AN OPTIMAL TRANSPORT VIEW ON SCHRÖDINGER S EQUATION

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1 AN OPTIAL TRANSPORT VIEW ON SCHRÖDINGER S EQUATION ax-k. von Renesse Abstract We show that the Schrödinger equation is a lift of Newton s third law of motion Ẇ µ µ = W F (µ) on the space of probability measures, where derivatives are taken w.r.t. 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 its Fisher information 2 8 ln µ 2 dµ. The precise relation is established via a well known ( adelung ) 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. 1. INTRODUCTION AND STATEENT OF RESULTS Recent applications of optimal transport theory have demonstrated that certain analytical and geometric problems on finite dimensional Riemannian manifolds (, g) or more general metric measure spaces (X, d, m) can nicely be treated in the corresponding ( Wasserstein ) space of probability measures P 2 (X) = {µ P(X) X d2 (x, o)µ(dx) < } equipped with the quadratic Wasserstein metric { } 1/2 d W (µ, ν) = inf d 2 (x, y)π(dx, dy) Π Γ(µ,ν) X 2 where Γ(µ, ν) = { Π P(X 2 ) Π(X A) = ν(a), Π(A X) = µ(a), A B(X) }. This metric corresponds to a relaxed version of onge s optimal transportation problem with cost function c(x, y) = d 2 (x, y) { } inf c(x, T y)µ(dx) T : X X, T µ = ν, X February , Technische Universität Berlin, mrenesse@math.tu-berlin.de, Keywords Schrödinger Equation, Optimal Transport, Newton s law, Symplectic Submersion. AS Subject Classification 81C25, 82C70, 37K05 1

2 with T µ denoting the image (push forward) measure of µ P(X) under the map T. The physical relevance of the Wasserstein distance was highlighted by the works of BENAOU-BRENIER [4] and ccann [13] who established in case X is Euclidean resp. smooth Riemannian and µ P 2 (X) is smooth that with d 2 W(µ, ν) = Φ(µ, ν) = inf (φ,µ) Φ(µ,ν) { 1 0 } φ t (x) 2 µ t (dx)dt, { φ C (]0, 1[ ), µ C([0, 1], P()) µ t = div( φ t µ t ), t ]0, 1[, µ 0 = µ, µ 1 = ν showing that d W is associated to a formal Riemannian structure on P() given by T µ P() = {ψ : R, ψ(x)dx = 0}, ψ 2 T = µp φ 2 dµ, for ψ = div(µ φ). In view of the continuity equation µ t = div( Φ t µ t ) for a smooth flow (t, x) Φ t (x) on, acting on measures µ through push forward µ t = (Φ t ) µ 0, this identifies the Riemannian energy of a curve t µ t P() with the minimal required kinetic energy E 0,t (µ) = t 0 t µ s 2 T µs P() ds = 0 Φ(x, s) 2 µ s (dx)ds. A major reason for the success of this framework is OTTO s interpretation [15] of evolution equations of type t u = div(u t F (u)), with F being the L 2 -Frechet derivative of some smooth functional F on L 2 (, dx), as d W -gradient ( steepest descent ) flow µ = W F (µ) for the measures µ(dx) = u(x)dx. Properties of the flow may thus be deduced from the geometry of the funtional F with respect to d W. An important case is the Boltzmann entropy F (u) = u ln u dx inducing the heat flow. 2 },

3 Here we propose an example of a second natural class of dynamical systems associated with the Riemannian metric on P() which can be written as Ẇ µ µ = W F (µ). (1) Equation (1) describes the Hamiltonian flow on T P() induced from the Lagrangian L F : T P() R; L F (ψ) = 1 2 ψ 2 T µp F (µ) for ψ T µp() with the functional F : P() R now playing the role of a potential field for the system. Apart from the closely related works [2, 7] it seems that a systematic approach to such Hamiltonian flows on P() is missing in the literature. The example we want to propose is obtained by choosing where F (µ) = I(µ) = V (x)µ(dx) + 2 I(µ), (2) 8 ln µ 2 dµ. We show that via an appropriate transform the flow (1) solves the Schrödinger equation i t Ψ = 2 /2 Ψ + Ψ V. (3) The functional I is known today as Fisher information. Physically I(µ) is the instantaneous kinetic energy required by the unperturbed heat flow at state µ. The prominent role of I for quantum behaviour was noticed long ago, e.g. in a classical paper by BOH [5], using the following well-known system of generalized Hamilton-Jacobi and transport equations t S S 2 + V + 2 ( ln µ µ µ) = 0 t µ + div(µ S) = 0. (4) This system was poposed by ADELUNG very early [12] as an equivalent description of the wave function Ψ = µe i S under the Schrödinger equation. In the sequel it will be referred to as adelung flow. Various attempts to derive it from first order principles can be found in the physics literature, e.g. most recently in [9]. In fact, using LOTT s recently proposed second order calculus on Wasserstein space, c.f. [11] we show that equations (4) and (1) are essentially the 3

