June  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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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. Google Scholar

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

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show all references

References:
[1]

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

[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. Google Scholar

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

[4]

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

[5]

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

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[10]

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[11]

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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. Google Scholar

[13]

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

[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. Google Scholar

[15]

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[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. Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[20]

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[21]

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

[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. Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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. Google Scholar

[30]

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

[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. Google Scholar

[32]

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

[33]

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

[34]

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

[35]

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

[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. Google Scholar

[37]

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

[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. Google Scholar

[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. Google Scholar

[40]

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

[41]

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

[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. Google Scholar

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[48]

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

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[52]

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

[53]

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

[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. Google Scholar

[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. Google Scholar

[56]

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

[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. Google Scholar

[58]

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

[59]

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

[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. Google Scholar

[61]

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

[62]

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

[63]

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

[64]

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

[65]

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

[66]

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