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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show all references

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

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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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[8]

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[14]

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[16]

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[17]

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[18]

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[19]

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[20]

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[24]

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[25]

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[26]

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

[27]

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[28]

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[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]

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[33]

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[34]

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[35]

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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.  Google Scholar

[40]

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

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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.  Google Scholar

[42]

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[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

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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.  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]

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