1007/978-0-387-21792-5.

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L. Masanes and M. P. Müller, A derivation of quantum theory from physical requirements,, New Journal of Physics, 13 (2011). doi: 10.1088/1367-2630/13/6/063001.

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A. Messiah, Quantum Mechanics,, North-Holland, (1965).

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P. W. Michor, Topics in Differential Geometry,, American Mathematical Society, (2008). doi: 10.1090/gsm/093.

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K. Modin, Generalized Hunter-Saxton equations, optimal information transport, and factorization of diffeomorphisms,, The Journal of Geometric Analysis, 25 (2015), 1306. doi: 10.1007/s12220-014-9469-2.

[55]

M. Molitor, The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations,, Differential Geom. Appl., 28 (2010), 543. doi: 10.1016/j.difgeo.2010.04.005.

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M. Molitor, Information geometry and the hydrodynamical formulation of quantum mechanics,, preprint, ().

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M. Molitor, Remarks on the statistical origin of the geometrical formulation of quantum mechanics,, Int. J. Geom. Methods Mod. Phys., 9 (2012). doi: 10.1142/S0219887812200010.

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M. Molitor, Exponential families, Kähler geometry and quantum mechanics,, J. Geom. Phys., 70 (2013), 54. doi: 10.1016/j.geomphys.2013.03.015.

[59]

M. Molitor, Gaussian distributions, Jacobi group, and Siegel-Jacobi space,, Journal of Mathematical Physics, 55 (2014). doi: 10.1063/1.4903182.

[60]

G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle,, Phys. Rep., 188 (1990), 147. doi: 10.1016/0370-1573(90)90137-Q.

[61]

M. K. Murray and J. W. Rice, Differential Geometry and Statistics,, Chapman & Hall, (1993). doi: 10.1007/978-1-4899-3306-5.

[62]

J. v. Neumann, Mathematische Begründung der Quantenmechanik,, Gött. Nachr., 1927 (1927), 1.

[63]

J. v. Neumann, Thermodynamik quantenmechanischer Gesamtheiten,, Gött. Nachr., 1927 (1927), 273.

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J. v. Neumann, Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik,, Gött. Nachr., 1927 (1927), 245.

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J. v. Neumann, Mathematische Grundlagen der Quantenmechanik,, Springer, (1968).

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H. Omori, Infinite-dimensional Lie Groups,, American Mathematical Society, (1997).

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F. Otto, The geometry of dissipative evolution equations: The porous medium equation,, Comm. Partial Differential Equations, 26 (2001), 101. doi: 10.1081/PDE-100002243.

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show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics,, $2^{nd}$ edition, (1978).

[2]

S. T. Ali, J.-P. Antoine, J.-P. Gazeau and U. A. Mueller, Coherent states and their generalizations: A mathematical overview,, Rev. Math. Phys., 7 (1995), 1013. doi: 10.1142/S0129055X95000396.

[3]

S.-I. Amari and H. Nagaoka, Methods of Information Geometry,, American Mathematical Society, (2000).

[4]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics,, Springer, (2013).

[5]

A. Ashtekar and T. A. Schilling, Geometrical formulation of quantum mechanics,, in On Einstein's Path (ed. A. Harvey), (1999), 23.

[6]

N. Ay, J. Jost, H. V. Lê and L. Schwachhöfer, Information geometry and sufficient statistics,, Probability Theory and Related Fields, (2014). doi: 10.1007/s00440-014-0574-8.

[7]

D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" variables. I,, Phys. Rev., 85 (1952), 166. doi: 10.1103/PhysRev.85.166.

[8]

D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden'' variables. II,, Physical Rev., 85 (1952), 180. doi: 10.1103/PhysRev.85.180.

[9]

D. C. Brody and E.-M. Graefe, Coherent states and rational surfaces,, J. Phys. A, 43 (2010). doi: 10.1088/1751-8113/43/25/255205.

[10]

N. N. Chentsov, Statistical Decision Rules and Optimal Inference,, Translations of Mathematical Monographs, (1982).

[11]

P. R. Chernoff and J. E. Marsden, Properties of Infinite Dimensional Hamiltonian systems,, Lecture Notes in Mathematics, (1974).

[12]

G. Chiribella, G. M. D'Ariano and P. Perinotti, Informational derivation of quantum theory,, Phys. Rev. A, 84 (2011). doi: 10.1103/PhysRevA.84.012311.

[13]

R. Cirelli and P. Lanzavecchia, Hamiltonian vector fields in quantum mechanics,, Nuovo Cimento B (11), 79 (1984), 271. doi: 10.1007/BF02748976.

