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Canadian Mathematical Bulletin

An Optimal Transport View of Schrödinger's Equation

Max von Renesse

Published Jul 9, 2012• 42 citations
Publisher

Abstract

Abstract We show that the Schrödinger equation is a lift of Newton's third law of motion on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potentialμ→F(μ) is the sum of the total classical potential energy (V, μ) of the extended system and its Fisher information . 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.

DOI
10.4153/cmb-2011-121-9

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Abstract We show that the Schrödinger equation is a lift of Newton's third law of motion on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potentialμ→F(μ) is the sum of the total classical potential energy (V, μ) of the extended system and its Fisher information . 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.
We show that the Schroedinger equation is a lift of Newton's law of motion on the space of probability measures, where derivatives are taken w.r.t. the Wasserstein Riemannian metric. Here the potential is the sum of the total classical potential energy of the extended system and its Fisher information. 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
A solution of the free Schrödinger equation is investigated by means of Optimal transport. The curve of probability measures μtμ_tμt​ this solution defines is shown to be an absolutely continuous curve in the Wasserstein space W2(R3)W_2(\mathbb{R}^3)W2​(R3). The optimal transport map from μtμ_tμt​ to μsμ_sμs​, the cost for this transport (i.e. the Wasserstein distance) and the value of the Fisher information along μtμ_tμt​ are being calculated. It is finally shown that this solution of the free Schrödinger equation can naturally be interpreted as a curve in so-called Shape space, which forgets any positioning in space but only describes properties of shapes. In Shape space, μtμ_tμt​ continues to be a shortest path geodesic.
In this work, we are in the position to view a measurement of a physical observable as an experiment in the sense of probability theory. To every physical observable, a sample space called the spectrum of the observable is therefore available. We have investigated the Wasserstein spaces over the spectrums of a quantum observables, i.e. the space of all probability measures defined on the suitable σ -algebra of subsets of the spectrum equipped with certain metric. It has been shown here the existence of one-to-one correspondence between the set of all quantum states and the set of all distribution functions defined on the spectrum of an observable. Thanks to the correspondence, every quantum state can be represented by a probability measure associated with the corresponding distribution function. It has been shown that the Wasserstein space over the spectrum of an observable is homeomorphic to the Wasserstein space over the spectrum of another observable. Such homeomorphism is referred to as generalized Fourier transform. By using the work of von Renesse, it has been shown that the Newtonian dynamics of the system can be formulated in the Wasserstein space over the spectrum of each observable which is equivalent to the Schroedinger dynamics, starting from the solution of Newton equation of motion in the Wasserstein space over the spectrum of a certain observable.
In this paper, we describe a possible generalization of the Wasserstein-2 metric, originally defined on the space of scalar probability densities, to the space of Hermitian matrices with trace one and to the space of matrix-valued probability densities. Our approach follows a control-theoretic optimization formulation of the Wasserstein-2 metric, having its roots in fluid dynamics, and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. It allows determining the gradient flow for the quantum entropy relative to this matricial Wasserstein metric.
This paper extends the causal reconstruction programme from static optimal states to finite-action dynamical histories. A reconstructive history is modelled as a curve of probability measures in the mass-scaled Wasserstein space (P(M), W2,m), and admissible histories are selected by a reconstructive action combining a Wasserstein kinetic term, a Fisher information term, and a causal constraint. The resulting Euler–Lagrange equations form a coupled system: a continuity equation for the evolving density and a generalized Hamilton–Jacobi equation for the velocity potential. The central result is the Fisher–Bohm identity. For any sufficiently regular positive density ρ, the functional derivative of Fisher information satisfies δI / δρ(x) = -4 Δ√ρ / √ρ, so that the Bohm–Madelung quantum potential is exactly Q(x) = (ℏ2 / 8m) · δI / δρ(x). Thus the quantum potential is the Fisher information gradient of the probability density, scaled by ℏ2/8m. When the Fisher coupling in the reconstructive action is set to λ = ℏ2/8m, the Euler–Lagrange equations coincide with the Madelung equations and are therefore equivalent to the Schrödinger equation, under the standard assumptions of positive density and irrotational flow. This establishes a structural correspondence between reconstructive dynamics and non-relativistic quantum mechanics. The result is not presented as a derivation of quantum mechanics: the coupling λ = ℏ2/8m is matched through the Madelung Hamiltonian rather than derived from the reconstruction hypotheses. The geometric origin of this coupling remains an open problem.
We develop an intrinsic geometric approach to the calculus of variations in the Wasserstein space. We show that the flows associated with the Schrödinger bridge with general prior, with optimal mass transport, and with the Madelung fluid can all be characterized as annihilating the first variation of a suitable action. We then discuss the implications of this unified framework for stochastic mechanics: It entails, in particular, a sort of fluid-dynamic reconciliation between Bohm’s and Nelson’s stochastic mechanics.