An Optimal Transport View of Schrodinger%26apos;s Equation

作者：von Renesse Max K^*

来源：Canadian Mathematical Bulletin, 2012, 55(4): 858-869.

DOI：10.4153/CMB-2011-121-9

• Schrodinger equation
• optimal transport
• Newton%26apos
• s law
• symplectic submersion

We show that the Schrodinger equation is a lift of Newton%26apos;s third law of motion del(W)((mu) over dot)(mu) over dot = -del F-W(mu) on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential mu -%26gt; F(mu) is the sum of the total classical potential energy %26lt; V, mu %26gt; of the extended system and its Fisher information h(2)/8 integral vertical bar del ln mu vertical bar(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%26apos;s formal Riemannian calculus for optimal transportation of probability measures.

• 出版日期2012-12