[14]

R. Cirelli, A. Manià and L. Pizzocchero, Quantum mechanics as an infinite-dimensional Hamiltonian system with uncertainty structure. I, II,, J. Math. Phys., 31 (1990), 2891. doi: 10.1063/1.528941.

[15]

R. Clifton, J. Bub and H. Halvorson, Characterizing quantum theory in terms of information-theoretic constraints,, Found. Phys., 33 (2003), 1561. doi: 10.1023/A:1026056716397.

[16]

M. Combescure and D. Robert, Coherent States and Applications in Mathematical Physics,, Theoretical and Mathematical Physics, (2012). doi: 10.1007/978-94-007-0196-0.

[17]

J. G. Cramer, An overview of the transactional interpretation of quantum mechanics,, Internat. J. Theoret. Phys., 27 (1988), 227. doi: 10.1007/BF00670751.

[18]

B. Dakić and Č. Brukner, Quantum theory and beyond: Is entanglement special?,, in Deep Beauty (ed. H. Halvorson), (2011), 365.

[19]

G. M. D'Ariano, Operational axioms for quantum mechanics,, in Foundations of Probability and Physics-4 (eds. G. Adenier, (2007), 79. doi: 10.1063/1.2713449.

[20]

J. Dieudonné, Treatise on Analysis. Vol. III,, Academic Press, (1972).

[21]

P. Dombrowski, On the geometry of the tangent bundle,, J. Reine Angew. Math., 210 (1962), 73.

[22]

D. Dürr, S. Goldstein and N. Zanghì, Bohmian mechanics as the foundation of quantum mechanics,, in Bohmian Mechanics and Quantum Theory: An Appraisal (eds. J. T. Cushing, (1996), 21. doi: 10.1007/978-94-015-8715-0_2.

[23]

C. A. Fuchs, Quantum mechanics as quantum information, mostly,, J. Modern Opt., 50 (2003), 987. doi: 10.1080/09500340308234548.

[24]

R. J. Glauber, Coherent and incoherent states of the radiation field,, Phys. Rev. (2), 131 (1963), 2766. doi: 10.1103/PhysRev.131.2766.

[25]

C. Godbillon, Géométrie Différentielle et Mécanique Analytique,, Hermann, (1969).

[26]

P. Goyal, Information-geometric reconstruction of quantum theory,, Phys. Rev. A (3), 78 (2008). doi: 10.1103/PhysRevA.78.052120.

[27]

P. Goyal, From information geometry to quantum theory,, New J. Phys., 12 (2010). doi: 10.1088/1367-2630/12/2/023012.

[28]

R. B. Griffiths and R. Omnes, Consistent histories and quantum measurements,, Physics Today, 52 (1999), 26.

[29]

A. Grinbaum, Elements of information-theoretic derivation of the formalism of quantum theory,, in Quantum Theory: Reconsideration of Foundations-2 (ed. A. Khrennikov), (2004), 205.

[30]

A. Grinbaum, Reconstruction of quantum theory,, British Journal for the Philosophy of Science, 58 (2007), 387. doi: 10.1093/bjps/axm028.

[31]

R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65. doi: 10.1090/S0273-0979-1982-15004-2.

[32]

L. Hardy, Quantum theory from five reasonable axioms,, preprint, ().

[33]

A. Heslot, Une caractérisation des espaces projectifs complexes,, C. R. Acad. Sci. Paris Sér. I Math., 298 (1984), 95.

[34]

A. Heslot, Quantum mechanics as a classical theory,, Phys. Rev. D (3), 31 (1985), 1341. doi: 10.1103/PhysRevD.31.1341.

[35]

D. Hilbert, J. von Neumann and L. Nordheim, Über die grundlagen der quantenmechanik,, Mathematische Annalen, 98 (1928), 1. doi: 10.1007/BF01451579.

[36]

P. Jordan, J. von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanical formalism,, Ann. of Math. (2), 35 (1934), 29. doi: 10.2307/1968117.

[37]

J. Jost, Riemannian Geometry and Geometric Analysis, $3^{nd}$ edition,, Springer-Verlag, (2002). doi: 10.1007/978-3-662-04672-2.

[38]

K.-K. Kan and J. J. Griffin, Single-particle Schrödinger fluid. I. Formulation,, Phys. Rev. C, 15 (1977), 1126. doi: 10.1103/PhysRevC.15.1126.

[39]

B. Khesin, J. Lenells, G. Misiołek and S. C. Preston, Geometry of diffeomorphism groups, complete integrability and geometric statistics,, Geom. Funct. Anal., 23 (2013), 334. doi: 10.1007/s00039-013-0210-2.

[40]

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

[41]

J. R. Klauder and B.-S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics,, World Scientific, (1985). doi: 10.1142/0096.