4 same (theorem 2.1). A virtue of formula (1) is its very intuitive physical interpretation as Newton s law for the motion of an extended system with inertia (we have put mass density equal to one). Acceleration comes from a a gradient field of a potential F which is the total mechanical potential of the extended system plus its kinetic potential w.r.t. the heat flow. (Note that the case of a classical Hamiltonian particle moving in a potential field F is embedded naturally in (1) if one puts = 0 and µ = δ x.) Secondly we show that the two equations (1) and (3) are, modulo constant phase shifts, symplectically equivalent. ore precisely, we compute the canonical symplectic form on the tangent bundle T P() induced from the Levi-Civita connection of the Wasserstein metric on P() (proposition 3.2) and show that the map Ψ = Ψ e i h S div( Ψ 2 S), which we shall call adelung transform, is a symplectic submersion of the standard Hamiltonian structure of the Schrödinger equation on the space of complex valued functions into the Hamiltonian structure associated to (1) on the tangent bundle T P() (theorem 4.3). Except for its curiosity in Wasserstein geometry this result seems to support the point of view of some authors that the familiar complex valued form (3) of the Schrödinger equation is the consequence of a smart choice of coordinates in which the intuitive but unhandy dynamical system (1) resp. (4) can be solved very efficiently. uch of what is presented below is well known in the literature, in particular in NELSON s theory of stochastic mechanics [14] and its follow-ups, e.g. most notably by LAFFERTY [10]. However, the aim here is to demonstrate that the Riemannian formalism of optimal transport yields a simple and compelling picture of the Schrödinger equation as a lift of Hamiltonian mechanics from point to diffuse systems. 2. SCHRÖDINGER EQUATION AND NEWTON s THIRD LAW ON (P(), d W) The computations below are conducted on the formal Riemannian manifold of fully supported smooth probability measures equipped with the Wasserstein metric tensor, as initiated in [15, 16] and extended in [11], ignoring full mathematical generality or rigor. (The basic background material taken from [11, 15] can be found in the appendix.) In the sequel we shall often identify µ P () with its density µ = dµ/dx. 4

5 Theorem 2.1. For V C () let F : P () R defined as in (2). Then any smooth local solution t µ(t) P() of (1) yields a local solution (µ t, S t ) of the adelung flow (4), where S(x, t) = S(x, t) + t 0 L F (S σ, µ σ )dσ and S(x, t) is the velocity potential of the flow µ, i.e. satisfying Sdµ = 0 and µ t = div( S t µ). Conversely, let (µ t, S t ) be smooth a local solution of (4) then t µ t P() solves (1). Proof. Let µ solve (1) where W is the Wasserstein gradient and Ẇ µ µ is the covariant derivative associated to the Levi-Civita connection on T P(). Let (x, t) S(x, t) denote the velocity potential of µ (cf. section ), then according to [11, proposition 4.24] the left hand side of (1) is computed as div ( µ ( t S + 12 S 2 )), where the right hand side of (1) equals (cf. section ) ( div µ (V + 2 ( ln µ µ µ))). Since µ t is fully supported on this implies t S S 2 + V + 2 ( ln µ µ µ) = c(t) for some function c(t). To compute c(t) note that due to the normalization S t, µ t = 0 0 = t S t, µ t = c(t) 1 2 S 2, dµ F (µ) + S, µ = c(t) 1 2 S 2, dµ F (µ) + S 2, µ = c(t) + L F (S t, µ t ). Hence the pair t (S t, µ t ) with S(x, t) = S(x, t) + t 0 L F (S σ, µ σ )dσ solves (4). The converse statement is now also obvious. Corollary 2.2. For V C () let F : P () R defined as in (2). Then any smooth local solution t µ(t) P() of Ẇ µ µ = W F (µ), 5

