2 Jun 2020 Madelung Transform and Probability Densities in Hybrid

Home , Madelung equations

arXiv:1907.06624v2 [math-ph] 2 Jun 2020 Conclusions 6 currents and densities quantum–classical Hybrid 5 equations Madelung Hybrid 4 dynamics quantum–classical Hybrid 3 mechanics classical Hove Koopman-van 2 Introduction 1 . uln n eut ...... results . . and . equations . Outline Madelung . and . maps mechanics . 1.4 Momentum quantum . in transform . 1.3 Madelung wavefunctions Hove 1.2 Koopman-van 1.1 . aitna tutr ...... positivity . . preserving . . . . . Hamiltonians . . . . . of . . class . . . . . A ...... structure . . 5.5 Hamiltonian . . equation . map continuity momentum . . . . quantum–classical a 5.4 . . The . . as . function . . . . . density . . . 5.3 . Hybrid ...... comments . . . 5.2 General ...... 5.1 ...... motion . . . . of . . . equations . . . and . . . structure form . . . . . Hamiltonian . symplectic . . . . . and . dynamics . . . . . 4.4 Hybrid ...... trajectories . . . . Bohmian . . 4.3 . Hybrid . . . . . transform . . . . Madelung . 4.2 . . Quantum–classical ...... 4.1 ...... operator . . density . . hybrid . . . . . the . of . . . . Equivariance . operator . . . density . . hybrid . . operators 3.4 The . Liouvillian . . hybrid . . . of 3.3 Algebra . . . . . equation . wave 3.2 . . Quantum–classical ...... 3.1 ...... density . . . Liouville transform the . Madelung . . and The . . representation Hove diffeomorphisms . . van symplectic 2.5 The . . of extension . diffeomorphisms central contact 2.4 . A strict . of group . 2.3 The equation Hove Koopman-van 2.2 The 2.1 vlto rsre h ino h rbblt est o h class the for density probability Hamilton hybrid the of of family sign infinite the an preserves presenting evolution allo by generally concludes is paper density the hybrid the of the operator construction, jo density assoc by the Neumann positive-definite is of von the equation flow While continuity this explicitly. the presented and addition, traj In coordinate Bohmian quantum subsystem. hybrid classical the the by of pr parameterized components is flow Poincaré integral phase-space the classical of the clas evolution the nontrivial As the theory. preserved, quantum The standard from L trajectories quantum–classical systems. hybrid Bohmian quantum identifying to and leads transform classical Madelung of interaction time-reversible 1 3 NSand CNRS ueia ehd iiin a lnkIsiuefrPlasm for Institute Planck Max Division, Methods Numerical hsppretnsteMdln-omfruaino unu mec quantum of formulation Madelung-Bohm the extends paper This aeugtasomadpoaiiydensities probability and transform Madelung 2 eateto ahmtc,Uiest fSre,Guildfor Surrey, of University Mathematics, of Department nhbi unu–lsia dynamics quantum–classical hybrid in cl oml u´rer,Lbrtied Météorologie Supérieure, de Laboratoire Normale Ecole ´ Fran¸cois Gay-Balmaz Abstract 1 1 ...... eaeTronci Cesare , ...... hsc,Grhn,Germany Garching, Physics, a ia ypetcfr sn longer no is form symplectic sical ...... sne xlcty Nevertheless, explicitly. esented n unu–lsia est is density quantum–classical int ...... clsubsystem. ical coyietf Hamiltonian a identify ectory gaga ah xedn the extending paths agrangian e ob nind However, unsigned. be to wed ...... unu usse salways is subsystem quantum ,Uie Kingdom United d, ypetcgoer fthe of geometry symplectic yaiu,Prs France Paris, Dynamique, ae otemto fthe of motion the to iated aswoecorresponding whose ians 2 , 3 aist eciethe describe to hanics 36 28 20 14 34 32 31 29 28 25 23 22 21 19 18 16 14 12 10 6 2 9 8 7 5 4 4 3 1 Introduction

This paper deals with the dynamics of coupled classical and quantum degrees of freedom. This topic has been attracting much attention since the early speculations on the role of the classical apparatus in the theory of quantum measurement [61, 65]. In the usual approach, one starts with a full quantum treatment for all degrees of freedom and then takes the semiclassical limit on some of them. Over the decades, this approach has led to several models differing in the way the semiclassical limit is performed. On the other hand, the alternative approach followed in the present work seeks a mathematically consistent description of hybrid quantum–classical systems that are not necessarily the limit of a fully quantum theory. In other words, classical motion is not regarded in this framework as an approximation of quantum mechanics. While this construction has led to the celebrated quantum–classical Liouville equation [1, 9, 23] in chemical physics [36], this equation suffers from the essential drawback of not preserving the quantum uncertainty relations. Indeed, the quantum–classical Liouville equation generally allows the quantum density matrix to become unsigned [17]. Other hybrid theories [52] also suffer from similar issues. Within the context of Hamiltonian dynamics, alternative theories also exist. In some cases [28] they retain quantum–classical correlations even in the absence of coupling. In some other cases [55], the emergence of further interpretative issues [2, 51] led some to exclude the possibility of a mathematically and physically consistent theory of quantum–classical coupling [53, 54]. Nevertheless, the search for a model of quantum–classical correlation dynamics is currently still open. Recently, a hybrid quantum–classical wave equation was formulated in [6] by using momentum map methods in symplectic geometry [26, 45] so that the system naturally inherits a standard Hamiltonian structure. General symplectic methods have been continuously success- ful in quantum theory [15], while momentum maps in geometric mechanics have been shown to be particularly advantageous for Gaussian quantum states [7, 8, 50, 49], quantum hydrodynamics [20], and mixed state dynamics [48, 58]. The Hamiltonian quantum–classical theory in [6] was largely inspired by the crucial contribution by George Sudarshan [13, 55, 56], who in 1976 proposed to describe hybrid quantum–classical systems by exploiting the Koopman- von Neumann (KvN) formulation of classical mechanics [40, 62]. The KvN theory proposes to describe classical mechanics in terms of wavefunctions, thereby allowing for a common Hilbert- space framework which is then shared by both classical and quantum mechanics. In the KvN construction, the classical probability density ρ(q,p) is represented in terms of a wavefunction Ψ(q,p) by setting ρ = Ψ 2. A direct verification shows that if Ψ satisfies the KvN equation | | i~∂ Ψ= L Ψ , with L := i~ H, , (1.1) t H H { } 2 then ρ = Ψ satisfies the Liouvilleb equation ∂tρ = H,b ρ from classical mechanics. Here, H is the Hamiltonian| | function, , denotes the canonical{ Poisson} bracket, while the Hermitian { } operator LH is often called the Liouvillian. The KvN equation has been rediscovered in several instances [16, 57] and it has been attracting some attention in recent years [5, 24, 39, 47, 60, 66]. For a broadb review of general applications of Koopman operators, see also [10]. Based on the KvN construction, Sudarshan’s theory invoked special superselection rules for physical consistency purposes. In turn, these superselection rules lead to interpretative problems which resulted in Sudarshan’s work being overly criticized [2, 51, 53, 56].

2 1.1 Koopman-van Hove wavefunctions As shown in recent work [6], the standard KvN theory fails to comprise the dynamics of classical phases and therefore it is somewhat incomplete. Indeed, it is evident that the KvN wavefunction in (1.1) is only deﬁned up to phase functions so that Ψ can be simply chosen to be real-valued. Over the years, phase factors have been suitably added to the standard KvN equation [5, 39], which was also related to van Hove’s work in prequantization [34, 9]. More particularly, equivalent variants of the equation

i~∂ Ψ= L Ψ (p∂ H H)Ψ (1.2) t H − p − made their first appearance in Kostant’sb work [42] from 1972 (see also [27]) under the name of “prequantized Schrödinger equation”. One recognizes that the phase term appearing in parenthesis identifies the phase-space expression of the Lagrangian. Only very recently it was shown [6] that this phase factor determines nontrivial contributions to the definition of the Liouville probability density, whose expression reads as follows:

ρ = Ψ 2 + ∂ (p Ψ 2) + i~ Ψ, Ψ¯ , (1.3) | | p | | { } where the bar symbol is used to denote complex conjugation. This result was found by applying standard momentum map methods to van Hove’s prequantization theory. Inspired by Kirillov [38], in [6] the resulting construction was referred to as the Koopman-van Hove (KvH) formulation of classical mechanics. Entirely based on prequantization, this theory allows the systematic application of geometric quantization [43]. Then, a hybrid quantum–classical theory was found in [6] by starting with the KvH equation for a two-particle wavefunction Ψ(q,p,x,s) and then quantizing one of the two particles by standard methods. This process yields an equation for a hybrid quantum–classical wavefunction Υ(q,p,x), where x denotes the quantum coordinate. As pointed out in [6], this partial quantization procedure leads to a Hamiltonian hybrid theory in which both quantum and classical pure states are lost. We recall that classical pure states are defined as extreme points of the convex set of classical probability densities [12] and these are given by delta-like Klimontovich distributions. Unlike quantum pure states, which may be entangled and not factorizable, classical pure states are completely factorizable. The absence of classical pure states in the general case of hybrid dynamics raises questions about the nature of Hamiltonian trajectories in quantum–classical coupling. Indeed, classical motion is given by a Hamiltonian flow producing characteristic curves representing particle trajectories and thus one is led to ask whether a Hamiltonian flow can still be identified in hybrid dynamics. In this paper, we address this question by extending the Lagrangian (or Bohmian) trajectories from quantum hydrodynamics to hybrid quantum–classical systems. To this purpose, we shall exploit the geometric structure of the Madelung transform. Another question emerging in the context of hybrid quantum–classical dynamics concerns the existence of a continuity equation for the hybrid density, which could then be used to define a hybrid current extending the probability current from standard quantum theory. This is the second question addressed in this paper, which exploits methods from Geometric Mechanics [26, 30, 45] to present the explicit hybrid continuity equation in terms of its underlying Hamiltonian structure. In turn, the existence of a continuity equation leads to the question whether the sign of the hybrid density is preserved in time. Here, we shall present an infinite family of hybrid systems for which this is indeed the case.

3 1.2 Madelung transform in quantum mechanics This paper, uses the polar form of the wavefunction in order to characterize the Madelung formulation of hybrid quantum–classical dynamics. This work is inspired by the Madelung- Bohm hydrodynamic formulation of quantum mechanics [44, 4], whose geometric features were recently revived in [37]. In order to obtain his equations of quantum hydrodynamics, Madelung replaced the polar form ψ(x, t)= R(x, t)e−iS(x,t)/~ of the wavefunction into Schrödinger’s equation i~∂ ψ = m−1~2∆ψ/2+ V ψ. This operation yields the following PDE system t − ∂S S 2 ~2 ∆R + |∇ | + V =0 , (1.4) ∂t 2m − 2m R ∂R 1 + div(R2 S)=0 . (1.5) ∂t 2mR ∇ The second equation yields the well-known continuity equation for the probability density D = R2. Madelung realized that defining the velocity vector field S v = ∇ m casts the above system into a set of hydrodynamic equations as follows:

∂v 1 ~2 ∆√D ∂D + v v = V + , + div(Dv)=0 . (1.6) ∂t ·∇ −m∇ 2m √ ∂t D Madelung’s equations were the point of departure for Bohm’s interpretation of quantum dynamics [4]. Following previous ideas by de Broglie [14], Bohm interpreted the integral curves of the velocity vector field v(t, x) as the genuine trajectories in space of the physical quantum particle. In this picture, particles are carried by a pilot wave transporting probability with a velocity v which itself changes in time according to the first equation in (1.6). Bohmian trajectories, however, are not exactly point particle trajectories: rather, they are trajectories in a fluid Lagrangian sense. More specifically, if the fluid label x0 is mapped to its current position x in terms of a smooth Lagrangian path χ, one writes x = χ(t, x ) and χ(t, ) is identified with t t 0 · a time-dependent diffeomorphism of the physical space M, that is χ(t, ) Diff(M). Then, Bohmian trajectories are fluid paths satisfying the reconstruction relation· ∈

∂tχ(t, x)= v(t, χ(t, x)) . (1.7)

While Bohmian mechanics and pilot wave theory have raised several fundamental interpretative questions, in this paper we shall not dwell upon these issues. The scope of this work is instead to extend the concept of Bohmian trajectories to hybrid quantum–classical systems and exploit the Madelung transform to draw conclusions about the dynamics of the joint quantum–classical density.

1.3 Momentum maps and Madelung equations In this work, we shall follow a geometric approach combining the geometric setting of the quantum Madelung transform with the prequantization theory of van Hove [59] and Kostant [43]. Indeed, both these constructions share momentum map structures which will serve as a unifying framework to describe quantum–classical coupling. The momentum maps appearing

4 in prequantization will be discussed in the next section, while those emerging in quantum hydrodynamics have recently been exploited in the context of quantum chemistry [20]. The Madelung momentum map takes the quantum Hilbert space L2(M) into the dual of the semidirect-product Lie algebra X(M) s (M), where X(M) denotes the Lie algebra of vector ﬁelds on M and (M) the space of real valuedF functions on M. More explicitly, the Madelung F momentum map J : L2(M) X∗(M) Den(M) is given by → × J(ψ)= ~ Im(ψ∗ ψ), ψ 2 =(mDv, D) . (1.8) ∇ | | Here, X∗(M) denotes the dual space of X(M), while Den(M) denotes the space of densities on ~ ¯ M. Upon considering the standard symplectic form Ω(ψ1, ψ2)=2 Im M ψ1ψ2 µ (here, µ is the volume form on physical space M), the above momentum map is generated´ by a unitary representation of the semidirect-product group Diﬀ(M) s (M,S1) which reads as follows: F

1 −iϕ/~ ψ χ∗(e ψ) , (1.9) 7→ Jac(χ)

1 p with (χ, ϕ) Diff(M) s (M,S ). Here, (N1, N2) generally denotes the space of mappings ∈ F F 1 1 from the manifold N1 to the manifold N2, so that (M,S ) denotes the space of S -valued functions on M. Moreover, Diff(M) denotes the groupF of diffeomorphisms of M while Jac(χ) denotes the Jacobian determinant of χ, and χ∗ denotes the push-forward. Upon denoting −iϕ/~ −iϕ/~ −1 composition by , one writes χ∗(e ψ)=(e ψ) χ . The representation (1.9) is typically constructed by◦ identifying the Hilbert space L2(M)◦ with the space of half-densities [3, 20], although here we shall not discuss this particular aspect. More importantly, throughout this paper we shall assume that the elements in Diff(M) s (M,S1) have sufficient regularity to ensure that this group is an infinite-dimensional manifoldF and a topological group with smooth right translation. Also we assume appropriate restrictions of the domain of the action (1.9) and the momentum map (1.8), so that all the operations are well-defined. The fact that (1.8) identifies a momentum map for the unitary representation (1.9) is a direct verification that makes use of the infinitesimal generator corresponding to (1.9), that is 1 ψ i~−1αψ u ψ (div u)ψ , (1.10) 7→ − − ·∇ − 2 with (u,α) X(M) s (M). In this paper, we shall exploit the Madelung momentum map (1.8) to present∈ the geometricF structure of the Madelung equations for hybrid quantum–classical systems. As mentioned previously, these will be described in terms of a hybrid wavefunction Υ(q,p,x), whose polar form will be used to define Bohmian trajectories in the context of hybrid systems.