[42]

S. Kochen and E. P. Specker, Logical structures arising in quantum theory,, in The Theory of Models (Proceeding of the 1963 International Symposium at Berkeley) (eds. W. Addison, (1963), 177.

[43]

A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis,, American Mathematical Society, (1997). doi: 10.1090/surv/053.

[44]

S. Lang, Introduction to Differentiable Manifolds, $2^{nd}$ edition,, Springer-Verlag, (2002).

[45]

J. Lott, Some geometric calculations on Wasserstein space,, Comm. Math. Phys., 277 (2008), 423. doi: 10.1007/s00220-007-0367-3.

[46]

G. W. Mackey, Mathematical Foundations of Quantum Mechanics,, Dover, (2004).

[47]

E. Madelung, Eine anschauliche deutung der gleichung von Schrödinger,, Naturwissenschaften, 14 (1926), 1004.

[48]

E. Madelung, Quantentheorie in hydrodynamischer form,, Zeitschrift für Physik, 40 (1927), 322. doi: 10.1007/BF01400372.

[49]

G. Marmo, E. J. Saletan, A. Simoni and B. Vitale, Dynamical Systems: A Differential Geometric Approach to Symmetry and Reduction,, John Wiley & Sons Ltd., (1985).

[50]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, $2^{nd}$ edition,, Springer-Verlag, (1999). doi: 10.1007/978-0-387-21792-5.

[51]

L. Masanes and M. P. Müller, A derivation of quantum theory from physical requirements,, New Journal of Physics, 13 (2011). doi: 10.1088/1367-2630/13/6/063001.

[52]

A. Messiah, Quantum Mechanics,, North-Holland, (1965).

[53]

P. W. Michor, Topics in Differential Geometry,, American Mathematical Society, (2008). doi: 10.1090/gsm/093.

[54]

K. Modin, Generalized Hunter-Saxton equations, optimal information transport, and factorization of diffeomorphisms,, The Journal of Geometric Analysis, 25 (2015), 1306. doi: 10.1007/s12220-014-9469-2.

[55]

M. Molitor, The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations,, Differential Geom. Appl., 28 (2010), 543. doi: 10.1016/j.difgeo.2010.04.005.

[56]

M. Molitor, Information geometry and the hydrodynamical formulation of quantum mechanics,, preprint, ().

[57]

M. Molitor, Remarks on the statistical origin of the geometrical formulation of quantum mechanics,, Int. J. Geom. Methods Mod. Phys., 9 (2012). doi: 10.1142/S0219887812200010.

[58]

M. Molitor, Exponential families, Kähler geometry and quantum mechanics,, J. Geom. Phys., 70 (2013), 54. doi: 10.1016/j.geomphys.2013.03.015.

[59]

M. Molitor, Gaussian distributions, Jacobi group, and Siegel-Jacobi space,, Journal of Mathematical Physics, 55 (2014). doi: 10.1063/1.4903182.

[60]

G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle,, Phys. Rep., 188 (1990), 147. doi: 10.1016/0370-1573(90)90137-Q.

[61]

M. K. Murray and J. W. Rice, Differential Geometry and Statistics,, Chapman & Hall, (1993). doi: 10.1007/978-1-4899-3306-5.

[62]

J. v. Neumann, Mathematische Begründung der Quantenmechanik,, Gött. Nachr., 1927 (1927), 1.

[63]

J. v. Neumann, Thermodynamik quantenmechanischer Gesamtheiten,, Gött. Nachr., 1927 (1927), 273.

[64]

J. v. Neumann, Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik,, Gött. Nachr., 1927 (1927), 245.

[65]

J. v. Neumann, Mathematische Grundlagen der Quantenmechanik,, Springer, (1968).

[66]

H. Omori, Infinite-dimensional Lie Groups,, American Mathematical Society, (1997).

[67]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation,, Comm. Partial Differential Equations, 26 (2001), 101. doi: 10.1081/PDE-100002243.

[68]

A. Perelomov, Generalized Coherent States and Their Applications,, Springer-Verlag, (1986). doi: 10.1007/978-3-642-61629-7.

[69]

C. Piron, Axiomatique quantique,, Helv. Phys. Acta, 37 (1964), 439.

[70]

C. Rovelli, Relational quantum mechanics,, Internat. J. Theoret. Phys., 35 (1996), 1637. doi: 10.1007/BF02302261.

[71]

E. Schrödinger, Der stetige Übergang von der Mikro- zur Makromechanik,, Naturwissenschaften, 14 (1926), 664.

[72]

R. Schack, Quantum theory from four of Hardy's axioms,, Found. Phys., 33 (2003), 1461. doi: 10.1023/A:1026044329659.

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