Momentum maps for mixed states in quantum and classical mechanics

This paper presents the momentum map structures which emerge in the dynamics of mixed states. Both quantum and classical mechanics are shown to possess analogous momentum map pairs. In the quantum setting, the right leg of the pair identifies the Berry curvature, while its left leg is shown to lead to more general realizations of the density operator which have recently appeared in quantum molecular dynamics. Finally, the paper shows how alternative representations of both the density matrix and the classical density are equivariant momentum maps generating new Clebsch representations for both quantum and classical dynamics. Uhlmann's density matrix and Koopman-von Neumann wavefunctions are shown to be special cases of this construction.

1 Pure vs. mixed states: quantum and classical The geometric setting of quantum mechanics has been attracting attention ever since the work of Kibble [35], who showed how Schrödinger's equation is a Hamiltonian system on the projective Hilbert space. Over the years, the geometric viewpoint of both pure and mixed states in quantum mechanics has been developed in several instances [5,13,14,17,40,44,53]. However, while the difference between pure and mixed quantum states is widely known, its classical correspondent is only rarely reported in the literature, see e.g. [15,50]. This difference is especially important when one considers the coexistence of quantum and classical systems. For example, in quantum molecular dynamics, the complexity of a full quantum treatment requires approximations which treat parts of the molecule (the nuclei) as classical particles interacting with a pure state wavefunction governing the electronic quantum dynamics [6,43].
This paper presents a correspondence between the geometric features underlying the dynamics of quantum and classical states in terms of momentum map structures. This is a relatively new perspective. Indeed, while the momentum map character of projection operators in quantum mechanics and point measures in the classical phase-space have long been known, a deeper investigation of the several other momentum maps appearing in quantum mechanics has begun only more recently. For example, in [48,49] momentum maps were used for multipartite systems to characterize entanglement, while in [8,9] momentum maps were related to expectation value dynamics. On the other hand, in the case of multipartite systems, partial traces of the density matrix have long been known to identify momentum maps [44].
In this paper, the concept of momentum map is applied to mixed states in both quantum and classical mechanics. While this section continues by reviewing the geometric setting of quantum and classical pure states, geometric extensions of the concept of quantum (and classical) mixture are presented later. In this generalized context, this paper shows that the celebrated Berry curvature [7] also identifies a momentum map, whose dynamics appears in recent molecular dynamics models [1,2] beyond the Born-Oppenheimer approximation [11,6]. In the last part of the paper, new momentum map structures are shown to recover and extend alternative representations of both quantum and classical mechanics, such as Uhlmann's density matrix of quantum states [53] and the Koopman wavefunction for classical dynamics [38].

Quantum states
Consider a physical system consisting of only one particle. In the quantum case, the particle dynamics may have two alternative descriptions depending on whether the system is in a pure or a mixed state. If the system is in a mixed state, then the particle dynamics is given in terms of a unit-trace positive-definite Hermitian operator ρ defined on the quantum Hilbert space H = L 2 (R 3 ) and obeying the quantum Liouville equation where H is the Hermitian Hamiltonian operator for the system. If we denote by S Q ⊂ Her(H ) the convex subset of density operators in the set of Hermitian operators, its extreme points define the pure states. The latter are realized in terms of projection operators of the type where ψ ∈ H is the usual wavefunction (normalized, ψ 2 = 1) satisfying the Schrödinger equation It is well known [14] that the map ψ → −i ψψ † is an equivariant momentum map for the left representation ψ → Uψ of operators U ∈ U(H ) in the unitary group U(H ) on the quantum Hilbert space. This is easily seen by considering the canonical symplectic structure on H , which is given as where ·|· denotes the standard inner product. The momentum map J : H → u(H ) * is given by the usual formula [41,30] for linear Hamiltonian actions, that is Here, we have used the following notation for the real-valued pairing and we have identified u(H ) * ≃ u(H ) via the inner product µ|ξ = Tr(µ † ξ) on operators.

