Zentralblatt MATH Database 1931 – 2013c© 2013 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag

Zbl 1256.81072

von Renesse, Max-K.An optimal transport view of Schrodinger’s equation. (English)Can. Math. Bull. 55, No. 4, 858-869 (2012). ISSN 0008-4395; ISSN 1496-4287/ehttp://dx.doi.org/10.4153/CMB-2011-121-9http://www.cms.math.ca/cmb/

Summary: We show that the Schrodinger equation is a lift of Newton’s third law ofmotion ∇Wµ µ = −∇WF (µ) on the space of probability measures, where derivatives aretaken with respect to the Wasserstein Riemannian metric. Here the potential µ → F (µ)is the sum of the total classical potential energy 〈V, µ〉 of the extended system and itsFisher information ~2

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∫|∇ lnµ|2dµ. The precise relation is established via a well-known

(Madelung) transform which is shown to be a symplectic submersion of the standardsymplectic structure of complex valued functions into the canonical symplectic spaceover the Wasserstein space. All computations are conducted in the framework of Otto’sformal Riemannian calculus for optimal transportation of probability measures.

Keywords : Schrodinger equation; optimal transport; Newton’s law; symplectic submer-sionClassification :

∗81S20 Stochastic quantization82C70 Transport processes37K05 Hamiltonian structures, etc.81Q70 Differential-geometric methods81Q05 Closed and approximate solutions to quantum-mechanical equations

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