6 yields a local solution of the Schrödinger equation (3) via Ψ(t, x) = µ(t, x)e i S(x,t) (5) where S(x, t) = S(x, t) + t 0 L F (S σ, µ σ )dσ and S(x, t) is the velocity potential of the flow µ, i.e. satisfying Sdµ = 0 and µ t = div( S t µ). Remark 2.3. The passage from S to S = S + const. does not bear any physical relevance, since two wave functions Ψ, Ψ with Ψ = e iκ Ψ for some κ R parameterize the same physical system. Remark 2.4. The d W -gradient flow on P() for F as in (2) corresponding to the overdamped limit of (1) gives a nonlinear 4th-order equation which is sometimes called the Derrida-Lebowitz-Speer-Spohn or quantum-driftdiffusion equation. A rigorous treatment of it can be found in [8]. The usual argument for the derivation of Euler-Lagrange equations yields the following statement. Corollary 2.5. For V C () let F : P () R defined as in (2). Then any smooth local Lagrangian flow [0, ɛ] t µ t T P () associated to L F yields a local solution of the Schrödinger equation i t Ψ = 2 /2 Ψ + Ψ V via Ψ(t, x) = µ(t, x)e i S(x,t) where S(x, t) = S(x, t) + t 0 L F (S σ, µ σ )dσ and S(x, t) is the velocity potential of the flow µ, i.e. satisfying Sdµ = 0 and µ t = div( S t µ). Remark 2.6. An equivalent version of theorem 2.1 puts Ψ = µ(x, t)e i S(x,t) where t ( div( S t µ t ), µ t ) is a Lagrangian flow for L F and S is chosen to satisfy for all t 0 S t, µ t S 0, µ 0 = t 0 L F ( µ s )ds. 6

7 3. HAILTONIAN STRUCTURE OF THE ADELUNG FLOW ON T P() In this section we show that the adelung flow (4) has a Hamiltonian structure w.r.t. the canonical symplectic form induced from the Wasserstein metric tensor on the tangent bundle T P(). To this aim we use the representation T P() = { div( fµ) f C (), µ P()}. Definition 3.1 (Standard Vector Fields on T P()). Each pair (ψ, φ) C () C () induces a vector field V φ,ψ on T P() via V ψ,φ ( div( fµ)) = γ where t γ ψ,φ (t) = γ(t) T P() is the curve satisfying γ(t) = div(µ(t) (f + tφ)) µ t = exp(t ψ) µ Recall that the standard symplectic form on the tangent bundle of a Riemannian manifold is given by ω = dθ, where the canonical 1-form Θ is defined as Θ(X) = ξ, π (X) Tπξ, X T ξ (T ), and where π denotes the projection map π : T. Proposition 3.2. Let ω W Λ 2 (T P()) be the standard symplectic form associated to the Wasserstein Riemannian structure on P(), then ω W (V ψ,φ, V ψ, φ)( div( fµ)) = ψ, φ µ ψ, φ µ (6) Proof. We use the formula ω W (V ψ,φ, V ψ, φ) = V ψ,φ Θ(V ψ, φ) V ψ, φθ(v ψ,φ ) Θ([V ψ,φ, V ψ, φ]), (7) where [V ψ,φ, V ψ, φ] denotes the Lie-bracket of the vector fields V ψ,φ and V ψ, φ. From the definition of Θ we obtain Hence Θ(V ψ, φ)( div( fµ)) = f, ψ µ. V ψ,φ (Θ(V ψ, φ)) = d dt t=0θ(v ψ, φ)(γ ψ,φ (t)) = d + tφ), dt t=0 (f ψ µ(t) = φ, ψ µ f ψ( div ψµ)dx = φ, ψ µ + ( f ψ) ψdµ (8) 7