1.4 Outline and results In Section 2, the Koopman-van Hove formulation of classical mechanics [6] is reviewed, along with its underlying geometric structure in terms of strict contact transformations, that is, connection-preserving automorphisms of the prequantum circle bundle T ∗Q S1 T ∗Q. This treatment is essentially equivalent to that presented by Kostant [42] in× the early→ 70’s. Following [33], we show how the group of strict contact diffeomorphisms is isomorphic to a central extension of the symplectic diffeomorphism group by S1, whose Lie algebra identifies the Poisson algebra of Hamiltonian functions on the classical phase-space T ∗Q. In Section 2.4

5 we review recent work [6] to show how the KvH formulation of classical mechanics produces the classical Liouville equation. This connection is established by a momentum map associated to the unitary action of strict contact diffeomorphisms on the sections of the prequantum bundle, which are here identified with complex wavefunctions on the classical phase-space. In Section 2.5, the Madelung transform is applied to the KvH equation (1.2) to show how the classical phase is naturally incorporated. Section 3.1 presents the mathematical setting of the hybrid wave equation for quantum– classical dynamics. The hybrid wavefunction on the hybrid coordinate space Γ = T ∗Q M × (here, M is the quantum configuration space) undergoes unitary evolution, whose Hermitian generator is called hybrid Liouvillian. The algebraic study of hybrid Liouvillian operators is presented in Section 3.2, along with a remarkable identity relating commutators and Poisson brackets. In the same section, hybrid Liouvillians are shown to be equivariant under both quantum unitary transformations and classical strict contact transformations. The same long sought equivariance properties [9] are shared by a hybrid density operator extending the quantum density matrix to quantum–classical dynamics, as shown in Section 3.3. While the density matrix of the quantum subsystem is positive-definite by construction in all cases, the hybrid density operator is generally unsigned and thus the sign of the classical Liouville density may require a case-by-case study. Classical and quantum pure states are shown to be both lost in the general case of hybrid dynamics thereby leading to questions about the existence of trajectories in the case of quantum– classical coupling. Section 4 addresses these questions by applying the Madelung transform to the hybrid wave equation, thereby leading to the identification of hybrid quantum–classical Bohmian trajectories and their generating vector field in Section 4.2. In the presence of a quantum–classical interaction potential, the symplectic form on the classical phase-space is not preserved by the hybrid flow and Section 4.3 characterizes explicitly the nontrivial dynamics of the Poincaréintegral on the hybrid coordinate space Γ. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectories identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the dynamics of the classical subsystem. Also, the Hamiltonian structure of the hybrid Madelung equations is presented in Section 4.4. In Section 5, we consider the geometric structure of the joint quantum–classical density on the hybrid coordinate space Γ. This hybrid density is found to be a momentum map in Section 5.2 and this ensures preservation of its sign in the special case when the quantum kinetic energy is absent in the hybrid Hamiltonian. Section 5.3 presents the continuity equation for the hybrid density, thereby leading to the identification of a hybrid quantum–classical current analogue to the probability current in quantum mechanics. The hybrid continuity equation is then shown to possess a Lie-Poisson Hamiltonian structure in Section 5.4. The paper closes with Section 5.5, which identifies an infinite family of hybrid Hamiltonians producing a quantum–classical dynamics that preserve the sign of the classical probability density.

2 Koopman-van Hove classical mechanics

In this Introduction, we shall review the KvH theory developed in [6] and present some of its features, along with its Madelung representation.

6 2.1 The Koopman-van Hove equation Let Q be the conﬁguration manifold of the classical mechanical system and T ∗Q its phase space, given by the cotangent bundle of Q. We assume that the manifold Q is connected. We shall ∗ i denote by z T Q an element of the phase space, and write z = (q ,pi) in local coordinates. ∈ i The phase space is canonically endowed with the one-form = pidq and the symplectic form i A ω = d = dq dpi, where d denotes the exterior derivative. For later purpose, it is also convenient− A to consider∧ the trivial principal circle bundle T ∗Q S1 T ∗Q (2.1) × → (known as prequantum bundle) in such a way that identiﬁes a principal connection + ds A A with curvature given by (minus) the symplectic form ω. A classical wave function Ψ is an element of the complex Hilbert space

2 ∗ HC = L (T Q) with standard Hermitian inner product

¯ H Ψ1 Ψ2 = Ψ1(z)Ψ2(z) Λ with Ψ1, Ψ2 C , h | i ˆT ∗Q ∈ deﬁned in terms of the Liouville volume form Λ = ( 1)n(n−1)/2ωn/n! (the multiplicative factor is such that in local coordinates one has Λ = dq1 ...− dqn dp ... dp ). The corresponding ∧ ∧ ∧ 1 ∧ ∧ n real-valued pairing and symplectic form on HC are given by

Ψ1, Ψ2 = Re Ψ¯ 1(z)Ψ2(z)Λ and Ω(Ψ1, Ψ2)=2~ Im Ψ¯ 1(z)Ψ2(z)Λ . (2.2) h i ˆT ∗Q ˆT ∗Q Given a classical Hamiltonian function H (T ∗Q), the KvH equation for classical wavefunctions was presented in [6, 27, 34, 39, 42] and∈F it reads i~∂ Ψ = i~ H, Ψ ( X H)Ψ . (2.3) t { } − A· H − Here, X is the Hamiltonian vector field associated to H, i.e. i ω = dH, and H,K = H XH { } ω(XH,XK) is the canonical Poisson bracket, extended in (2.3) to C-valued functions by C- linearity. Note that XH = pi∂pi H in local coordinates, thereby recovering the KvH equation (1.2) for a one-dimensionalA· configuration manifold Q. Then, X H identifies the Lagrangian A· H− associated to H and the entire right hand side of (2.3) defines the covariant Liouvillian operator = i~ H, ( X H) . (2.4) LH { } − A· H − Also known as prequantum operator, this is easily seen to be an unbounded Hermitian operator b on HC . As a consequence, the KvH equation (2.3) comprises a Hamiltonian system with respect to the symplectic form (2.2) and Hamiltonian functional

h(Ψ) = Ψ¯ H ΨΛ . ˆT ∗Q L b ∞ ∗ The correspondence H H satisfies [ H , F ] = i~ {H,F }, for all H, F C (T Q). Hence, 7→ L −1L L L ∈ on its domain, the operator Ψ i~ H Ψ defines a skew-Hermitian (or, equivalently, symplectic) left representation ofb the7→ Lie − algebraLb (b (T ∗Q)b, , ) on the classical Hilbert space H . F { } C Note that, unlike the map H L = i~b H, in (1.1), the correspondence H is now 7→ H { } 7→ LH injective, i.e. H = F H = F . L L ⇐⇒ b b b b 7 2.2 The group of strict contact diffeomorphisms In this Section, we shall assume that the first cohomology group H1(T ∗Q, R) = 0 (or, equivalently, H1(Q, R) = 0). Under this assumption, we shall follow van Hove [59] and show that the operator i~−1 integrates to a unitary left representation, whose corresponding group − LH is characterized below. As a preliminaryb step, given the trivial circle bundle (2.1), we consider its automorphism group given by the semidirect product Diff(T ∗Q) s (T ∗Q, S1). As explained in Section 1.3, F this group carries a natural unitary representation on the classical Hilbert space HC , which reads 1 −iϕ/~ Ψ η∗(e Ψ) , (2.5) 7→ Jac(η) with (η,ϕ) Diff(T ∗Q) s (T ∗Q, S1).p This is essentially the same representation as in (1.9), upon replacing∈ the quantumF configuration space M with the classical phase space T ∗Q. Like- wise, the infinitesimal generator corresponding to the unitary action (2.5) is again the analogue of (1.10) and one gets 1 Ψ i~−1νΨ X Ψ (div X)Ψ , (2.6) 7→ − − ·∇ − 2 where (X, ν) X(T ∗Q) s (T ∗Q) and div is the divergence with respect to the Liouville ∈ F volume form Λ on T ∗Q. As already anticipated, the representation (2.5) and its infinitesimal generator (2.6) will be of fundamental importance in later sections. A relevant subgroup of the semidirect product Diff(T ∗Q) s (T ∗Q, S1) is given by those F transformations preserving the connection one-form + ds on T ∗Q S1, that is the group ∗ 1 A × AutA(T Q S ) of connection-preserving automorphisms of the principal bundle (2.1). More explicitly, one× has

Aut (T ∗Q S1) := (η, eiϕ) Diff(T ∗Q) s (T ∗Q, S1) η∗ + dϕ = , (2.7) A × ∈ F A A n o ∗ where η denotes pullback. The above transformations were studied extensively in van Hove’s thesis [59] and are known as forming the group of strict contact diffeomorphisms [25]. This ∗ group is related to the more familiar group Diffω(T Q) of symplectic diffeomorphisms (canonical transformations) and this relation will be discussed in detail in the next section. For the moment, we simply notice that the relation η∗ + dϕ = implies A A z η∗(d )=0 , ϕ(z)= θ + ( η∗ ) , (2.8) A ˆz0 A − A

∗ so that η Diffω(T Q) and ϕ is determined up to a constant phase θ = ϕ(z0). Since 1 ∗ ∈ H (T Q, R) = 0, the line integral above does not depend on the curve connecting z0 to z. ∗ ∗ 1 ∗ 1 As a subgroup of the semidirect product Diff(T Q) s (T Q, S ), the group AutA(T Q S ) inherits from (2.5) a unitary representation, which is obtainedF essentially by replacing (2.8×) in (2.5). As we shall show in the next section, the operator i~−1 emerges as the infinitesimal LH generator of this representation. ∗ The relations (2.8) have an immediate correspondent at the levelb of the Lie algebra autA(T Q S1), which can be initially defined by using Lie derivatives as ×

aut (T ∗Q S1)= (X, ν) X(T ∗Q) s (T ∗Q) £ + dν =0 . A × ∈ F X A n o

8 We notice that the relation £X + dν = 0 implies £X (d ) = 0 thereby identifying a Hamil- tonian vector ﬁeld X = X , forA some H (T ∗Q). InA turn, Cartan’s magic formula yields H ∈ F £ = d( X H), so that d(ν + X H) = 0 and eventually one is left with X A A· H − A· H − X = X , ν = H X , H −A· H where an integration constant has been absorbed into the Hamiltonian H. For later purpose, here we introduce the notation F = X F, (2.9) A A· F − for the Lagrangian associated to the function F . It is now evident that any smooth Hamiltonian ∗ ∗ 1 function H (T Q) determines a Lie algebra element in (X, ν) autA(T Q S ) via the map ∈ F ∈ × H (T ∗Q) (X , H ) aut (T ∗Q S1). (2.10) ∈F → H − A ∈ A × Analogously, any pair (η, θ) Diﬀ (T ∗Q) S1 determines a group element in Aut (T ∗Q S1). ∈ ω × A × This picture can be given an equivalent and more convenient geometric structure in terms of central extensions.

2.3 A central extension of symplectic diffeomorphisms Given the group Diff (T ∗Q) of symplectic diffeomorphisms, let us fix a point z T ∗Q and ω 0 ∈ introduce the group 2-cocycle

η2(z0) ∗ Bz0 (η1, η2) := ( η1 ) , (2.11) ˆz0 A − A

∗ given by the line integral of the one form η1 along a path connecting z0 to η2(z0). As ∗ AH1 − ∗ A ∗ discussed previously, η1 is exact since (T Q, R) = 0. Thus, the relation η1 = dϕ1 leads to B (η , η ) =A−η∗ϕ A(z ) ϕ (z ) whose value is independent of the integrationA− A path. z0 1 2 2 1 0 − 1 0 Also, as reported in [33], the cohomology class of Bz0 is independent of both choices for the point z0 and the 1-form , where we recall d = ω. Here, we shall use theA group 2-cocycle (−2.11A) to express the group (2.7) of strict contact ∗ diffeomorphisms as a convenient central extension of the group Diffω(T Q) of symplectic diffeomorphisms by the circle group S1. In particular, we use the group 2-cocycle (2.11) to construct the following central extension: Diff (T ∗Q) = Diff (T ∗Q) S1 , (2.12) ω ω ×Bz0 endowed with the group productd structure [22, 33] (η , eiθ1 )(η , eiθ2 )=(η η , eiθ1+iθ2+iBz0 (η1,η2)), (2.13) 1 2 1 ◦ 2 where z T ∗Q is some fixed point. 0 ∈ At this stage, since the prequantum bundle (2.1) is trivial we have the following statement. Proposition 2.1 ([33]) Given an element (η, eiθ) Diff (T ∗Q), the following mapping de- ∈ ω fines a group isomorphism Diff (T ∗Q) Aut (T ∗Q S1): ω → A × z d iθ iθ i (A−η∗A) d (η, e ) η, e e z0 . (2.14) 7→ ´ The inverse isomorphism Aut (T ∗Q S1) Diff (T ∗Q) is given by (η, eiϕ) (η, eiϕ(z0)). A × → ω 7→ 9d Once the group structure of the central extension (2.12) is characterized, one can find its corresponding Lie algebra structure. The latter is given by the central extension of the Lie ∗ algebra of symplectic (hence Hamiltonian) vector fields Xω(T Q) denoted

X (T ∗Q) := X (T ∗Q) R . (2.15) ω ω ×Cz0

Here, Cz0 is the Lie algebra 2-cocyleb associated to Bz0 so that

(XH , κ), (XF ,γ) = X{H,F },Cz0 (XH ,XF ) ,

∗ for all (XH , κ), (XF ,γ) Xω(T Q). The Lie algebra 2-cocyle is constructed as follows: given two Hamiltonian vector∈ ﬁelds X ,X X (T ∗Q) whose ﬂows are denoted respectively by η (t) H F ∈ ω 1 and η2(s), we have b d d C (X ,X )= B (η (t)−1, η (s)) B (η (s), η (t)−1) . (2.16) z0 H F ds dt z0 1 2 − z0 2 1 s=0 t=0

After a direct calculation using the notation introduced in (2.9), we obtain [22] C (X ,X )= X (z ) F,H (z )= F,H (z ) . z0 H F A· {F,H} 0 −{ } 0 { }A 0 Then, as the Lie algebra structure of (2.15) is now characterized, proposition 2.1 leads naturally to the following result which is the analogue of (2.10): Proposition 2.2 ([22]) Given a point z T ∗Q, the following mapping 0 ∈ H (T ∗Q) (X , H (z )) X (T ∗Q) (2.17) ∈F 7→ H − A 0 ∈ ω ∗ ∗ is a Lie algebra isomorphism. The inverse isomorphism Xbω(T Q) (T Q) is given by (X, a) H, where H is the unique Hamiltonian of X with H(z )= →X F(z )+ a. 7→ 0 A· 0 b Proof. Taking the derivative of the group isomorphism (2.14) at the identity yields Lie algebra isomorphism

(X , κ) X (T ∗Q) (X , κ + H (z ) H ) aut (T ∗Q S1) , (2.18) H ∈ ω 7→ H A 0 − A ∈ A × whose inverse is simply given by (X , ν) aut (T ∗Q S1) (X , ν(z )) X (T ∗Q). By b H ∈ A × 7→ H 0 ∈ ω composing with the isomorphism (2.10) we do get (2.17). The Lie algebra isomorphism property of (2.17) b H, F (T ∗Q) X , H, F (z ) X (T ∗Q) (2.19) { }∈F 7→ {H,F } −{ }A 0 ∈ ω follows from a direct computation. b 2.4 The van Hove representation and the Liouville density