Classical states
Let us now consider the situation in the classical case. For one-particle systems, mixed states are identified with probability distributions in the space of densities Den(R 6 ). The dynamics of a classical probability distribution is then given by the classical Liouville equation where {·, ·} denotes the canonical Poisson bracket on R 6 and H denotes the classical Hamiltonian function. Similarly to the quantum case, if we denote by S C ⊂ Den(R 6 ) the convex subset of positive-definite (probability) distributions in the set of phase-space densities, its extreme points define the pure states. In the classical setting, the latter coincide with point measures of the type f (q, p) = δ(q −q)δ(p −p), so that the Liouville equation (4) returns Hamilton's equationṡ for the motion of the single particle in the system. For further details, see [15] and [50]. Again, the mapping (q,p) → δ(q −q)δ(p −p) is the general case of an equivariant momentum map which first appeared in [42] and was later studied in [31,23,24]. The geometry underlying this momentum map is somewhat involved and goes back to van Hove's thesis [54] in 1951. As mentioned, the map (6) takes phase-space in the space Den(R 6 ) of densities, which is identified with the dual of phase-space functions in C ∞ (R 6 ). In turn, the latter space is a Lie algebra under the canonical Poisson bracket {·, ·} and this Lie algebra integrates to an infinite dimensional group that was studied in detail by van Hove. Previously called "contact transformations" by Dirac [18,19] (following Lie [39]), the elements of this group were later named strict contact transformations [25] as they apply only to autonomous Hamiltonian systems (i.e. with a time-independent Hamiltonian). This group is a central extension of canonical transformations (Hamiltonian diffeomorphisms, Diff Ham (R 6 )) (see e.g. [22,23,33]) by the real numbers and its group multiplication reads as follows Here, • denotes composition, * denotes pullback, and A = −p · dq is the canonical one form. This group possesses the natural left action on R 6 , given by (p, q) → η(p, q), and whose infinitesimal generator is (p, q) → X H (p, q) (here, X H denotes the Hamiltonian vector field generating the canonical transformation η). Notice that, in this case, the group action is not linear and thus relations such as the first equality in (3) cannot be used. However, (6) is easily seen to be a momentum map [31,23] upon verifying the Poisson bracket formula for nonlinear symplectic actions. Indeed, this relation is verified immediately by setting J(q,p) = δ(q −q)δ(p −p). In addition, this momentum map is manifestly equivariant since det ∇η = 1 and thus (J • η)(q,p) = η * J(q,p).
The above correspondence between pure and mixed states in quantum and classical mechanics is the departure point for this paper, which shows how the fundamental momentum maps reported above can be immediately generalized to yield momentum map pair structures in different contexts.

Mixed states and momentum maps
The setting outlined in the previous section can be immediately generalized to what is usually called mixtures in the context of quantum mechanical states. The same concept also applies to classical mechanics within the so-called Klimontovich method of kinetic theory [37]. Before beginning the discussion, it is important to remark that here we continue to consider a oneparticle system in both the quantum and the classical setting.

Quantum mixtures as momentum maps
In quantum mechanics, mixed states are often expressed in terms of mixtures of (non-orthogonal) pure states as follows: where ψ i ∈ H and evidently Tr ρ = k w k = 1. Here, the number N has nothing to do with the number of particles in the system, since here we only deal with one particle. In standard textbooks [46,55], the relation (9) is usually interpreted as a mixture of pure states for the one-particle system, for which w i indicates the probability of the k−th pure state ψ i in the mixture.
In order to unfold the momentum map features of (9), let us define the following symplectic form on the Cartesian product H × · · · × H : Then, it is immediate to see that the quantity −i ρ = −i k w k ψ k ψ † k identifies an equivariant momentum map of the type for the natural right action {ψ k } → {Uψ k } of unitary operators U ∈ U(H ). In turn, this momentum map leads to a sequence of Schrödinger equations for each pure state ψ k , that is This can be verified by simply replacing (9) in the quantum Liouville equation (1) for ρ. The above picture can be further extended by replacing the sequence {ψ k } k=1...N by a continuous family of wavefunctions ψ(r) parameterized by a set of coordinates r ∈ R n . In this setting, the normalization condition becomes ψ(r) 2 = 1 and the weights w k are replaced by the measure w(r) ∈ Den(R n ), so that (R n , w) becomes a volume vector space and (9) generalizes to ρ =ˆw(r) ψ(r)ψ † (r) d n r .
This type of expression has recently emerged in dynamical models for nonadiabatic molecular dynamics [33,20], where it determines the density operator for the electronic dynamics. As it is shown below, this expression determines the left leg of a dual pair of momentum maps underlying quantum dynamics. The momentum map character of the quantity −i ´w (r) ψ(r)ψ † (r) d n r is easy to see. If we denote by C ∞ (R n , H ) the set of wavefunctions in H that are parameterized by r ∈ R n , it suffices to construct the symplectic form to observe that the generalized mixture (12) identifies a momentum map for the natural right action ψ(r) → Uψ(r) of unitary operators U ∈ U(H ). In terms of matrix elements notation, this action reads It is obvious that the above picture can be generalized further to consider a sequence of volume forms {w k (r)d n r} on R n so that the Cartesian product C ∞ (R n , H ) × · · · × C ∞ (R n , H ) can be endowed with the symplectic form thereby leading to an equivariant momentum map associated to the density matrix which generalizes the previous expressions.

