An Optimal Transport View of Schrödinger's Equation

Renesse, Max-K. von

Canadian mathematical bulletin, Tome 55 (2012) no. 4, pp. 858-869

Résumé

We show that the Schrödinger equation is a lift of Newton's third law of motion $\nabla _{{\dot{\mu }}}^{\mathcal{W}}\,\dot{\mu }\,=\,-{{\nabla }^{\mathcal{W}}}\,F\left( \mu\right)$ on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential $\mu \,\to \,F\left( \mu\right)$ is the sum of the total classical potential energy $\left\langle V,\,\mu\right\rangle $ of the extended system and its Fisher information $\frac{{{\hbar }^{2}}}{8}\,{{\int{\left| \nabla \,\text{1n}\,\mu\right|}}^{2}}\,d\mu $ . The precise relation is established via a well-known (Madelung) transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures.

Détail

DOI : 10.4153/CMB-2011-121-9

Mots-clés : 81C25, 82C70, 37K05, Schrödinger equation, optimal transport, Newton's law, symplectic submersion

Bibliographie

[1] [1] Abraham, R. and Marsden, J. E., Foundations of Mechanics. Second edition. Benjamin/Cummings Publishing, Reading, MA, 1978. Google Scholar

[2] [2] Ambrosio, L. and Gangbo, W., Hamiltonian ODEs in the Wasserstein space of probability measures. Comm. Pure Appl. Math. 61(2008), no. 1, 18–53. Google Scholar | DOI

[3] [3] Ambrosio, L., Gigli, N., and Savaré, G., Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH ZÜrich. Birkhäuser Verlag, Basel, 2005. Google Scholar

[4] [4] Benamou, J.-D. and Brenier, Y., A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84(2000), no. 3, 375–393. Google Scholar | DOI

[5] [5] Bohm, D.. A suggested interpretation of the quantum theory in terms of “hidden” variables. I, II. Physical Rev. 85(1952), 166–193. Google Scholar | DOI

[6] [6] Cordero-Erausquin, D., McCann, R. J., and Schmuckenschläger, M., A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146(2001), no. 2, 219–257. Google Scholar | DOI

[7] [7] Gangbo, W., Nguyen, T., and Tudorascu, A., Hamilton-Jacobi equations in the Wasserstein space. Methods Appl. Anal. 15(2008), no. 2, 155–183. Google Scholar

[8] [8] Gianazza, U., Savare, G., and Toscani, G., The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation. Arch. Rat. Mech. Anal. To appear. Google Scholar | DOI

[9] [9] Hall, M. J. W. and Reginatto, M., Schrödinger equation from an exact uncertainty principle. J. Phys. A 35(2002), no. 14, 3289–3303. Google Scholar | DOI

[10] [10] Lafferty, J. D., The density manifold and configuration space quantization. Trans. Amer. Math. Soc. 305(1988), no. 2, 699–741. Google Scholar | DOI

[11] [11] Lott, J., Some geometric calculations on Wasserstein space. Comm. Math. Phys. 277(2008), no. 2, 423–437. Google Scholar | DOI

[12] [12] Madelung, E., Quantentheorie in hydrodynamischer Form. Z. Phys. 40(1926), 322–326. Google Scholar

[13] [13] McCann, R. J., Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(2001), no. 3, 589–608. Google Scholar | DOI

[14] [14] Nelson, E., Quantum fluctuations. Princeton Series in Physics. Princeton University Press, Princeton, NJ, 1985. Google Scholar

[15] [15] Otto, F., The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26(2001), no. 1-2, 101–174. Google Scholar | DOI

[16] [16] Otto, F. and Villani, C., Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2000), no. 2, 361–400. Google Scholar | DOI

[17] [17] Villani, C., Topics in optimal transportation. Graduate Studies in Mathematics 58. American Mathematical Society, Providence, RI, 2003. Google Scholar

[18] [18] Völlinger, S., Geometry of the Schrödinger equation and stochastic mass transportation. J. Math. Phys. 46(2005), no. 8, 082105. Google Scholar | DOI

[19] [19] von Renesse, M.-K. and Sturm, K.-T., Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl.Math. 58(2005), no. 7, 923–940. Google Scholar | DOI

[20] [20] von Renesse, M.-K. and Sturm, K.-T., Entropic measure and Wasserstein diffusion. Ann. Probab. 37(2009), no. 3, 1114–1191. Google Scholar | DOI