∗ 1 At this stage, we can rewrite the unitary (left) representation of the group AutA(T Q S ) ∗ × on the classical Hilbert space as an action of the central extension Diffω(T Q). Indeed, if iθ ∗ H (η, e ) Diffω(T Q), its representation on C is obtained by using (2.14) in (2.5). We have ∈ d z −1 i d U iθ Ψ (z)=Ψ(η (z)) exp θ + (η ) . (2.20) (η,e ) −~ ˆ ∗A−A η(z0) 10 i i i i The property U(η1,e θ1 ) U(η2,e θ2 ) = U(η1,e θ1 )(η2,e θ2 ) can be directly verified by using (2.13). This representation made its◦ first appearance in [59] as a unitary representation of the group (2.7) and here we shall call it the van Hove representation. ∗ Analogously, the Lie algebra representation of (XH , κ) Xω(T Q) associated to (2.20) is computed as ∈ b d −1 u Ψ= U iθǫ Ψ = i~ (κ + H (z ) H )Ψ X Ψ (X,κ) dǫ (ηǫ,e ) − A 0 − A − H ·∇ ǫ=0 ~−1 = i (H XH)Ψ XH Ψ , (2.21) − −A· − ·∇ where (η , eiθǫ ) Diff (T ∗Q) is a path tangent to (X , κ)at(id, 1), H (T ∗Q) is an arbitrary ǫ ∈ ω H ∈F function, and HA is obtained from the notation (2.9). Then, the Lie algebra representation (2.21) coincides withd i~−1 as claimed previously. We emphasize that, since i~−1 is the − LH − LH infinitesimal generator of the representation (2.20), the prequantum operator H is equivariant ∗ L with respect to the action ofb Diffω(T Q), namely b b † i d U iθ H U(η,e θ ) = H◦η. (2.22) (η,e )L L This relation was used in [6] to write Koopman-vanb Hoveb dynamics in the Heisenberg picture. In the present work, we shall extend this result to the case of hybrid quantum–classical systems; see Section 3.2. So far, nothing has been said about how the KvH equation (2.3) is related to classical me- ∗ chanics. As shown in [6], this relation is given in terms of a momentum map HC Den(T Q), where Den(T ∗Q) denotes the space of densities on T ∗Q. Since the van Hove representation→ (2.20) is unitary, it is symplectic with respect to the symplectic form (2.2) and thus admits a momentum map ρ(Ψ) via the standard formula [26, 30, 45] 1 ρ(Ψ),H = Ω i~−1 Ψ, Ψ . (2.23) h i 2 − LH Here, , denotes the duality pairing between (T ∗Qb) and its dual Den(T ∗Q). Throughout h i F this paper, the angle brackets always denote a duality pairing, whose explicit expression may differ depending on the particular vector space under consideration. A direct calculation [6] leads to the momentum map

ρ(Ψ) = Ψ 2 div J Ψ 2 + i~ Ψ, Ψ¯ | | − A| | { } = Ψ 2 div Ψ¯ J( Ψ + i~ Ψ) , (2.24) | | − A ∇ where the divergence is associated to the Liouville form and where J : T ∗(T ∗Q) T (T ∗Q) is → deﬁned by F,H = dF, J(dH) . In local coordinates the second term reads div J Ψ 2 = ∂ (p Ψ 2).{ This} momentumh mapi is formally a Poisson map with respect− to theA| canonical| pi i Poisson| | structure 1 δf δh f, h (Ψ) = Im Λ {{ }} 2~ ˆT ∗Q δψ δψ on HC and the Lie-Poisson structure

δf δh f, h (ρ)= ρ , Λ {{ }} ˆ ∗ δρ δψ T Q 11 on Den(T ∗Q). Hence, if Ψ(t) is a solution of the KvH equation, the density (2.24) solves the Liouville equation ∂ ρ = H, ρ . As remarked in [6], a density of the form (2.24) is t { } not necessarily positive deﬁnite. However, the Liouville equation generates the sign-preserving evolution ρ(t)= η(t)∗ρ0, where η(t) is the ﬂow of XH , thereby recovering the usual probabilistic interpretation. Notice that the momentum map (2.24) yields the following relation for classical expectation ∗ values: given a classical observable A (T Q), its expectation value A := T ∗Q Aρ Λ is expressed as ∈ F h i ´

A = Ψ¯ AΨΛ , (2.25) h i ˆT ∗Q L which is diﬀerent from the usual expressions appearingb in quantum theory. At this stage, the meaning of the KvH equation (2.3) is still somewhat obscure and we shall try to shed some new light by applying the Madelung transform.

2.5 The Madelung transform The Madelung transform of the KvH equation (2.3) is obtained by writing Ψ in polar form

Ψ(t, z)= R(t, z)eiS(t,z)/~ , thereby leading to the following equations for R and S

∂ S + S,H = L (2.26) t { } ∂ R + R,H = 0 . (2.27) t { } Here, we have introduced the Lagrangian L = p ∂ H H (T ∗Q), or equivalently, using the i pi − ∈F notation in (2.9), L := HA. Thus, while (2.27) recovers the standard Koopman-von Neumann equation for the amplitude Ψ , the KvH construction comprises also the dynamics (2.26) of | | the classical phase. Notice that (2.26) is equivalently written as

d S(η(t, z), t)= L(η(t, z)) (2.28) dt where η(t) is the ﬂow of XH . If the right-hand side in (2.28) is set to zero, one recovers the phase evolution arising from the Koopman-von Naumann equation (1.1). However, in the Koopman-van Hove construction under consideration, the Lagrangian function L = XH H is retained in the expression (2.4) of the covariant Liouvillian and equation (2.28A·) is solved− LH formally as follows: b t S(z, t)= L(η(τ t, z)) dτ + S(η(t0 t, z), t0) . (2.29) ˆt0 − − We remark that, since £ = dH = dL, the phase dynamics also produces the relation XH A A (∂ + £ )(dS )=0 , (2.30) t XH −A which is written in terms of the Lie derivative £XH = diXH + iXH d. Notice that the relation dS = would be preserved in time thereby recovering the KvN prescription ρ = Ψ 2 via the momentumA map (2.24). However, as pointed out in [35], this possibility would introduce| |

12 topological singularities which we shall not treat on this occasion. Instead, here we notice that the relation (2.30) implies η(t)∗(dS(t) ) = dS , or, equivalently, −A 0 −A d(S(t) η(t) S )= η(t) . − ∗ 0 A − ∗A This is the customary relation for generating functions [45] and it is consistent with d η(t)∗ω =0 , (2.31) dt which follows directly from the fact that η(t) is the flow of XH . The amplitude equation also retains some interesting features. Indeed, we notice that defining D = R2 yields the Liouville-type equation ∂ D + D,H =0 , t { } which formally allows for the singular solution D(z, t)= δ(z ζ(t)) − where the curve ζ(t) T ∗Q satisfies the Hamilton equations dζ/dt = X (ζ). The particle ∈ H phase along ζ(t) is deduced from (2.28) by writing ζ(t)= η(t, z ) for some z T ∗Q as 0 0 ∈ t S(ζ(t), t)= L(ζ(τ))dτ + S(ζ(t0), t0). ˆt0 While this process is only formal (the relation D = R2 prevents D from being a delta function), these relations are somewhat revealing of a finite-dimensional correspondent of KvH theory. To conclude this section, we present the relation between the momentum map (2.24) for the classical Liouville equation and the KvH analogue of the hydrodynamic momentum map (1.8) associated to the Madelung transform. This KvH analogue reads J(Ψ) = (~ Im(Ψ¯ Ψ), Ψ 2)=:(σ, D) (2.32) ∇ | | and is associated to the representation (2.5) of the prequantum bundle automorphisms Aut(T ∗Q 1 × S ) on HC . As we have seen in Section 2.3, this representation reduces to (2.20) upon restrict- ∗ ∗ 1 ing to the subgroup of connection-preserving automorphisms Diffω(T Q) AutA(T Q S ) Aut(T ∗Q S1). Thus, the momentum map ρ(Ψ) in (2.24) for the classical≃ Liouville× equa-⊂ × ֒ (tion can be related to J(Ψ) in terms of the dual of the Lied algebra inclusion ι : (T ∗Q ∗ ∗ ∗ F ∗ → X(T Q) s (T Q). Here we recall the Lie algebra isomoprhism (T Q) Diffω(T Q) from PropositionF2.2. More explicitly, using the notation (2.9), the LieF algebra inclusion≃ is given by d ι(H)=(X , H ) (2.33) H − A and thus its dual map ι∗ : X(T ∗Q)∗ Den(T ∗Q) Den(T ∗Q) reads ι∗(σ, D)= D div(J D × →∗ − A − Jσ). Then, as one verifies explicitly, one obtains ι [J(Ψ)] = ρ(Ψ) for all Ψ HC . This indeed provides an important relation between the momentum map (2.24) for the∈ classical Liouville equation and the momentum map (2.32) associated to the KvH Madelung transform. In more generality, we have seen how several quantities appearing in KvH classical mechanics are all interconnected between them and their relations are most often given by specific momentum maps associated to particular diffeomorphisms of the prequantum bundle. The general picture of the variables appearing in KvH classical mechanics is presented in Figure 1, which displays also the role of the various momentum maps presented so far.

13 Classical Liouville description ρ Den(T ∗Q) ∈ 7 O ♦♦♦ ♦♦♦ ♦♦♦ Momentum map (2.24) for the subgroup Diﬀ (T ∗Q) ω Classical KvH ♦♦♦♦ ∗ d Ψ HC , H (T Q) ∈ ∈F Dual to the Lie algebra inclusion (2.33) ¯ ∗ ∗ ∗ (h(Ψ) = Ψ H ΨΛ ι : (T Q) ֒ X(T Q) s (T Q ˆT ∗Q L F → F i~∂tΨ= H Ψ b ◆ L ◆◆◆ Momentum map (2.32) Ψ=bReiS/~ for the group Diﬀ(T ∗Q) s (T ∗Q,S1) F ◆◆ ◆◆◆ ◆◆◆ ◆' KvH-Madelung Madelung variables equations (2.26) / ∗ (σ, D) X(T ∗Q) s (T ∗Q) for (S, R) ∈ F Figure 1: Schematic description of the various quantities appearing in Koopman-van Hove classical mechanics and some of the mappings between them.

3 Hybrid quantum–classical dynamics

3.1 Quantum–classical wave equation As mentioned earlier, the KvH framework leads naturally to the hybrid description of a coupled quantum–classical system. Indeed, one may start with the KvH equation for two particles and then apply geometric quantization to quantize one of them. Here, instead of quantizing observables, we follow an alternative procedure. As outlined by Klein [39] in the case of one particle, this method transforms the KvH equation (2.3) into the Schrödinger equation and here we restrict to consider Hamiltonians for the type H = T + V (i.e. given by the sum of kinetic and potential energy). In one dimension Klein’s method proceeds as follows: (1) write the one-particle KvH equation for Ψ(x, ν) with H = m−1ν2/2+ V (x) and = νdx, (2) A restrict to consider solutions ∂ν Ψ = 0, and (3) replace ν i~∂x. A direct verification shows that this yields the standard Schrödinger equation i~∂ Ψ(→x −) = (m−1~2/2)∆Ψ + V Ψ. The t − condition ∂ν Ψ = 0 corresponds to fixing a polarization in geometric quantization [38], while the replacement ν i~∂x corresponds to the usual canonical quantization prescription. At this point, a→ hybrid − theory can be obtained by starting with the KvH equation for two particles and then applying Klein’s method to quantize one of the particles. This is precisely the approach adopted in [6], which led to the following quantum–classical wave equation

i~∂ Υ= i~H, Υ + H X b Υ . (3.1) t { } −A· H Similar equations already appeared in [9b] and wereb rejected by the authors. Here, the phase- space function H(z) takes values in the space of unbounded Hermitian operators on the quan-

b 14 2 2 ∗ tum Hilbert space HQ := L (M). In addition, Υ L (T Q M) is a hybrid wavefunction depending on both the classical and the quantum coordinates,∈ den× oted by z T ∗Q and x M ∈ ∈ respectively. Here, we assume that M is endowed with a volume form µ so that the inner product and symplectic form on L2(T ∗Q M) are defined by the immediate generalization of the classical definitions (2.2). For convenience,× here we shall denote the hybrid quantum–classical Hilbert space by H := L2(T ∗Q M) . (3.2) QC × It is useful to recall that the identification Υ(z, x) (Υ(z))(x) yields the isometric isomor- ≃ phism H L2(T ∗Q; H ) , (3.3) QC ≃ Q where L2(T ∗Q; H )= H H is the Bochner-Lebesgue space of L2 functions on T ∗Q taking Q C ⊗ Q values in the quantum Hilbert space HQ; see, e.g., [32, 1.2]. Notice that the same approach yields the alternative isomorphism §

H L2(M; H )= H H . (3.4) QC ≃ C C ⊗ Q Both isomorphisms (3.3) and (3.4) will be useful to compute integrals of the form

Υ(¯ z′, x)Υ(z, x) µ := (Υ(z′))(x)(Υ(z))(x) µ (3.5) ˆM ˆM Υ(¯ z, x′)Υ(z, x) Λ := (Υ(x′))(z)(Υ(x))(z)Λ . (3.6) ˆT ∗Q ˆT ∗Q

In addition, these isomorphisms lead to defining the quantum adjoint Υ†(z) as follows. If we ∗ evaluate the square-integrable function Υ HQC at a fixed point z T Q, the isomorphism (3.3) yields another square-integrable function∈ Υ(z) H in the quantum∈ Hilbert space so ∈ Q that for each fixed z the standard inner product on H induces a linear form Υ†(z) on h|i Q HQ given by Υ†(z)ψ := Υ(z) ψ = Υ(¯ z, x)ψ(x) µ , (3.7) h | i ˆM † 2 with ψ HQ. For example, one writes Υ (z)Υ(z) = Υ(z) , where the norm is induced ∈ † k k k k by the inner product on HQ. We shall call Υ (z) the quantum adjoint of Υ, while an analogous procedure canh|i evidently be used to define the classical variant. By construction, the hybrid Liouvillian operator

b = i~H, + H X b (3.8) LH { } −A· H H is an unbounded Hermitian operatorb on bQC so thatb the the quantum–classical wave equation (3.1) reads i~∂ Υ = b Υ and the hybrid wavefunction Υ undergoes unitary dynamics. Using t LH local coordinates on T ∗Q, the operator (3.8) is written as b b Υ = i~ ∂ i H∂ Υ ∂ H∂ i Υ + H p ∂ H Υ. LH q pi − pi q − i pi Similarly to the injectiveb correspondenceb H b underlyingb the covariantb Liouvillian oper- 7→ LH ator (2.4), the correspondence H b is also injective. Importantly, upon considering the → LH immediate generalization of the symplectic formb in (2.2), we notice that the quantum–classical b b

15 wave equation (3.1) is Hamiltonian with the following Hamiltonian functional expressed in terms of the quantum Liouvillian:

b ¯ b h(Υ) = Υ H Υ Λ= Υ H Υ Λ µ . (3.9) ˆT ∗Q L ˆT ∗QˆM L ∧ While hybrid Liouvillian operators do b not comprise a Lie algebrab structure, the next section presents their general algebraic properties.

3.2 Algebra of hybrid Liouvillian operators While the covariant Liouvillian operators deﬁned in (2.4) possess a Lie algebra structure given by [ , ] = i~ , no such structure is available for the hybrid Liouvillian operators. The LF LK L{F,K} latter satisfy obvious identities that can be written upon introducing the convenient notation ∗ A b b(T Q) forb classical observables and A Her(HQ) for quantum observables, while C ∈ F Q ∈ A (T ∗Q, Her(H )) denotes a hybrid observable. Here, Her(H ) denotes the space of ∈ F Q Q Hermitian operators on H , so that (T ∗Q, Her(b H )) is the space of phase-space functions Q F Q takingb values in the space Her(HQ) of quantum observables. For example, with this notation we have b = AQ. More generally, one has the obvious identities LAQ

b b = AbQ b , b , b = b b , A , b = i~ b , LAQB LB LAQ LB L[AQ,B] L C LB L{AC ,B} as well asb b b b b b b b b

b , b = b b , b , B = i~ b . LAQAC LBQ L [AQ,BQ]AC LAQAC L C L{AC ,BC }AQ In additionb to the aboveb algebraicb rules, here it mayb be usefulb to repbort a further remarkable relation. To this end, we start by introducing the conjugate of an operator as follows.