8 Next, since Θ measures tangential variations only one gets that Θ([V ψ,φ, V ψ, φ])( div( fµ)) = f, [ ψ, ψ] µ. (9) Finally, it is easy to check that ( f ψ) ψdµ ( f ψ) ψdµ f, [ ψ, ψ] µ = 0, which together with (7), (8) and (9) establishes the claim. Remark 3.3. Proposition 3.2 shows that ω W is the lift of the standard symplectic form on T to T P(). This corresponds to the result in [11, section 6], which however is less explicit than formula (6). Using the the Riemannian inner product in each fiber of T P() the Hamiltonian associated with L F is H F : T P() R; H F ( div( fµ)) = 1 f 2 dµ + F (µ) (10) 2 Proposition 3.4. Let X F denote the Hamiltonian vector field X F induced on T P() from H F and ω W, then X F ( div( fµ)) = V f, ( 1 2 f 2 +V + h2 8 ( ln µ 2 2 µ µ ))( div( fµ)) Proof. Fix ψ, φ C () and let V ψ,φ (.) denote the corresponding standard vector field. Let t γ(t) = div(( f + tφ)µ t ), where µ t = exp(t ψ) µ, denote the corresponding curve on T P(), then V ψ,φ (H F )( div( fµ)) = t t=0 H F (γ(t)) (1 = t t=0 (f + tφ) 2 dµ t + V, µ t + h2 2 8 I(µ t) ) = I + II + III, where I = f φdµ = f, φ µ + ψ, ( 1 2 f 2 ) f 2 ( div( ψµ)) II = V ( div( ψµ)) = V, ψ µ 8

9 and III = 2 8 = ln µ t ( div( ψµ) )dµ + 2 µ 8 ( 2 µ ψ, ( µ ) ) µ + ψ, ln µ 2 µ Hence, collecting terms V ψ,φ (H F )( div( fµ)) = f, φ µ ln µ 2 ( div( ψµ)) ( ( 1 2 f 2 + V ( ln µ 2 2 µ µ ))), ψ µ. From this and formula (6) the claim follows. Corollary 3.5. The pair t (S t, µ t ) C () P() solves the adelung flow equation (4) if and only if t div( S t µ t ) T P() is an integral curve for X F. 4. THE ADELUNG TRANSFOR AS A SYPLECTIC SUBERSION In this section we prove that the two equations (1) and (3) are related via a symplectic submersion. Definition 4.1. A smooth map s : (, ω) (N, η) between two symplectic manifolds is called a symplectic submersion if its differential s : T T N is surjective and satisfies η(s X, s Y ) = ω(x, Y ) for all X, Y T. Note that this definition implies in particular that the map s itself is surjective. The following proposition is easily verified. Its meaning is that in order to solve a Hamiltonian system on N we may look for solutions for the lifted Hamiltonian g s on the larger state space and project them via s back again to N. Proposition 4.2. Let s : (, ω) (N, η) be a symplectic submersion and let f C () and g C (N) with g s = f, then s maps Hamiltonian flows associated to f on (, ω) to Hamiltonian flows associated to g on (N, η). Let now C() = C (; C) denote the linear space of smooth complex valued functions on. Identifying as usual the tangent space over an element Ψ C with C, T C is naturally equipped with the symplectic form ω C (F, G) = 2 Im(F G)(x)dx. 9

10 It is a well-known fact that the Schrödinger equation (3) is the Hamiltonian flow induced from the symplectic form ω C and the Hamiltonian function on C H S (Ψ) = 2 Ψ 2 dx + Ψ(x) 2 V (x)dx. 2 Let C () denote the subset of nowhere vanishing functions from C such that Ψ(x) 2 dx = 1 and note that C () is invariant under the Schrödinger flow. Assuming simple connectedness of implies (via a standard lifting theorem of algebraic topology) that each function Ψ C admits a decomposition Ψ = Ψ e i S, where the smooth field S : R is uniquely defined up to an additive constant 2πk, k N. Hence we may define a the adelung transform σ : C () T P (), σ(ψ) = div( Ψ 2 S). (11) For the next theorem recall that in our definition of T P() we assume that the supporting measures are smooth and strictly positive on. Theorem 4.3. Let be simply connected. Then the adelung transform σ : C () T P(), σ( Ψ e i S ) = div( Ψ 2 S) defines symplectic submersion from (C (), ω C ) to (T P(), ω W ) which preserves the Hamiltonian, i.e. H S = H F σ. Remark 4.4. Together with proposition 4.2 this result presents the Schrödinger equation (3) as a symplectic lifting of Newton s law on Wasserstein space (1) to the larger space C (), and which can be solved much easier because it is linear. Projecting the solution down to T P() via σ yields the desired solution to (1). Going in inverse direction from (1) to (3) requires a scalar correction term in the phase field, c.f. remark 2.3. Proof of theorem 4.3. Clearly, σ(c ()) = T P(). To see that σ : C () T P() is a submersion fix a reference point 0, then for each r [0, 2π [ the map τ = τ (r) τ : T P() C (), div( Sµ) µe i (S (S(0) r)), is a bijection from T P() to the subset {Ψ C, Ψ Ψ (0) = e i r } which satisfies σ τ = Id T P(). This proves that the differential s of s is surjective. 10