Klimontovich approach to classical mechanics
The arguments in the previous section transfer immediately to the classical setting. For example, the momentum map (5) immediately extends to the sampling distribution where again´f d 3 q d 3 p = k w k = 1. Analogously to the quantum case, one defines the following symplectic form on R 6N : determines a momentum map which coincides with (16). Again, notice that the number N has generally nothing to do with the number of particles: indeed, in our setting the system under consideration comprises only one particle whose probability density is given by the distribution f (q, p). As mentioned above, the expression (16) can be interpreted as a classical mixture in terms of a standard sampling process in statistics. Nevertheless, we observe that replacing (16) in the classical Liouville equation (4) does return a multi-body system obeying canonical equationsq as it is prescribed by the collectivization theorem of Guillemin and Sternberg [26].
In previous work [23,31], the author considered the following extension of the above construction. Upon replacing the sequence {(q k ,p k )} k=1...N by a continuous family of points (q(r),p(r)) parameterized by a set of coordinates r ∈ R n , one can construct the distribution where w(r) ∈ Den(R n ). Once again, one can construct a symplectic form [23,24] where J ab is the canonical symplectic form and X(R 6 ) denotes the space of vector fields on R 6 . Then, the relation (17) identifies an equivariant momentum map for the natural action To continue the analogy with the previous section, we can also generalize further the construction above to consider a sequence of volume forms {w k (r)d n r} on R n so that the Cartesian product C ∞ (R n , R 6 )×· · ·×C ∞ (R n , R 6 ) can be endowed with a suitable symplectic form thereby leading to the momentum map For example, expressions of this type were considered in [31], where they were also related to the singular solutions emerging in certain types of hydrodynamic PDEs, known as EPDiff equations [29].

Right actions and diffeomorphisms
In the previous sections, all momentum maps appearing in mixed states for both quantum and classical mechanics were associated to specific left actions of U(H ) and Diff Ham (R 6 ) × R, respectively. In the particular case when the representation spaces are C ∞ (R n , H ) and C ∞ (R n , R 6 ), with R n carrying the volume form w(r) d n r, additional momentum maps can be constructed by considering the pullback action of the group Diff vol (R n ) of volume-preserving diffeomorphisms of R n . In the case of classical mechanics, this fact led Marsden and Weinstein [42] to construct a dual pair of momentum maps underlying planar incompressible fluid flows. As reported also in the sections below, this construction has recently been developed further in [23,24], while the application to Liouville-type (Vlasov) equations was presented in [31]. The following section shows that an analogue construction also underlies quantum mixed states.