Definition 3.1 (Conjugate operator) Let H a complex Hilbert space and let A : H H → be a linear operator. Then, the conjugate operator A¯ of A is defined by the relation Au¯ := Au¯ for any u H . b ∈ b b Here, no confusion should arise from adopting the notation A¯ in place of A . For example, if H H ′ = Q and A is an integral operator with kernel Ab(x, x ) (or ‘matrix element’, in physics ′ K terminology), then we have ¯(x, x ) = b(x, x′). Also, in analogy with theb definition of the KA KA transpose of a linearb mapping, recalling that H = H H is a tensor product space, we QC C ⊗ Q define the quantum transpose of a linear operator : H H as the partial transpose with L QC → QC respect to the factor HC . The intrinsic definition is given by b

Deﬁnition 3.2 (Quantum transpose) Let : HQC HQC be a linear operator on the tensor- L → T product Hilbert space HQC = HC HQ. Then, the quantum transpose of is the operator ⊗ b L L T ¯ ¯ ¯ (Ψ2ψ2) Ψ1ψ1 := Ψ1ψ2 (Ψ2ψ1) , b b (3.10) L | | L for all Ψ , Ψ H and ψ , ψ H . 1 2 ∈ C 1 b 2 ∈ Q

16 Notice the position of the indices 1 and 2 in this definition. In the case of an integral operator ′ ′ ′ ′ ′ ′ with kernel b(z, z , x, x ), we have b T (z, z , x, x )= b(z, z , x , x). At this point, with the L KL KL KL definitions above, we have the following result: b Theorem 3.3 (Remarkable algebraic relation) Let A(z) and B(z) be two hybrid observ- ∗ ables in the space (T Q, Her(HQ)) of phase-space functions taking values in the space of F H H Hermitian operators Her( Q) on the quantum Hilbert spaceb Q. Then,b their associated Liou- villian operators b and b satisfy the following identity: LA LB T b b b, b + ¯, ¯ = i~ b b b b . (3.11) LA LB LA LB L{A,B}−{B,A} Proof. It will be convenientb tob distinguishb b betweenb the the inner products , , h|iC h|iQ and QC on the different Hilbert spaces HC , HQ, and HQC , respectively. We note that for Θ, Ψh|iH and θ, ψ H , we have ∈ C ∈ Q

Ψψ Ab (Θθ) = Ψ hψ|Aθb i Θ , (3.12) L QC L Q C D E D E where on the right hand side b b denotes the classical b covariant Liouvillian operator (2.4) Lhψ|AθiQ ∗ associated to the function z T Q ψ H(z)θ Q C. In the remainder of this section, it ∈b 7→ h | i ∈ will be convenient to use Dirac’s notation for vectors in the quantum Hilbert space HQ; for example, we replace ψ Aθ by ψ A θ inb (3.12). Then, choosing a sequence α H such h | iQ h | | iQ | i ∈ Q that α α dα = idH , we can write | ih | Q ´ b b

b b b b Ψψ Ab, Bb (Θθ) = Ψ hψ|A|αi hα|B|θi hψ|B|αi hα|A|θi Θ dα L L QC ˆ L Q L Q − L Q L Q C D E D E b b b b b b = Ψ bhψ|A|αi bhα|B|θi bhα|B|θi bhψ|A|αi Θ dα ˆ L Q L Q − L Q L Q C D E

+ Ψ b hα|Bb|θib hψ|Ab|αi b hψ|Bb|αbi hα|Ab|θi Θ dα ˆ L Q L Q − L Q L Q C D E b b b b + Ψ bhα|A|θi bhψ|B|αi bhα|A|θi bhψ|B|αi Θ dα ˆ L Q L Q − L Q L Q C D E ~ b b b b b~ bb b = Ψ i hψ|A|αi , hα|B|θi Θ i hψ|B|αi , hα|A|θi Θ dα ˆ L Q Q − L Q Q C D { } { } E b b b b + Ψ b hα|B|θi hψ|A|αi Θ hαb|A|θi hψ|B|αi Θ dα, ˆ L Q L Q − L Q L Q C D E where in the third equality we used [ b, ] =b i~ forb givenb in (2.4). The ﬁrst two LH LF L{H,F } L terms in the last equality can be written as b b b b ~ b b b b ~ b b b b Ψ i hψ|{A,B}−{B,A}|θi Θ = Ψψ i {A,B}−{B,A}Θθ L Q C L QC D E D E while the last two terms b are b

Ψ ¯ ¯ ¯ ¯ Θ ¯ ¯ ¯ ¯ Θ dα = Ψθ¯ ¯ , ¯ (Θψ¯) hθ|B|α¯iQ hα¯|A|ψiQ hθ|A|α¯iQ hα¯|B|ψiQ B A ˆ L L − L L C L L QC D E T D E ¯ ¯ b = bB¯ , A¯ (Θψ) Ψbθ =b Ψψ B¯ , A¯ Θθ b b L L QC L L QC D E D E b b b b 17 by using (3.10). These relations are satisﬁed for all Ψ, Θ HC and all ψ, θ HQ. In particular ∈ H H ∈ they hold for any orthonormal bases (Ψi)i∈I and (ψj)j∈J of C and Q, respectively. Since (Ψ ψ ) is an orthonormal basis of H = H H , we obtain (3.11). i ⊗ j (i,j)∈I×J QC C ⊗ Q We conclude this section by presenting the equivariance properties of hybrid Liouvillians. These properties will be used in the next section to obtain a physically more relevant result.

Lemma 3.4 (Equivariance) Let A(z) be a hybrid observable in the space (T ∗Q, Her(H )) F Q of phase-space functions taking values in the space of Hermitian operators Her(HQ) on the quantum Hilbert space HQ. Also, letb (HQ) denote the group of unitary operators on HQ and U U iθ denote the van Hove unitary operator (2.20). Then, the Liouvillian b associated to (η,e ) LA A(z) satisfies † iθ ∗ b iθ b b U(η,eiθ ) AU(η,e ) = η∗A , (η, e ) Diffω(T Q) ; (3.13) b L L ∀ ∈ † U bU = † b , U (H ) . (3.14) b LA LbU AU ∀ ∈ U dQ Proof. The relation (3.13) isb proved inb Appendix Bb, while (3.14) follows by immediate verification. On one hand, the relation (3.13) is a direct extension of (2.22) to the case of hybrid Liouvillians. This is a natural consequence of the fact that the van Hove representation U(η,eiθ ) in (2.20) does not involve quantum degrees of freedom. On the other hand, one also has equivariance under unitary transformations of the quantum Hilbert space space. In the next section, these equivariance relations will be shown to apply also to a hybrid density operator extending the classical Liouville density as well as the von Neumann’s celebrated density matrix in quantum theory.

3.3 The hybrid density operator As shown in [6], the Hamiltonian structure of the quantum–classical wave equation (3.1) leads to deﬁning a hybrid density operator for the computation of expectation values. Indeed, the latter can be identiﬁed by rewriting the Hamiltonian functional (3.9) by using integration by parts as follows:

h(Υ) = Υ Hb Υ Λ = Tr H(z) (z)Λ . (3.15) ˆT ∗Q L ˆT ∗Q D

Here, in analogy to the expression (2.24 b) of the classical Liouvilleb b density, the hybrid density operator is given as D = ΥΥ† div J ΥΥ† + i~ Υ, Υ† , (3.16) D − A { } b so that Tr ∗ Λ = 1. Again, here the divergence is taken relative to the Liouville volume T Q b form Λ on ´T ∗QD. The hybrid density operator is deﬁned in such a way that its application to a H quantum wavefunctionb ψ Q reads ∈ (z)ψ = Υ(z) Υ(z) ψ + ∂ (p Υ(z) ψ Υ(z)) + i~ Υ(z), Υ(z) ψ , (3.17) D h | i pi ih | i { h | i} where we recallb (3.7). As usual, by appropriately restricting the space of wavefunctions Υ, the associated hybrid density operator is an operator-valued density on T ∗Q taking values in D b

18 the space of trace-class Hermitian operators on L2(M). We write Den(T ∗Q, Her(H )). In D ∈ Q particular, is an integral operator with kernel (or ‘matrix element’) D b ′ ′ ′ b(z; x, x ) =Υ(z, x)Υ(¯ z, x ) ∂ p Υ(z, x)Υ(¯ z, x ) b KD − pi i ′ ′ + i~ ∂ i Υ(z, x)∂ Υ(¯ z, x ) ∂ Υ(z, x)∂ i Υ(¯ z, x ) . (3.18) q pi − pi q Given the hybrid density operator , one computes the quantum density operator D ρˆ := b(z)Λ= Υ(z)Υ†(z)Λ , (3.19) ˆT ∗Q D ˆT ∗Q so that the quantum probability densityb in conﬁguration space is obtained as

2 b ρq(x)= D(z; x, x)Λ = Υ(z, x) Λ . (3.20) ˆT ∗Q K ˆT ∗Q | |

b On the other hand, the classical density reads ρc(z) = Tr (z)= M D(z; x, x)µ. Here the trace is computed only with respect to the quantum degrees ofD freedom´ K so that, upon using (3.18), b 2 2 ~ ¯ ρc(z) =Tr (z)= Υ(z, x) ∂pi pi Υ(z, x) + i Υ, Υ (z, x) µ. (3.21) D ˆM | | − | | { } h i We now move onb to discuss expectation values. It is evident that the second equality in (3.15) holds upon replacing H(z) by any hybrid quantum–classical observable A(z), whose expectation value A can then be written as h i b b

b b A = Tr A(z) (z)Λ= Υ AΥ Λ . h i ˆT ∗Q D ˆT ∗Q L

This relation extends the classicalb case (b2.25).b Again, we notice b the difference from the relations appearing in the purely quantum formalism. The usual quantum expectation value is however recovered naturally in the purely quantum case, since the relation b = AQ from Section 3.2 LAQ H for a quantum observable AQ Her( Q) implies AQ = T ∗Q Υ AQΥ Λ. ∈ h i ´ h | b i b 3.4 Equivariance ofb the hybrid densityb operatorb Another feature of the hybrid density operator is the equivariance property of its defining mapping Υ (Υ) in (3.16) under both quantumD and classical transformations. More 7→ D specifically, we have the following result. b b Theorem 3.5 (Equivariance of the hybrid density operator) Let (H ) denote the U Q H i group of unitary operators on Q and U(η,e θ) denote the unitary operator (2.20) corresponding iθ ∗ H to the van Hove representation of (η, e ) Diffω(T Q) on QC . Also, let the hybrid density ∗ ∗ ∈ operator ( (T Q, Her(HQ))) be defined as in (3.17). Then, one has D ∈ F d iθ ∗ b (U iθ Υ) = η∗ (Υ) , (η, e ) Diffω(T Q) . (3.22) D (η,e ) D ∀ ∈ † b (UΥ) = Ub (Υ)U , U (HdQ) . (3.23) D D ∀ ∈ U b b b b b b 19 Proof. The relation (3.22) can be verified by pairing (U(η,eiθ )Υ) against a hybrid observable ∗ D A (T Q, Her(HQ)) as follows: ∈F b b iθ iθ b iθ Tr A (U(η,e )Υ)Λ = U(η,e )Υ AU(η,e )Υ Λ ˆT ∗Q D ˆT ∗Q L

† b b b b iθ = Υ U(η,eiθ) AU(η,e )Υ Λ ˆT ∗Q L

b b = Υ η∗AΥ Λ ˆT ∗Q L

= Tr b (Υ) η∗A Λ ˆT ∗Q D b b = Tr A η∗ (Υ) Λ , ˆT ∗Q D where we used the relation (3.13). In addition, (3.23) followsb byb construction from the deﬁnition (3.16). The equivariance properties (3.22)-(3.23) of the hybrid density operator under both classical and quantum transformations have long been sought in the theory of hybrid quantum–classical systems [9] and stand as one of the key geometric properties of the present construction. The equivariance properties (3.22)-(3.23) also reﬂect in the dynamics of both the classical distribution (3.21) and the quantum density matrix (3.19), which read respectively [6]

∂ρ ∂ρˆ c = Tr H, , i~ = [H, ] Λ (3.24) ∂t { D} ∂t ˆT ∗Q D

As pointed out in [6], pure state solutionsb b are prevented by theb densityb matrix evolution and this property is known as decoherence in the physics terminology. In addition, classical point trajectories are also lost in the general case of quantum–classical interaction, since the first equation in (3.24) does not possess delta-like Klimontovich solutions (that is, classical pure states). While not completely surprising, the absence of classical particle trajectories in hybrid dynamics raises questions about the meaning of the word ‘classical’ in this context. Classical motion is identified with a Hamiltonian flow producing characteristic curves representing particle trajectories. Then, the question emerges whether any feature of a classical Hamiltonian flow can still be identified in hybrid dynamics. In this paper, we address this question by extending the Lagrangian trajectories from quantum hydrodynamics to hybrid quantum–classical systems. To this purpose, the following sections will apply the Madelung transform to equation (4.2). As a result, we shall present a hybrid generalization of Bohmian trajectories in terms of Lagrangian paths, which will be discussed in terms of their Hamiltonian structure and the corresponding momentum maps.

4 Hybrid Madelung equations

In the remainder of this paper, we shall restrict to consider hybrid Hamiltonians of the type

~2 1 H(q,p,x)= ∆ + p 2 + V (q, x) , (4.1) −2m x 2M | | b 20 thereby ignoring the possible presence of magnetic ﬁelds. Here ∆x is the Laplacian on M associated to a given Riemannian metric and the norm p is given with respect to a Riemannian | | metric on Q. In this case, the hybrid quantum–classical wave equation (3.1) reads

~2 i~∂ Υ= L + ∆ Υ + i~ H , Υ , (4.2) t − I 2m x { I } where we have deﬁned the following scalar functions L ,H on the hybrid space T ∗Q M: I I × 1 1 H (q,p,x) := p 2 + V (q, x) , L (q,p,x) := p 2 V (q, x) . (4.3) I 2M | | I 2M | | − These are respectively the classical Hamiltonian and Lagrangian both augmented by the presence of the interaction potential. As we shall see, these quantities play a key role in the geometry of hybrid quantum–classical systems.