11 To prove that σ is symplectic let Ψ = µe i f C with f(0) = r [0, 2π [ and let η = div(µ f) = σ(ψ) T P(). Again due to the identity σ τ = Id T P() it suffices to prove that τ ω C = 1/ ω W on T η (T P()). Since the set {V ψ,φ ( div(µ f)) ψ, φ C ()} spans the full tangent space T η (T P()), it remains to verify ω C (τ V ψ,φ, τ V ψ, φ) = 1 ω W(V ψ,φ, V ψ,φ ) for all ψ, φ, ψ, φ C (). By definition of V ψ,φ and τ = τ (r) for µ t := exp(t ψ) (µ) and c(t) := f(0) + tφ(0) r ( τ V ψ,φ = t t=0 µt e i (f+tφ c(t)) = e i 1 f 2 µ ( div( ψµ)) + µ i ) (φ ċ) Hence ω C (τ V ψ,φ, τ V ψ, φ) = 2 ( 1 2 µ ( div( ψµ)) ( µ 1 ( φ + c)) + µ 1 (φ + ċ) 1 2 µ ( div( ψµ)) ) dx = 1 ( ψ, φ µ φ, ψ µ ) = 1 ω W(V ψ,φ, V ψ, φ) Finally, for Ψ = τ( ( div fµ)), Ψ = µe i f ( 1 ln µ + i f) such that 2 2 Ψ 2 = 1 f 2 dµ I(µ) and Ψ(x) 2 V (x)dx = V, µ which establishes the third claim H S = H F σ of the theorem. 5. APPENDIX: FORAL RIEANNIAN CALCULUS ON P() Let P 2 () denote the set of Borel probability measures µ on a smooth closed finite dimensional Riemannian manifold (, g) having finite second moment d2 (o, x)µ(dx) <. As argued in [11] the subsequent calculations make strict mathematical sense on the d W -dense subset of smooth fully supported probabilities P () P 2 () which shall often be identified with their corresponding density µ = dµ/dx. Vector Fields on P() and Velocity Potentials. A function φ Cc () induces a flow on P() via push forward t µ t = (Φ φ t ) µ 0, 11

12 where t Φ t is the local flow of difformorphisms on induced from the vector field φ Γ() starting from Φ 0 = Id. The continuity equation yields the infinitesimal variation of µ P() as µ = t t=0 µ t = div( φµ) T µ (P). Hence the function φ induces a vector field V φ Γ(P()) by V φ (µ) = div( φµ), acting on smooth functionals F : P() R via V φ (F )(µ) = ɛ ɛ=0 F (µ ɛ div( φµ)) = t t=0 F ((Φ φ t ) µ) with Riemannian norm V φ (µ) 2 T µp = φ 2 (x)µ(dx). Conversely, each smooth variation ψ T µ (P) can be identified with ψ = V φ (µ) with φ = G µ ψ, where G µ is the Green operator for µ : φ div(µ φ) on L 2 0(, dx) = L 2 0(, dx) { f, dx = 0}. Hence, for each ψ T µ P there exists a unique φ C L 2 (, dx) such that ψ = div(µ φ) and φ, µ = 0, which we call velocity potential for ψ T µ P(). Riemannian Gradient on P(). The Riemannian gradient of a smooth functional F : Dom(F ) P() R is computed to be W F µ = µ (DF µ ), where x DF µ (x) is the L 2 (, dx)-frechet-derivative of F in µ, which is defined through the relation ɛ ɛ=0 F (µ + ɛξ) = DF µ (x)ξ(x)dx, for all ξ chosen from a suitable dense set of test functions in L 2 (, dx). The following examples are easily obtained. F (µ) = φ(x)µ(dx), F (µ) = µ log µdx, W F (µ) = ln µ 2 dµ, W F µ = V φ (µ) = div( φµ) F µ = div(µ log µ) = µ W F µ = div(µ ( ln µ 2 2 µ)). µ 12

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