4.1 Quantum–classical Madelung transform In this section we extend the usual Madelung transformation from quantum mechanics to the more general setting of coupled quantum–classical systems. The Madelung transform was already applied to KvH classical mechanics in Section 2.5, while the equations (1.6) for standard quantum hydrodynamics appeared originally in Madelung’s work [44]. We emphasize that here we focus on Madelung’s original approach by invoking a single-valued phase function. The possibility of multi-valued quantum phase functions leading to topological singularities [18] and nontrivial holonomy was emphasized in [63] and will not be considered in the present context. See also [21] for a geometric dynamical treatment of nontrivial holonomy in quantum hydrodynamics. In order to apply the Madelung transform to the hybrid setting, we write the hybrid wavefunction in polar form, that is

Υ(t, z, x)= (t, z, x)eiS(t,z,x)/~, (4.4) R where calligraphic fonts are used to distinguish the hybrid case from the previous purely classical case treated in Section 2.5. Then, the quantum–classical wave equation (4.2) produces the following dynamics for the hybrid amplitude and phase

∂ 2 ~2 ∆ S + |∇xS| xR = L + H , (4.5) ∂t 2m − 2m I { I S} ∂ 1 R R + div ( 2 )= H , , (4.6) ∂t 2m x R ∇xS { I R} R where the operators , div , and ∆ = div are deﬁned in terms of the Riemannian metric ∇x x x x ∇x on M. Each equation carries the usual quantum terms in (1.4)-(1.5) on the left-hand side, while the terms arising from KvH classical dynamics appear on the right-hand side (see equations (2.26)-(2.27)). In the absence of classical degrees of freedom, the ﬁrst equation simply recovers the so called quantum Hamilton-Jacobi equation (1.4). We observe that (4.5) can be written in Lie derivative form as follows:

∂ + £X = L . (4.7) ∂t S

21 Here, X is the hybrid velocity vector field on T ∗Q M given by × (z, x) X(z, x)= X (z, x), ∇xS , (4.8) HI m ∗ XHI being the x-dependent Hamiltonian vector field associated to HI on (T Q, ω). Moreover, we have defined the (time-dependent) hybrid quantum–classical Lagrangian

2 2 2 x ~ ∆x√ L (t, z, x) := LI + |∇ S| + R , 2m 2m √ 2 R in analogy with the so-called quantum Lagrangian [67] for a free quantum particle, given by the last two terms above. Then, upon taking the total differential d (on T ∗Q M) of (4.7) and rewriting (4.6) in terms of the density 2, we may rewrite (4.5)-(4.6) as follows:× R ∂ + £X d = dL , (4.9) ∂t S ∂ 2 R + div( 2X)=0 . (4.10) ∂t R In (4.10), the operator div denotes the divergence operator induced on T ∗Q M by the Liouville × form on T ∗Q and the Riemannian metric on M. As we shall see, the above form of the hybrid Madelung equations will be crucially important. Notice that, in the absence of coupling, we have ∂qj ∂xk V = 0 in (4.1) and the mean-field factorization Υ(z, x)=ΨC (z)ΨQ(x) becomes an exact solution of (3.1). Indeed, in this case we have (z, x) = S (z)+ S (x) and (z, x) = R (z)R (x), so that equation (4.7) recovers the S C Q R C Q purely classical case (2.26) for SC (z) and the purely quantum case (1.4) for SQ(x). Analogously, equation (4.10) leads to (2.26) and (1.5) for RC (z) and RQ(x), respectively. Before closing this section, we emphasize that the hybrid vector field (4.8) cannot be directly used to construct a probability current for the hybrid quantum–classical system. Indeed, while the vector field X X(Γ) transports the density 2 = Υ 2 appearing in (3.20), it does not ∈ R | | transport the hybrid probability density, which instead must be constructed from the hybrid operator-valued density (3.16). This topic is developed in the second part of the paper; see equation (5.2) in Section 5.2.

4.2 Hybrid Bohmian trajectories As commented at the end of Section 3.4, the absence of classical particle trajectories in hybrid dynamics raises the question whether a Hamiltonian ﬂow can still be deﬁned as incorporating the motion of the classical subsystem. In this section, a positive answer is provided in terms of Lagrangian paths extending quantum Bohmian trajectories to hybrid quantum–classical systems. For later use, it will be convenient to introduce the shorthand notation (z, x) Γ, ∈ where Γ := T ∗Q M × represents the hybrid quantum–classical coordinate manifold. Evidently, this is a volume manifold with volume form given by µ =Λ µ . (4.11) Γ ∧ Although equation (4.9) is not in the typical hydrodynamic form, the hybrid Madelung equations (4.9)-(4.10) still lead to a similar continuum description to that obtained in the

22 quantum case. For example, the density equation (4.10) still yields hybrid trajectories, which may be defined by considering the evolution equation 2(t)=( 2 Φ(t)−1)/ Jac(Φ(t)), where R R0 ◦ Φ(t) is the flow of the vector field X and Jac(Φ) is the Jacobian relative to the volume form p (4.11) on Γ. Then, this flow is regarded as a Lagrangian trajectory obeying the equation d Φ(t, z, x)= X(Φ(t, z, x)) , (4.12) dt which is the hybrid quantum–classical extension of the quantum Bohmian trajectories [4, 67] in (1.7). In turn, hybrid Bohmian trajectories (4.12) are also useful to express (4.5) in the form d (t, Φ(t, z, x)) = L (t, Φ(t, z, x)) , dtS which is the hybrid analogue of the classical phase evolution (2.28). Additionally, in the absence of classical degrees of freedom, this picture recovers the quantum Bohmian trajectories since in that case the coordinate z plays no role. At this point, we address the question of how the symplectic property (2.31) of the flow is affected by quantum–classical coupling. In other words, here we shall unfold the geometric features underlying the dynamics of the classical canonical symplectic form ω = dqi dp . ∧ i By construction, we observe that the phase-space components XHI of the hybrid vector field X in (4.8) identify a Hamiltonian vector field parameterized by the coordinate x M, that is X M, X (T ∗Q) . Indeed, upon denoting by d the differential on T ∗Q, the∈ relation HI ∈ F ω z iX ω = dzHI follows from a straightforward verification. Then, the flowη ˜ of XH is symplectic HI I so that d η˜(t) ω =0 . (4.13) dt ∗ Therefore, despite the absence of point particle trajectories in quantum–classical coupling, a Hamiltonian flow preserving the classical symplectic form can still be defined as the flow ∗ associated to the Hamiltonian HI . Notice thatη ˜(t) M, Diffω(T Q) differs from the La- grangian trajectory Φ(t) Diff(T ∗Q M) of the hybrid∈F system (4.12), unless the quantum ∈ 2 × kinetic energy operator (~ /2m)∆x is dropped from the hybrid Hamiltonian (4.1). Indeed, − X in the latter case the hybrid vector field (4.8) drops to = (XHI , 0) and one is left with Φ(t)(z, x)=(˜η(t)(z, x), x), so that the hybrid Lagrangian trajectory Φ(t) is essentially equivalent to the pathη ˜(t).

4.3 Hybrid dynamics and symplectic form When the quantum kinetic energy is retained, the classical symplectic form is not preserved in the whole hybrid space T ∗Q M and here we shall present its corresponding dynamics. In particular, we have the following× result:

∗ Theorem 4.1 (Hybrid trajectories and symplectic form) Let πT ∗Q : Γ T Q be the standard projection map and let X X(Γ) be the hybrid vector field (4.8),→ with H given ∈ I by (4.3). Also, denote by Φ Diff(Γ) the flow of X as in (4.12) and define the two-form ∗ 2 ∈ Ω := πT ∗Q ω Ω (Γ) naturally induced on the hybrid coordinate space Γ. Then, we have the statements below.∈

∗ ∗ X 1. The mapping XHI : Γ T T Q given by XHI = T πT Q defines a parameterized Hamiltonian vector field,→ i.e. X M, X (T ∗Q) ; ◦ HI ∈F ω 23 ∗ ∗ 2. the map η˜ : Γ T Q given by η˜ := Φ πT ∗Q defines a parameterized symplectic diffeomorphism, i.e. →η˜ M, Diff (T ∗Q) , which identifies the flow of X ; ∈F ω HI 3. the following relation holds: d ∂2V Φ(t)∗Ω= d Φ(t)∗π∗ d V , with d(π∗ d V )= dqj dxk . (4.14) dt − M x M x ∂qj∂xk ∧ Proof. The first two statement follow from the discussion at the end of the previous section. ∗ For the third point, if X X(Γ) is the hybrid vector field (4.8) and πT ∗Q :Γ T Q, πM :Γ ∈ A ∗ → 1 → M are the projection maps, then we may consider the one-form := πT ∗Q Ω (Γ) naturally induced by the canonical one-form on T ∗Q and we have A ∈ A A ∗ £X = dLI + πM dxV, where dx denotes the differential on M. This relation is obtained by a direct verification as follows:

£XA = iXdA + d(iXA) ∗ ∗ = iXdπ ∗ + d(iXπ ∗ ) T QA T QA ∗ = iXπ ∗ ω + d(iX ) − T Q HI A ∗ = π ∗ dzHI + d(iX ) − T Q HI A ∗ = dHI + π dxHI + d(iX ) − M HI A ∗ = dLI + πM dxV. Inserting this relation in (4.9) yields

1 2 2 ∆x√D ∗ (∂ + £X) (d A)= d + ~ π d V, (4.15) t S − 2m |∇xS| √ − M x D which is equivalently written as d 1 Φ(t)∗ d A = d Φ(t)∗(L L ) Φ(t)∗π∗ d V. (4.16) dt S − 2m − I − M x Then, taking the exterior differential d on Γ of the relation (4.16) yields (4.14). The relation (4.14) describes the evolution of the classical canonical form under the whole hybrid flow Φ. As expected, the canonical symplectic form is not preserved by this flow unless 2 the quantum–classical coupling vanishes, that is ∂qjxk V = 0. Notice that, in local coordinates, A i ∗ i = pidq and πM dxV = ∂xi V dx , so that (4.16) produces the following equation for the Poincaréintegral in the hybrid coordinate space Γ:

d i ∂V i pi dq = i dx dt ˛γ(t) − ˛γ(t) ∂x where γ(t)=Φ(t) γ and γ is a time-independent loop in Γ. ◦ 0 0 In summary, we have found that, although the classical canonical form is not preserved by the overall hybrid flow Φ(t) of X X(Γ), it is preserved by the Hamiltonian flowη ˜(t) M, Diff (T ∗Q) of X M, X(∈T ∗Q) , given by the first component of the hybrid flow∈ F ω HI ∈ F Φ(t)(z, x)=(˜η(t)(z, x),ζ(t)(z, x)). In turn, the flowη ˜(t) determines an important subgroup of the group Diff(Γ) s (Γ,S1) advancing the full hybrid dynamics. F 24 Lemma 4.2 (A subgroup of the hybrid flow) We have the following group inclusion

iθ+i z (A−η∗A) M, Diff (T ∗Q) Diff(Γ) s (Γ,S1), (η, eiθ) η,˜ e z0 (4.17) F ω ⊂ F 7→ ´ ∗ 1 where η d(M, Diffω(T Q)), θ (M,S ), and with η˜ Diff(Γ) defined by η˜(z, x) = (η(x)(z), x∈). F The associated Lie algebra∈ F inclusion reads ∈

(ι : (Γ) M, X (T ∗Q) ֒ X(Γ) s (Γ F ≃ F ω → F ξ (Xξ, ξA(z0)) (Xξ, 0), ξA . (4.18) ≃ −b 7→ − Proof. The group inclusion is more naturally written by inserting an intermediate subgroup as follows

M, Diff (T ∗Q) M, Diff(T ∗Q) s (T ∗Q, S1) Diff(Γ) s (Γ,S1). F ω ⊂F F ⊂ F ∗ ∗ ∗ 1 The first inclusiond is associated to the group inclusion Diffω(T Q) Diff(T Q) s (T Q, S ), which follows from (2.7) and (2.14). The second inclusion is (η,⊂ f) (˜η, f˜) withF η ˜(x, z) = 7→ (x, η(x)(z)) and f˜(x, z)= f(x)(z), for η(x) Diff(T ∗Q)d and f(x) (T ∗Q, S1), which can be checked to be a group homomorphism. The∈ composition of these gro∈Fup inclusions yields (4.17). ∗ The Lie algebra isomorphism (Γ) M, Xω(T Q) given by ξ (Xξ, ξA(z0)) follows from the Lie algebra isomorphismF (≃2.17 F), where we identify (Γ) with≃ (−M, (T ∗Q)). F F F Taking the derivative of (4.17) at the identity yieldsb the Lie algebra inclusion (X , κ) ξ ∈ M, X (T ∗Q) ((X , 0), κ + ξ (z ) ξ ) X(Γ) s (Γ) similarly as computed in (2.18). F ω 7→ ξ A 0 − A ∈ F X ∗ By composing with the Lie algebra isomorphism (Γ) M, ω(T Q) we get the desired formula.b F ≃ F Section 5 will show that the group M, Diff (T ∗Q) is of fundamentalb importance in the § F ω probabilistic interpretation of the hybrid quantum–classical wave equation (4.2). In the next section, we shall study thed hybrid Madelung equations (4.9)-(4.10) in terms of its Hamiltonian structure, which arises naturally from the momentum map property of the Madelung transform [20].

4.4 Hamiltonian structure and equations of motion The equations (4.9)-(4.10) possess a Hamiltonian structure whose Lie-Poisson bracket is iden- tical to that governing compressible barotropic ﬂuids [46]. This is due to the fact that the mapping

J : L2(Γ) X(Γ)∗ Den(Γ) → × Υ J(Υ) = ~ Im(ΥdΥ)¯ , Υ 2 = 2d , 2 (4.19) 7→ | | R S R comprise a momentum map structure that is the hybrid analogue of its quantum and classical variants given by (1.8) and (2.32), respectively. Here, Den(Γ) and X(Γ)∗ denote respectively the space of densities and one-form densities on Γ = T ∗Q M. Upon identifying L2(Γ, C) with × the space of half-densities, the momentum map (4.19) is produced by the (left) representation

1 −iϕ/~ Υ Φ∗ e Υ (4.20) 7→ Jac(Φ) p 25 of the semidirect-product group Diﬀ(Γ) s (Γ) (Φ,ϕ). Since the momentum map (4.19) is equivariant,F ∋ it is also a Poisson map, thereby producing the Lie-Poisson structure on the dual of the semidirect-product Lie algebra X(Γ) s (Γ). More explicitly, upon deﬁning the one form σ = 2d and the density D = 2 on Γ, theF Lie-Poisson bracket reads R S R δf δh δh δf f , h LP(σ, D)= σ { } − ˆ j δσ ·∇δσ − δσ ·∇δσ Γ j j δf δh δh δf + D µ . (4.21) δσ ·∇δD − δσ ·∇δD Γ With the above bracket, the Hamiltonian functional producing the Madelung equations (4.9)- (4.10) is

1 σ 2 ~2 D 2 h(σ, D)= | x| + |∇x | DL + σ X µ , (4.22) ˆ 2m D 8m D − I z · HI Γ Γ where we have used the notation

σ = 2d = 2d , 2d =: (σ , σ ) R S R zS R xS z x to split classical and quantum components. Here, the symbols dz and dx denote the exterior diﬀerentials on T ∗Q and M, respectively. The Hamiltonian (4.22) is obtained by rewriting the Hamiltonian (3.9) in terms of σ and D. Then, equations (4.9)-(4.10) are obtained from a direct 2 veriﬁcation via the Poisson bracket formulation df/dt = f, h LP and by writing D = and { } R d = σ/D. The Hamiltonian structure (4.21)-(4.22) yields the equations (4.9)-(4.10) in the followingS Lie-Poisson form:

∂ ∂D + £X σ = DdL , + div(DX)=0 . (4.23) ∂t ∂t The general picture summarizing the various steps in hybrid quantum–classical mechanics is shown in Fig. 2. At this point, it may be interesting to project equation (4.9) for d = σ/D onto both its S phase-space and quantum components. This task can be quickly achieved by taking differentials of equation (4.7). First, we consider the phase function in (4.4) as a function on T ∗Q parameterized by the quantum coordinate, that is, instead ofS (T ∗Q M), we interpret ∗ S∈F × ∗ the phase as a function T Q, (M) . Then, defining Q := xS/m T Q, X(M) and using the Lie derivativeS∈F£ on MF leads to rewriting (4.7V) in the∇ equivalent∈F form VQ ∂ + £ = L + H , , ∂t VQ S { I S} so that taking the differential d yields ∂ + £ d = d (L X ). At a second x t VQ xS x − HI ·∇zS stage, we may proceed alternatively by setting M, (T ∗Q) . Then, defining S∈F F 2 ~2 x ∆x√D ∗ Q := |∇ S| M, (T Q) H 2m − 2m √D ∈F F

26 Hybrid wave equation (3.1) Hybrid density operator (3.16) Υ HCQ ∈ ∗ H / Den(T Q, Her( Q)) b h(Υ) = Υ H Υ Λ D ∈ ˆT ∗Q L Hamiltonian h( ) b D i~∂tΥ= b Υ b ✤ LH ▼▼▼ ✤ b b Momentum map (4.19) ✤ Υ= eiS/~ for the group See Section 5 1 ✤ R Diﬀ(Γ) s (Γ,S ) F ▼ ✤ ▼▼▼ ✤ ▼& Hybrid Madelung variables ∗ Hybrid Madelung (σ, D) X(Γ) s (Γ) equations (4.5)-(4.6) / ∈ F Hamiltonian h(σ, D) in (4.22) for ( , ) S R Hamiltonian system (4.23)

Figure 2: Schematic description of the various quantities appearing in quantum–classical mechanics and some of the mappings between them. The quantities in the boxes on the left will be related in Section 5. takes equation (4.7) into the equivalent form ∂ + £X + Q = LI , ∂t HI S H ∗ ∗ where we recall that XH M, X(T Q) and £X is the Lie derivative on T Q. Then, I ∈ F HI taking the diﬀerential dz of the above yields (∂t + £X )dz = dz(LI Q). Then, the HI Lie-Poisson equations (4.23) are equivalently written as S − H ∂ + £ d = d (L X ) , (4.24) ∂t VQ xS x − HI ·∇zS ∂ + £X dz = dz(LI Q) , (4.25) ∂t HI S − H ∂ + £X D =0 . (4.26) ∂t Despite its similarities with standard quantum hydrodynamics, equation (4.24) diﬀers from the

first in (1.6) in that it retains the important term XHI z , which persists in the absence of quantum–classical coupling. This is due to the fact that·∇ S the function (t, z, x) is not the quantum phase, but rather it is a phase-like quantity associated to the compoundS quantum– classical system evolving along the hybrid Lagrangian trajectory Φ with Eulerian velocity X = (X , /m). In order to understand the information carried by equation (4.25), we are going HI ∇xS to rewrite it in a more convenient form. Similarly to above, we may redefine the canonical one- i ∗ form = pidq as a one-form on T Q parameterized by the quantum coordinate space, that A 1 ∗ is M, Ω (T Q) , and we observe that £X = dzLI so that XH is Hamiltonian, i.e. A∈F HI A I X M, X (T ∗Q) . Consequently, equation (4.25) may also be rewritten as HI ∈F ω ∂ + £X (dz )= dz Q (4.27) ∂t HI S−A − H 27 which implies ∂ + £ d = 0, thereby recovering (4.13) for the flowη ˜(t) of X . t XHI z HI While the quantum–classicalA Madelung transform has been characterized in its geometric content and related to hybrid Bohmian trajectories, the second part of this paper considers the role of the Madelung transform in the context of probability densities. The latter will be studied in terms of their continuity equation and the associated quantum–classical currents. As we shall see, this study will establish a relation between the two boxes on the right in Fig. 2.

5 Hybrid quantum–classical densities and currents

While the previous sections presented the main geometric properties of the hybrid Madelung equations (4.9)-(4.10), here we want to focus on their physical interpretation in terms of probability densities and currents.

5.1 General comments As presented in Section 3.1, the general hybrid density associated to the quantum–classical wavefunction in (3.1) is given by the operator-valued distribution (z) in (3.16). At present, D there is no criterion available to establish whether the dynamics of (z) preserves its sign [6], unless one considers the trivial case in which the quantum–classicab Dl interaction is absent.

Indeed, in the latter case, the hybrid wave equation (3.1) with H = HbQ + HC produces the following evolution equation for : D b b ∂ = i~−1[H , ]+ H , . (5.1) tDb − Q D { C D}

Here, we have assumed a potentialbV (q, x)= bVQ(xb)+VC (q) inb (4.1) so that the hybrid Hamilto- nian is written as H(q,p)= HQ + HC (q,p), where the subscripts Q and C refer respectively to quantum and classical. It is obvious that the evolution (5.1) preserves the sign of , which then D remains positive-definiteb in time.b Indeed, upon using the notation in (3.23)-(3.22), equation † (5.1) leads to the sign-preserving evolution (t) = U(t)(η(t)∗ 0)U(t) , with i~ dbU/dt = HQU and dη/dt = X η(t)−1. D D HC ◦ However, in the general case the equationb of motionb for b isb sensibly moreb complicatedb b as it involves the hybrid wavefunction Υ as well as its gradientsD [6]. Then, the study of the evolution of the sign of becomes very challenging. So far,b all we know is that currently D the hybrid theory in Section 3.1 is the only Hamiltonian quantum–classical correlation theory that 1) retains the quantumb uncertainty principle (since (3.19) is positive definite) and 2) allows the mean-field factorization as an exact solution in the absence the quantum–classical coupling. On the other hand, no statement is yet available on the sign of the classical density ρ (z) = Tr( (z)) and one is led to consider the possibility that ρ may assume negative values. c D c Following Feynman’s work [19], this point was justified in [6] by using arguments involving the Wigner functionb for a harmonic oscillator coupled to a nonlinear quantum system: even in that simple case, the oscillator distribution must be allowed to acquire negative values. Still, in [6] an example of hybrid dynamics was provided in which the classical density remains positive. Then, the question arises of characterizing possible cases in which the classical positivity is preserved in time.

28 5.2 Hybrid density function as a momentum map Instead of considering the evolution of the classical density ρ as it arises from the equation, c D− the remainder of this paper focuses on the dynamics of the diagonal elements b(z; x, x) of the KD kernel (3.18) of , which we denote as b D 2 2 (z, x) := b(z; x, x)= Υ(z, x) ∂ p Υ(z, x) + i~ Υ, Υ¯ (z, x) . (5.2) D b KD | | − pi i| | { } ֒ (Notice that the mapping is given by the dual of the vector space inclusion (Γ D 7→ D F → (T ∗Q, Her(H )) given by ξ(z, x) ξ(z) and underlying the hybrid Hamiltonian H (z, x). In F Q 7→ I terms of the polar form ofb the hybrid wavefunction (4.4), one has b = 2 + ∂ (p 2)+ 2, . (5.3) D R pi iR {R S} This quantity represents a joint density function for the position of the system in the hybrid coordinate space T ∗Q M, in such a way that the quantum and the classical probabilities deﬁned in (3.20) and (3.21× ) can be computed from as D

ρq(x)= (z, x) Λ and ρc(z)= (z, x) µ . (5.4) ˆT ∗Q D ˆM D Thus, finding the evolution equation of allows characterizing a hybrid current J such that D ∂t = div J. While this will be the subject of the next section, here we show how the D − ∗ quantity (5.2) is actually a momentum map for the action of the group M, Diffω(T Q) on the space H = L2(T ∗Q M) of hybrid wavefunctions. F QC × Given an element (˜η(x), eiκ(x)) M, Diff (T ∗Q) , its (left) action on ΥdH can be ∈ F ω ∈ QC constructed by suitably adapting the the van Hove representation (2.20) as follows: d i z Υ(z, x) Υ η˜−1(z; x), x exp κ(x)+ (˜η(x) ) . (5.5) 7→ −~ ˆ ∗A−A η˜(z0;x) Here, the notation is such thatη ˜(x) identifies a symplectic diffeomorphism z η˜(x)(z) = η˜(z; x). We shall drop the explicit dependence on the phase-space coordinates7→ where convenient. Then, the KvH construction summarized in Section 2 is naturally transferred to the case of parameterized transformations: the Lie algebra of M, Diff (T ∗Q) is the space F ω M,C∞(T ∗Q)) (T ∗Q M) of x dependent phase-space Hamiltonians endowed with F ≃ F × − the canonical Poisson bracket on T ∗Q, the dual space Den(T ∗Q Md) is the space of joint distributions on the hybrid space T ∗Q M, and the infinitesimal generator× of a parameterized × ∞ ∗ −1 Hamiltonian function ξ(z, x) M,C (T Q) is Υ i~ ξ(x)Υ, see Section 2.3. The momentum map for the action∈ (5.5 F ) is found from the relation7→ − L b 1 (Υ),ξ = Ω i~−1 Υ, Υ , (5.6) hD i 2 − Lξ where , denotes the duality pairing between (T ∗bQ M) and its dual Den(T ∗Q M) h i F ~ ×¯ × and the symplectic form is given by Ω(Υ1, Υ2)=2 Im Γ Υ1Υ2µΓ. Then, (5.6) leads to the momentum map ´ (Υ) = Υ 2 + ∂ (p Υ 2) + i~ Υ, Υ¯ , (5.7) D | | pi i| | { } which is the natural extension of expression (2.24) and recovers precisely (5.2).

29 The momentum map structure of the hybrid density provides much insight into the geometry of its evolution. For example, dropping the quantumD kinetic energy operator (~2/2m)∆ − x from the hybrid Hamiltonian (4.1) produces a classical Liouville equation parameterized by x, that is ∂ = H , , (5.8) tD { I D} which can be deduced from (5.3) by using that (4.5) and (4.10) reduce to ∂t = LI + HI , and ∂ 2 = H , 2 , respectively. Equation (5.8) does not come as a surprise,S since{ S} we tR { I R } already observed in Section 4.2 that dropping the quantum kinetic energy makes the hybrid

Lagrangian path Φ coincide with the ﬂow of the x dependent Hamiltonian vector ﬁeld XHI . In turn, given the characteristic nature of equation (−5.8), the latter possesses Klimontovich-like solutions of the form (z, x, t)= w(x)δ(z ζ(x, t)) , (5.9) D − where w(x) Den(M) and dζ/dt = X (ζ). Then, the classical Liouville density (5.4) reads ∈ HI

ρc(z, t)= w(x)δ(z ζ(x, t)) µ . ˆM − As shown in [31], this expression of the classical Liouville density identifies the left leg of a dual pair of momentum maps [64]. Furthermore, when the quantum kinetic energy is drpped in (4.1), a possibly relevant consequence of equation (5.8) is that the sign of the joint probability density is preserved in time D even if the same conclusion cannot be generally reached about the operator-valued density . Evidently, the sign of is also preserved in the absence of quantum–classical coupling, thaDt 2 D is when ∂ j k V = 0. In this trivial case, equation (5.1) preserves the sign of and thereforeb q x I D also the sign of its diagonal elements. At present, similar statements about sign conservation are unavailable in the more general case of the hybrid Hamiltonian (4.1). However,b it may still be interesting to write down the continuity equation for in order to characterize the corresponding quantum–classical current. This is the focus of thDe next section. We conclude this section by extending the discussion at the end of Section 2.5 to the case of hybrid quantum–classical dynamics. In analogy to the representations (2.5) and (2.20) and in agreement with Lemma 4.2, the representation (4.20) of Diff(Γ) s (Γ,S1) on H F QC reduces to the representation (5.5) when restricted to the subgroup M, Diff (T ∗Q) F ω ⊂ Diff(Γ) s (Γ,S1). Thus, their corresponding momentum maps J(Υ) = ~ Im(ΥdΥ)¯ , Υ 2 F | | in (4.19) and (Υ) in (5.7), are related by the dual map to the correspondingd Lie algebra inclusion ι : D(Γ) ֒ X(Γ) s (Γ) given in (4.18). Upon denoting (σ, D)=( 2d , 2) and F → F R S R recalling (5.3), this dual map ι∗ : X(Γ)∗ Den(Γ) Den(Γ) is computed as × → ι∗(σ , σ ,D)= D div (J D Jσ ) . (5.10) z x − z A − z Then, this enables us to write

ι∗[J(Υ)] = (Υ), for all Υ H , D ∈ QC which indeed provides an important relation between the momentum map (5.7) for the joint hybrid density and the momentum map J(Υ) in (4.19) associated to the hybrid Madelung transform. The overall picture is represented in Fig. 3, which extends Fig. 2 by presenting the role of the quantum–classical density function and its relations to the hybrid density operator and the Madelung variables (σ, D). D D b 30 Hybrid density operator (3.16) ∗ Den(T Q, Her(HQ)) D ∈ Hamiltonian h( ) b 8 D rrr rrr b rrr rrr rrr rrr Map (5.2), dual to the inclusion rrr ∗ H ((rr (Γ) ֒ (T Q, Her( Q rrr F →F rrr rrr rr Hybrid wave equation (3.1) H Hybrid density (5.2) Υ CQ Momentum map (5.7) ∈ for the subgroup / (Υ) Den(Γ) b h(Υ) = Υ H Υ Λ ∗ D ∈ ˆT ∗Q L M, Diﬀω(T Q) Equation (5.11) F ~ b O i ∂tΥ= H Υ b L d ▲▲▲ ▲▲▲ b ▲▲▲ Momentum map (4.19) Dual to the inclusion Υ= eiS/~ for the group 1 s (R Diﬀ(Γ) s (Γ,S ) ι : (Γ) ֒ X(Γ) (Γ F ▲ F → F ▲▲▲ ▲▲▲ ▲▲▲ ▲% Hybrid Madelung variables ∗ Hybrid Madelung (σ, D) X(Γ) s (Γ) equations (4.5)-(4.6) / ∈ F Hamiltonian h(σ, D) in (4.22) for ( , ) S R Hamiltonian system (4.23)

Figure 3: Extended version of Fig. 2 including the role of the hybrid density function and its accompanying geometric structures. D

5.3 The quantum–classical continuity equation Although a more geometric picture for the hybrid continuity equation ∂ = div J will be tD − developed in the next section, here we shall simply present the explicit expression of the hybrid current J(z, x) obtained by using the equations (4.5) and (4.10) when taking the time derivative of (5.3). This calculation is particularly simpliﬁed by noticing that all the terms involving HI and LI in (4.5) and (4.10) combine by construction into the right-hand side of equation (5.8). Thus, we can initially drop all the H terms from the equations (4.5) and (4.10) (as well as I − the LI term in (4.5)) and restore the corresponding term in the equation at a later stage. Upon applying− the Leibniz product rule and noticing that D−

√ 2 1 2 ∆x , R = divx , x , 2 R √ 2 {R ∇ R} R this process leads to ∂ = div J = div J div J , (5.11) tD − − z C − x Q

31 with the following classical and quantum component of the hybrid current J =(JC ,JQ):

J := X , (5.12) C D HI J := m−1 2 + ∂ (p 2 )+ 2 , ~2 , . (5.13) Q R ∇xS pi iR ∇xS {R ∇xS S} − {R ∇xR} We observe that the usual quantum continuity equation is written by simply integrating (5.11) over the phase-space coordinates and using ρ (x) = (z, x)Λ = 2(x, z)Λ, thereby q T ∗Q D T ∗Q R obtaining ´ ´ ∂ρq 1 2 = divx ( x )Λ , ∂t −m ˆT ∗Q R ∇ S whereas the classical density ρc(z)= M (z, x)µ evolves according to ´ D ∂ρc = HI , µ . ∂t ˆM { D}

While the geometric origin of the quantum current JQ will be considered in the next section, here we emphasize that the quantum current JQ is produced only by the quantum kinetic energy operator (~2/2m)∆ in the hybrid Hamiltonian (4.1), while H produces essentially classical − x I dynamics as we discussed in Section 5.2. Moreover, we point out that it is not known whether JQ can be divided by to form a well-deﬁned vector ﬁeld. This is only possible if does not D 2 D change its sign. As long as (~ /2m)∆x is retained in (4.1), the sign of is certainly preserved − 2 D in the absence of quantum–classical coupling (that is ∂qjxk VI = 0), as discussed at the end of Section 5.2. However, it is not known whether this happens also in the general case.

5.4 Hamiltonian structure

We have seen that the hybrid Hamiltonian h(Υ) = Υ, LHb Υ , for H given as in (4.1), can be written uniquely in terms of (σ, D), see (4.22). In order to characterize the continuity equation for , it is useful to express the hybrid equations in a wayb that makesb appear explicitly as D D an independent variable. To do this, we shall make use of the fact that Υ (Υ) in (5.7), or alternatively, (σ, D) (σ, D) in (5.10), are momentum maps and we will7→ apply D the following 7→ D lemma. This produces the explicit Poisson bracket governing the combined dynamics of Υ and . In this section, we prefer to express Υ in terms of the variables (σ, D). D Lemma 5.1 Consider a Poisson manifold (P, , P ), and an equivariant momentum map J : P g∗ with respect to a left canonical action of{ the} Lie group G on P . Then the map → P g∗ P, p (J(p),p) → × 7→ is a Poisson map with respect to the Poisson bracket , on P and the Poisson bracket { }P ∂g ∂f ∂f ∂g f,g = f,g + + f,g P , + , (5.14) { } { } { } − ∂p ∂σ P ∂p ∂σ P on g∗ P . In particular, given a Hamiltonian H : P R, if p(t) P is a solution of Hamilton’s equations× for H, then (ν(t),p(t))=(J(p(t)),p(t)) →g∗ P is a solution∈ of Hamilton’s equation ∈ × for h : g∗ P R with respect to the Poisson bracket (5.14), where h is a function such that h(J(p),p)=× H→(p), for all p P . ∈

32 Proof. This follows from a result of [41] (see Prop. 2.2 therein) stating that the map

(ν,p) g∗ P (ν + J(p),p) g∗ P (5.15) ∈ × 7→ ∈ × is a Poisson diﬀeomorphism sending the Poisson bracket , + + , P to the Poisson bracket { } { }∗ (5.14). Then, we note that (P, , P ) is a Poisson submanifold of (g P, , + + , P ) with respect to the inclusion p (0,p{),} as a direct computation shows. By composing× { } this{ inclusion} 7→ with the Poisson map (5.15), the result follows.

This result can be applied to P = HQC Υ endowed with the symplectic Poisson bracket. In ∋ ∗ this case the Lie group G = M, Diﬀω(T Q) acts canonically on the left as in (5.5) with associated equivariant momentumF map Υ (Υ) given in (5.7). Alternatively, we have the 7→ D following result. d

Theorem 5.2 (Hamiltonian structure) With the definitions (5.12)-(5.13) and (σ, D) := ( 2d , 2), the system comprised by equations (4.23) and (5.11) is Lie-Poisson for the semidi- R S R ∗ 1 rect-product group G s S, with G = M, Diffω(T Q) , S = Diff(Γ) s (Γ,S ), and the inclusion G S given in Lemma 4.2.F In particular, if ι : g ֒ s is theF corresponding Lie ⊂ → algebra inclusion and d ad∗ (σ, D)= £ σ + D f, div (Dv ) (5.16) (v,f) v ∇ x x is the infinitesimal coadjoint action for s with respect to the standard L2 duality pairing , : s∗ s R, then equations (4.23) and (5.11) are Lie-Poisson with respect to the bracketh· ·i × → δf δh δf δh δh δf f, h ( , σ, D)= , σ { } D ˆ D δ δ − j δσ ·∇δσ − δσ ·∇δσ Γ D D j j δf δh δh δf + D µ δσ ·∇δD − δσ ·∇δD Γ δh δh δf δf + , , ad∗ (σ, D) , , ad∗ (σ, D) (5.17) δσ δD ι(δf/δD) − δσ δD ι(δg/δD) and the Hamiltonian 1 σ 2 ~2 D 2 h(σ, D, )= | x| + |∇x | + H µ . (5.18) D ˆ 2m D 8m D I D Γ Γ

Proof. In Lemma 5.1, we choose P = X(Γ) s (Γ) ∗ (σ, D) endowed with the Lie- ∗F ∋ Poisson bracket (4.21). Then, G = M,Diffω(T Q) acts canonically from the left on P by the coadjoint action of Diff(Γ) sF (Γ,S1), suitably restricted to the subgroup G. See F the discussion at the end of Section 5.2. Noticed that the Poisson manifold P is the dual space s∗ of the semidirect-product Lie algebra s = X(Γ) s (Γ) of the automorphism group S = Aut(Γ S1) = Diff(Γ) s (Γ,S1) of the trivial circleF bundle Γ S1 Γ. Then, since G = M, Aut× (T ∗Q S1) isF a subgroup of S, the corresponding group× inclusion→ generates a F A × semidirect-product structure G s S, so that Lemma 5.1 leads to a Lie-Poisson bracket on the dual Lie algebra (g s s)∗. In this context, the momentum map associated to subgroup action of G on S is given by (5.10), that is the dual of the Lie algebra inclusion ι : g ֒ s. In this case, the infinitesimal generator associated to the Lie algebra element ξ g =→ (Γ) acting on (σ, D) P = s∗ reads ad∗ (σ, D), where ad∗ is the infinitesimal coadjoint∈ actionF (5.16). ∈ − ι(ξ) Using this, the Lie-Poisson bracket (5.14) on (Γ) s X(Γ) s (Γ) ∗ gives (5.17). F F 33 The hybrid Hamiltonian h(Υ) in (3.9), with H given as in (4.1), can be written in terms of (σ, D) and = ι∗(σ, D) as (5.18). Then, the bracket (5.17) yields the Lie-Poisson equations D ∗ b ∗ (∂tσ, ∂tD)= ad δh , δh (σ, D) adι δh (σ, D) − ( δσ δD ) − ( δD ) δh ∗ ∗ ∂t = , ι ad δh , δh (σ, D) , D − D δ − ( δσ δD ) n Do The first line above recovers the equations (4.23). This follows from evaluating δh/δσz = 0 and δh/δ = HI and recalling the expression ι(ξ)=((Xξ, 0), ξA) of the Lie algebra inclusion (see the endD of Section 5.2). On the other hand, the quantum–classical− continuity equation (5.11) emerges from the equation above by recognizing that the two terms on the right-hand side identify exactly theD− contributions of the classical and quantum currents as

∗ ∗ ,HI = divz JC and ι ad δh , δh (σ, D) = divx JQ . (5.19) {D } ( δσ δD ) The proof is completed after verifying the second equality as in Appendix A.

2 Notice that if the quantum kinetic energy operator (~ /2m)∆x is absent in the hybrid Hamil- tonian (4.1), then the Hamiltonian h(Υ) in (3.9) collectivizes− (in the sense of Guillemin and Sternberg [26]) with respect to the momentum map (5.7). Indeed, the previous expression (5.18) reduces to

h(Υ) = Tr(H(z) (z))Λ = HI (z, x) (z, x)µΓ. ˆT ∗Q D ˆΓ D In this case, the Lie-Poisson equationsb decoupleb thereby recovering the previous result (5.8).

5.5 A class of Hamiltonians preserving positivity In this section we identify an inﬁnite family of hybrid systems for which both the quantum density matrix and the classical Liouville density are positive in time. Indeed, while the quantum density matrix (3.19) is always positive-deﬁnite by construction, the sign of the classical Liou- ville density (3.21) requires further study. In this section, we shall consider hybrid Hamiltonians of the form H(z)= H(z, α) , (5.20) where α is a purely quantum observable, i.e. it is an Hermitian operator on HQ. Here, we assume that the dependence of H on αbis analytic. Asb we shall see, any hybrid wave equation (3.1) associatedb to the type of Hamiltonian (5.20) leads to the positivity of both quantum and classical densities. In what follows,b it is convenient to use Dirac’s notation so that, upon recalling the isomorphism (3.3), a hybrid state vector is denoted by

Υ(z) := Υ(z) H H . | i ∈ C ⊗ Q By a slight generalization of (5.20), the following statement replaces α by any set of mutually commuting quantum observables. b Proposition 5.3 Consider a hybrid Hamiltonian H(z) depending on a sequence αi i=1,...,N of mutually commuting quantum observables, that is { } b b H(z)= H(z, α ) , (5.21) { i} b 34 b where [αi, αj]=0 for all i, j = 1,...,N. Also, consider a solution Υ(z, t) of the associated quantum–classical wave equation (3.1) and deﬁne the projection Υ(z,α,t| )= iα Υ(z, t) . Then, h | i the signb ofb the joint distribution (z,α,t) := Υ(z,α,t) 2 ∂ p Υ(z,α,t) 2 + i~ Υ(z,α,t), Υ(¯ z,α,t) (5.22) D | | − pi i| | { } is preservede in time.

Proof. Since mutually commuting Hermitian operators share the same eigenvalue problem, here we loose no generality by restricting to the case (5.20) of only one quantum observable α Her(H ). Upon denoting Λ = i~ , the quantum–classical wave equation (3.1) reads ∈ Q − ∇z i~∂ Υ(z) = X (z, α) Λ Υ(z) L(z, α) Υ(z) , b t| i H · | i − | i where L(z, α)= X (z, α) H(z, α)= L (z)αn. Now, consider the spectrum of α, that A· H − b n n b is α α = α α and write | i | i P b b b b b b i~∂ α Υ(z) = α X (z, α) α′ Λ α′ Υ(z) α L(z, α) α′ α′ Υ(z) dα′ , (5.23) th | i ˆ h | H | i· h | i−h | | ih | i where α := α † for all α. The termb α L(z, α) α′ can be rewrittenb according to h | | i h | | i α L(z, α) α′ = L (z) α αn α′ h | | i n bh | | i n X ′ n ′ b = Ln(z)(α )b α α h | i n X = L(z,α′)δ(α α′) − = (z) X (z,α′) H(z,α′) δ(α α′) A · H − − ′ ′ ′ and by proceeding analogously one also has α XH(z, α) α = XH (z,α )δ(α α ). Then, upon writing Υ(z,α) := α Υ(z) , equation (5.23)h becomes| | i − h | i b i~∂ Υ(z,α)= X (z,α) ΛΥ(z,α) X (z,α) H(z,α) Υ(z,α) t H · − A· H − = Υ(z,α) . LH(z,α) At this point, we constructb the joint quantum–classical density (5.22) for the classical position z in phase-space and the quantum degree of freedom α. Upon following the same arguments as in Section 5.2, one shows that the joint density (z,α) is a momentum map L2(T ∗Q σ(α)) D × → Den(T ∗Q σ(α)), where σ(α) denotes the spectrum of α. Consequently, the hybrid density × (z,α) satisﬁes the Liouville equation e b D b b b e ∂t (z,α)= H, (z,α)= £XH , D { D} − D which indeed preserves the signe of (z,α). Thus,e if at the initiale time D (t =0,z,αe) 0 , (z,α) T ∗Q σ(α), (5.24) D ≥ ∀ ∈ × then (t,z,α) 0 for alle times and for all (z,α) T ∗Q σ(α). D ≥ ∈ × b e 35 b For example, (t = 0,z,α) is positive whenever the hybrid density operator (z) is positive- deﬁnite at theD initial time, that is D e b ψ (t =0, z) ψ 0 , ψ H . h |D | i ≥ ∀| i ∈ Q

Since the classical density in (5.4b) can be written as ρc(t, z)= (t, z, x)µ = (t,z,α)dα 0, then we obtain the following result: ´ D ´ D ≥ e Corollary 5.4 Assume that the hybrid density operator (z) is positive at the initial time, then D the density ρc is also positive at initial time and its sign is preserved by the quantum–classical wave equation (3.1) with Hamiltonian of the type (5.21).b In the simplest case, we can consider the position operator x on M = Rn such that [xi, xj] = 0; then, one recovers the results in Section 5.2 for the joint probability density (z, x). Analogously, one can consider the momentum operator p = i~ so that [p , p ]=0 D b− ∇x i j andb b construct a Hamiltonian of the type b b b H(z)= H(z, p) .

In this case the eigenvectors are k = (2π~)−n/2 eik·x/~ and Υ(z, k)= k Υ(z) are the Fourier b b b transforms | i h | i 1 −ik·x/~ Υ(z, k)= n Υ(z, x)e µ. √2π~ ˆM H Another case of possible interest is that of a ﬁnite-dimensional quantum Hilbert space Q, for which one repeats the same steps and eventually is left with

i~∂ Υ = Υ t n LH(αn) n so that the density

(z) := Υ (z) 2 ∂ p Υ (z) 2 + i~ Υ , Υ¯ (z) Dn | n | − pi i| n | { n n} ˜ satisﬁes the Liouvillee equation ∂t n(z) = H(z,αn), n(z) and thus the same conclusion as D { D } in the continuum case holds for the classical density ρ = . In the case H = C2 of c n Dn Q two-level quantum subsystems, a proofe of this result already appeared in [6]. P e 6 Conclusions

Despite the absence of classical particle trajectories in quantum–classical dynamics, this paper has addressed the problem of identifying a Hamiltonian flow governing the motion of the classical subsystem within the entire hybrid system. In more generality, hybrid Bohmian trajectories were identified by applying the Madelung-Bohm picture to the quantum–classical wavefunction Υ(q,p,x). In addition, the continuity equation (5.11) for the quantum–classical density (5.2) was presented explicitly, along with the hybrid current distribution (5.12)-(5.13) extending the probability current from standard quantum mechanics. The results in this paper shed a new light on the 40-year old problem of quantum–classical coupling. Indeed, while several general ideas about phenomenological aspects have emerged over a century of continuing efforts, a mathematical foundation of quantum measurement is still absent. A theory of quantum–classical coupling represents a relevant step forward as a

36 prelude to a measurement theory. For example, hybrid Bohmian trajectories may lead to a new understanding of the measurement process without the need of invoking the wavefunction collapse postulate, which is indeed avoided in the pilot-wave interpretation of standard quantum mechanics [4]. Alternatively, hybrid Bohmian trajectories may also be used to design new reduced models for nonadiabatic molecular dynamics (see [20] for a geometric hydrodynamic treatment thereof), of paramount importance in chemical physics. In this context, the difficul- ties of a full quantum treatment lead to the necessity of modeling nuclei as classical particles while retaining the full quantum treatment of electron dynamics. Such models are typically formulated by taking semiclassical limits of a full quantum treatment and in most cases this process suffers from not capturing the quantum backreaction beyond mean-field effects. As the quantum backreaction is intrinsically built in the approach formulated in this paper, hybrid Bohmian trajectories may serve as a point of departure for formulating closure models overcom- ing the issues present in conventional molecular dynamics simulations. We intend to develop this particular direction in the near future. The present hybrid theory is formulated by starting from the Koopman-van Hove equation for two classical particles and then applying a partial quantization procedure leading to the quantum–classical wavefunction Υ(q,p,x), where (q,p) are classical phase-space coordinates while x is the coordinate on the quantum configuration space. This wavefunction undergoes a unitary evolution generated by a hybrid Liouvillian operator associated to the quantum– classical Hamiltonian. The long-sought equivariance properties of hybrid Liouvillians under both quantum and classical transformations were studied in Section 3.2, which also presented a remarkable relation relating commutators and Poisson brackets. Moreover, Section 3.3 formulated a hybrid density operator extending the quantum density matrix to the quantum–classical setting; while the density matrix of the quantum subsystem is always positive-definite by construction, the hybrid quantum–classical density is generally allowed to be unsigned and this point was developed further in the second part of the paper. In Section 4, we applied the symplectic geometry of the Madelung transform to hybrid wavefunctions and obtained fluid-like Lagrangian paths providing a hybrid quantum–classical extension of the celebrated Bohmian trajectories in quantum mechanics. In the presence of quantum–classical coupling, the symplectic form on the classical phase-space is not preserved by the hybrid flow and explicit equations of motion were presented for the Poincaréintegral, which is no longer a dynamical invariant. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectories identify a Hamiltonian flow parameterized by the quantum coordinate. This flow is associated to the motion of the classical subsystem and it was indeed shown to preserve the classical symplectic form. In addition, the Hamiltonian structure of the hybrid Madelung equations was also characterized explicitly in terms of reduction by symmetry in Section 4.4. In the last part of the paper, the joint quantum–classical density is considered in terms of its underlying momentum map structure. A hybrid continuity equation was presented in Section 5.3, thereby identifying hybrid quantum–classical current mimicking the quantum probability current. The hybrid continuity equation (5.11) and its current distribution (5.12)-(5.13) were also shown to emerge from a Lie-Poisson Hamiltonian structure, which sheds more light on the geometry underling the hybrid density evolution. While the latter does not generally preserve the sign of the distribution, the paper concludes by characterizing an infinite family of hybrid systems preserving the sign of the classical probability density.

37 A Quantum component of the hybrid current

Here, we prove the second equality in (5.19). Using the expression of ι∗ in (5.10), we get

∗ ∗ ∗ δh δh £ δh ι ad δh , δh (σ, D)= ι σ + D , divx D ( δσ δD ) δσ ∇δD δσ x δh δh δh = divx D divz J divx D J £ δh σ + D . δσx − A δσx − δσ ∇δD z Since D δh/δσ = m−1 2d , the ﬁrst term is m−1 div 2d ). The second term is x R xS x R xS δh δh div J div D = div div J D = m−1 div ∂ (p 2 S) . − z A x δσ − x z A δσ x pi iR ∇x x x Then, we apply divz J to the z-component of £δh/δσσ, with δh/δσz = 0. We ﬁnd

δh δh δh δh div J div σ + σ = div div Jσ + J σ z δσ ⊗ z ∇z δσ · z x δσ ⊗ z ∇z δσ · x x x δh δh = div div Jσ + div J σ x z z ⊗ δσ z ∇z δσ · x x x 2 δh = div div J R + σ , x z ∇zS ⊗ m ∇xS x δσ x δh = m−1 div 2 , + σ , . x{R ∇xS S} x δσ x Finally, we compute δh δh div J D = D, . z ∇z δD δD The result follows by noting that

2 δh δh σx 1 σx 1 ∆x√D σx, + D, = σx, D, + D, δσx δD mD − 2m D 2 ( √D ) n o = div √D, √D x{ ∇x } and using the expression (5.13) for JQ.

B Proof of the equivariance lemma 3.4

In this Appendix, we present a proof the equivariance property (3.13) of the hybrid Liouvillian under strict contact transformations. Upon using the notation ϕ(z) deﬁned on the right-hand

38 side of (2.8), we write

† U i bU iθ Υ (z) (η,e θ)LA (η,e ) −1 = bU iθ Υ (η(z)) exp i~ ϕ(z) LA b(η,e ) = i~ A, Υ η−1 exp i~−1ϕ η−1 (η(z)) b ◦ − ◦ −1 −1 (η(z)) X b(η(z)) A(η(z)) Υ(z) exp i~ ϕ(z) exp i~ ϕ(z) − bA · A − − = i~ A, Υ η−1 (η(z)) + i~ A, exp i~−1ϕ η−1 (η(z))Υ(z) exp i~−1ϕ(z) { ◦ } { b − ◦ } XAb◦η(z) A η) (z)Υ(z) + dϕ(z) XAb◦η(z)Υ(z) −b A· − ◦ b · −1 −1 = b Υ (z) + i~ A η, exp i~ ϕ (z)Υ(z) exp i~ ϕ(z) + dϕ(z) X b Υ(z) LA◦η { b◦ − } · A◦η(z) −1 −1 = b Υ (z) i~ d exp i~ ϕ (z) X b (z)Υ(z) exp i~ ϕ(z) + dϕ(z) X b Υ(z) LbA◦η − b − · A◦η · A◦η(z) = b Υ (z). LbA◦η In theb third equality we used η∗ + dϕ = and we emphasize that the symplectic potential should not be confused with theA hybrid observableA A (T ∗Q, Her(H )). A ∈F Q

Acknowledgements. This material is partially basedb upon work supported by the NSF Grant No. DMS-1440140 while CT was in residence at the MSRI, during the Fall 2018 semester. In addition, CT acknowledges support from the Alexander von Humboldt Foundation (Hum- boldt Research Fellowship for Experienced Researchers) as well as from the German Federal Ministry for Education and Research. FGB is partially supported by the ANR project GE- OMFLUID, ANR-14-CE23-0002-01.

References

[1] Aleksandrov, I.V. The statistical dynamics of a system consisting of a classical and a quantum subsystem. Z. Naturforsch. 36a (1981), 902-908

[2] Barcel´o, C.; Carballo-Rubio, R.; Garay, L.J.; G´omez-Escalante, R. Hybrid quantum–classical formulations ask for hybrid notions. Phys. Rev. A 86 (2012), 042120

[3] Bates, S.; Weinstein, A. Lectures on the Geometry of Quantization. Berkeley Mathematics Lec- ture Notes. 8, AMS, Providence, 1997.

[4] Bohm, D. A Suggested interpretation of the quantum theory in terms of “hidden” variables, I. Phys. Rev. 85 1952, 166-179

[5] Bondar, D.I.; Cabrera, R.; Lompay, R.R.; Ivanov, M.Yu.; Rabitz, H.A. Operational dynamic modeling transcending quantum and classical mechanics. Phys. Rev. Lett. 109 (2012), 190403

[6] Bondar, D.I.; Gay-Balmaz, F.; Tronci, C. Koopman wavefunctions and quantum–classical correlation dynamics. Proc. R. Soc. A 475 (2019), 20180879.

[7] Bonet Luz, E.; Tronci, C. Geometry and symmetry of quantum and quantum–classical variational principles. J. Math. Phys. 56 (2015), no. 8, 082104

39 [8] Bonet Luz, E.; Tronci, C. Hamiltonian approach to Ehrenfest expectation values and Gaussian quantum states. Proc. R. Soc. A 472 (2016): 20150777

[9] Boucher, W.; Traschen, J. Semiclassical physics and quantum ﬂuctuations. Phys. Rev. D 37 (1988), 3522-3532

[10] Budiˇsi´c, N.; Mohr, R.; Mezi´c, I. Applied Koopmanism. Chaos 22 (2012), 047510.

[11] Cendra, H.; Marsden, J.E.; Pekarsky, S.; Ratiu, T.S. Variational principles for LiePoisson and HamiltonPoincar´eequations, Mosc. Math. J. 3 (2003), 833-867

[12] Chernoﬀ, P.R.; Marsden, J.E. Some remarks on Hamiltonian systems and quantum mechanics. In “Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science” Harper and Hooker (eds.) Vol. III, 35-53. Reidel Publishing Company, Dordrecht-Holland. 1976

[13] Chru´sci´nski, D.; Kossakowski, A.; Marmo, G.; Sudarshan, E.C.G. Dynamics of interacting classical and quantum systems. Open. Syst. Inf. Dyn. 18 (2011), 339

[14] de Broglie, L. La m´ecanique ondulatoire et la structure atomique de la mati`ere et du rayonnement. J. Phys. Radium 8 (1927), n. 5, 225-241

[15] de Gosson, M. Symplectic geometry and quantum mechanics. Birkh¨auser-Verlag, 2006.

[16] Della Riccia, G.; Wiener, N. Wave mechanics in classical phase space, Brownian motion, and quantum theory. J. Math. Phys. 6 (1966), 1372-1383

[17] Di´osi, L.; Gisin, N.; Strunz W.T. Quantum approach to coupling classical and quantum dynamics. Phys. Rev. A 61 (2000), 022108

[18] Dirac, P.A.M. Quantised singularities in the electromagnetic ﬁeld. Proc. R. Soc. A 133 (1931), n. 821, 60–72

[19] Feynman, R. Negative probability. In “Quantum Implications: Essays in Honour of David Bohm”. Hiley and Peat (eds.) 235–248. Routledge & Kegan Paul Ltd. 1987.

[20] Foskett, M.S.; Holm, D.D.; Tronci, C. Geometry of nonadiabatic quantum hydrodynamics. Acta. Appl. Math. 162 (2019), n. 1, 63–103.

[21] Foskett, M.S.; Tronci, C. Holonomy and vortex structures in quantum hydrodynamics. arXiv:2003.08664 (2020)

[22] Gay-Balmaz, F.; Tronci, C. Vlasov moment ﬂows and geodesics on the Jacobi group. J. Math. Phys. 53 (2012), 123502

[23] Gerasimenko, V. Dynamical equations of quantum–classical systems. Theor. Math. Phys. 50 (1982), 49-55

[24] Ghose, P. The unﬁnished search for wave-particle and quantum–classical harmony. J. Adv. Phys. 4 (2015), 236-251

[25] Grey, J.W. Some global properties of contact structures. Ann. Math. 69 (1959), n. 2, 421-450

[26] Guillemin, V; Sternberg, S. Symplectic Techniques in Physics. Cambridge University Press. 1984.

40 [27] G¨unther, P. Presymplectic manifolds and the quantization of relativistic particle systems. In “Diﬀerential geometrical methods in mathematical physics” (Proc. Conf., Aix-en- Provence/Salamanca, 1979), 383400, Lecture Notes in Math. 836, Springer, 1980

[28] Hall, M.J.W.; Reginatto, M. Ensembles on Conﬁguration Space. Springer. 2016

[29] Holm, D.D.; Marsden, J.E.; Ratiu, T.S. The Euler–Poincar´eequations and semidirect products with applications to continuum theories. Adv. Math., 137 (1998), 1-81

[30] Holm, D.D.; Schmah, T.; Stoica, C. Geometric Mechanics and Symmetry: From Finite to Inﬁnite Dimensions. Oxford University Press. 2009

[31] Holm, D.D.; Tronci, C. Geodesic Vlasov equations and their integrable moment closures. J. Geom. Mech. 1 (2009), n. 2, 181-208

[32] Hyt¨onen T.; van Neerven J.; Veraar M.; Weis L. Bochner spaces. In: Analysis in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 63. Springer, Cham, 2016.

[33] Ismagilov, R.S.; Losik, M.; Michor, P. A 2-cocycle on a group of symplectomorphisms. Mosc. Math. J. 6 (2006), 307-315

[34] Jauslin, H.R.; Sugny, D. Dynamics of mixed quantum–classical systems, geometric quantization and coherent states. In “Mathematical Horizons for Quantum Physics”. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 20, 65-96. World Scientiﬁc. 2010.

[35] Joseph, I. Koopman-von Neumann approach to quantum simulation of nonlinear classical dynamics. arXiv:2003.09980 (2020)

[36] Kapral, R. Progress in the theory of mixed quantum–classical dynamics. Annu. Rev. Phys. Chem. 57 (2006), 129-57

[37] Khesin, B., Misiolek, G.; Modin, K. Geometric hydrodynamics via Madelung transform. PNAS, 115 (2018), n. 24, 6165-6170.

[38] Kirillov, A.A. Geometric quantization. In “Dynamical Systems IV”, 139-176, Encyclopaedia Math. Sci., 4, Springer, 2001.

[39] Klein, U. From Koopmanvon Neumann theory to quantum theory. Quantum Stud.: Math. Found. 5 (2018), 219-227

[40] Koopman, B.O. Hamiltonian systems and transformations in Hilbert space. Proc. Nat. Acad. Sci. 17 (1931), 315

[41] Krishnaprasad, P.S.; Marsden, J.E. Hamiltonian structure and stability for rigid bodies with ﬂexible attachments, Arch. Rational Mech. Anal., 98 (1987), 71–93.

[42] Kostant, B. Line bundles and the prequantized Schr¨odinger equation. In “Colloquium on Group Theoretical Methods in Physics”. Centre de Physique Th´eorique, Marseille, June 1972, IV.1-IV.22

[43] Kostant, B. Quantization and unitary representations, In “Lectures in modern analysis and applications III”, 87–208. Lecture Notes in Math. 170, Springer, 1970

[44] Madelung, E. Quantentheorie in hydrodynamischer Form. Z. Phys. 40 (1927), n. 3-4, 322-326

[45] Marsden, J.E.; Ratiu, T.S. Introduction to Mechanics and Symmetry. Springer. 1998

41 [46] Marsden, J.E.; Ratiu, T.S.; Weinstein, A. Semidirect products and reduction in mechanics. Trans. Amer. Math. Soc. 281 (1984), 147-177

[47] Mauro, D. On Koopman-von Neumann waves. Int. J. Mod. Phys. A 17 (2002), 1301

[48] Montgomery, R. Heisenberg and isoholonomic inequalities. In “Symplectic Geometry and Math- ematical Physics”, P. Donato et al. (eds.) Birkh¨auser, Boston, pages 303–325, 1991.

[49] Ohsawa, T.; Leok, M. Symplectic semiclassical wave packet dynamics. J. Phys. A: Math. Theor. 46 (2013) 405201

[50] Ohsawa, T.; Tronci, C. Geometry and dynamics of Gaussian wave packets and their Wigner transforms. J. Math. Phys. 58 (2017), 092105

[51] Peres, A.; Terno, D.R. Hybrid quantum–classical dynamics. Phys. Rev. A 63 (2001), 022101

[52] Prezhdo, O.V.; Kisil, V.V. Mixing quantum and classical mechanics. Phys. Rev. A 56 (1997), 162-175

[53] Terno, D.R. Inconsistency of quantumclassical dynamics, and what it implies. Found. Phys. 36 (2006), 102

[54] Salcedo, L.L. Absence of classical and quantum mixing. Phys. Rev. A 54 (1996), n. 4, 3657-3660

[55] Sudarshan, E.C.G. Interaction between classical and quantum systems and the measurement of quantum observables. Pr¯aman. a 6 (1976), 117 [56] Sudarshan, E.C.G. Consistent measurement of a quantum dynamical variable using classical apparatus. arXiv:quant-ph/0402134 (2004)

[57] ’t Hooft, G. Quantummechanical behaviour in a deterministic model. Found. Phys. Lett. 10 (1997), 105-111

[58] Tronci, C. Momentum maps for mixed states in quantum and classical mechanics. J. Geom. Mech. 11 (2019), n. 4, 639-656

[59] van Hove, L. On certain unitary representations of an inﬁnite group of transformations. PhD Thesis (1951). Word Scientiﬁc 2001

[60] Viennot, D.; Aubourg, L. Schr¨odinger-Koopman quasienergy states of quantum systems driven by classical ﬂow. J. Phys, A: Math. Theor. 51 (2018), 335201

[61] von Neumann, J. Mathematical Foundations of Quantum Mechanics. Princeton University Press, 1955

[62] von Neumann, J. Zur Operatorenmethode in der klassischen Mechanik. Ann. Math. 33 (1932). 587-642

[63] Wallstrom, T.C. Inequivalence between the Schrodinger equation and the Madelung hydrodynamic equations. Phys. Rev. A 49 (1994), n. 3, 1613–1617

[64] Weinstein, A. The local structure of Poisson manifolds. J. Diﬀ. Geom. 18 (1983), 523-557

[65] Wheeler, J.A.; Zurek, W.H. Quantum Theory and Measurement. Princeton University Press. 1983

42 [66] Wilczek, F. Notes on Koopman von Neumann mechanics, and a step beyond. Unpublished. 2015

[67] Wyatt, R.E. Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics. Springer. 2005.