The Berry curvature as a momentum map
As mentioned above, the space C ∞ (R n , H ) comprising the wavefunctions ψ(r) (also known as electronic wavefunctions in molecular dynamics [6,43]) carries two different representations. On one hand, the group U(H ) of unitary operators acts from the left, thereby generating the momentum map associated to (12). On the other hand, the group Diff vol (R n ) of volumepreserving diffeomorphisms of R n carries a right (linear) action given by the pullback operation.
In turn, this momentum map is given as Here d is the differential on R n and the one-form is the celebrated Berry connection [7], so that the equivariant momentum map for the pullback representation of Diff vol (R n ) on C ∞ (R n , H ) is given by the Berry curvature B = dA. The proof that the mapping (19) is a momentum map is a direct verification upon applying the formula where the symplectic form Ω is given in (13) and the minus sign is now due to the fact that we are dealing with a right action. Here, the Lie algebra element ξ is given by a volume-preseving vector field acting on ψ(r) by Lie derivative, that is ψ → ı ξ dψ, where ı ξ denotes the insertion of a vector field into a one-form. Since ξ is such that div(wξ) = 0, then where ♭ is the index lowering (flat) operator, δ denotes the co-differential [3] and the two-form γ ∈ Λ 2 (R n ) is usually known in fluid dynamics as the stream-function. For more details, see [42,24]. Since we are in R n , we can drop the flat symbol by using the Euclidean metric. In the general case, the relation (22) defines a Lie algebra isomorphism between the space X vol (R n ) of incompressible vector fields and the space Λ 2 (R n )/R of two-forms modulo real numbers. At this point, it suffices to expand the right hand side of (21) to get where we have used the Hodge star operator * : Λ k (R n ) → Λ n−k (R n ) and the property A, δγ = dA, γ of the codifferential under the Hodge pairing. Then, since the space ξ ∈ X vol (R n ) is identified with γ ∈ Λ 2 (R n )/R, the dual space X vol (R n ) * of incompressible vector fields can be identified with the space of (vorticity) two-forms and thus the relation (21) returns the momentum map indeed coinciding with the Berry curvature B := dA. Here, it may be useful to remark that this picture may also be extended to the generalized case associated to the density matrix expression (15) without essential modifications. To summarize, the space C ∞ (R n , H ) of parameterized (electronic) wavefunctions is a representation space for two different groups, that is U(H ) (acting from the left) and Diff vol (R n ) (acting from the right). Both these groups carry Hamiltonian actions producing momentum maps summarized as follows: where the left leg corresponds to the relation (12) and the right leg is given by (19). Special cases of similar constructions are provided by dual pairs, in which the kernels of the two momentum maps enjoy a symplectic orthogonality condition [24,23,56]. For example, a different pair of momentum maps in the context of quantum mixed states was found to be a dual pair in [44]. In this case, ψ ∈ H = H (1) ⊗ H (2) and the partial traces ρ 2 = Tr H (1) ψψ † and ρ 1 = Tr H (2) ψψ † were found to identify momentum maps for the natural left actions of U(H (2) ) and U(H (1) ), respectively. Then, the momentum map pair u(H (2) ) * ← H → u(H (1) ) * was found to be a dual pair.
We do not know whether the momentum maps in (25) identify a genuine dual pair; this question is left for future work. In the classical case, a similar construction does lead to a dual pair of momentum maps [42,23,24] and this is reported in the following section.

The dual pair of classical mechanics
As discussed above, the group of volume-preserving diffeomorphisms has a natural pullback action on the space of parameterized wavefunctions. Likewise, in the classical setting the same group acts by pullback on the space C ∞ (R n , R 6 ) of generalized coordinates q(r),p(r) from Section 2.2. Since this representation is also Hamiltonian, it leads to an equivariant momentum map that is expressed as [23,24,31] q(r),p(r) → −d p a (r)dq a (r) = dq a (r) ∧ dp a (r) ∈ Λ 2 (R n ) ≃ X vol (R n ) * , which is the immediate classical analogue of the Berry curvature from the previous section. Then, we are left with a similar picture to that found in the quantum case, which may be summarized as follows: Here, the Lie algebra X Ham (R 6 ) × R of the group Diff Ham (R 6 ) × R can be identified with the Poisson algebra C ∞ (R 6 ) via the isomorphism [22] X H (q, p), γ → H(q, p) − H(0, 0) + γ , so that the dual space X Ham (R 6 ) * × R can be replaced by the space of densities Den(R 6 ). For further details about this identification and other features of the group Diff Ham (R 6 ) × R, we refer the reader to [33,22]. In the above construction, the left leg is given by the generalized Klimontovich solution (17), while the right leg is given as above. Interestingly enough, in the classical case, the momentum maps in (26) are known to produce a dual pair structure, as discussed in [23,24,31]. We conclude this section by emphasizing its main result: analogous momentum maps occur in both quantum and classical mechanics. While the left leg reproduces quantum mixtures and Klimontovich solutions (respectively, in the quantum and the classical case), the right leg yields the Berry curvature and its classical analogue. The next section will apply the momentum maps occurring in the quantum case to certain models currently used in molecular dynamics simulations.

Momentum maps in quantum molecular dynamics
This section unfolds how the above momentum maps for quantum mixtures appear in quantum chemistry, with special focus on molecular dynamics models. In this context, one wants to solve the Schrödinger equation for an ensemble of particles comprising a different number of nuclei and electrons. Given the computational costs of full quantum simulations, different strategies have been designed over almost a century to approximate the nuclei as classical particles while treating the electrons in a full quantum setting.

Born-Oppenheimer approximation and electron mixtures
In quantum molecular dynamics, the most celebrated model is the Born-Oppenheimer approximation [11]. This is based on the following decomposition for the molecular wavefunction: which is then replaced in the multi-body Schrödinger equation. Here, the treatment has been simplified to consider only one electron (coordinates x) and one nucleus (coordinates r). While χ(r, t) is a genuine wavefunction, ψ(x; r) is considered as an r−dependent wavefunction with respect to x. As such, one has the so-called partial normalization condition (PNC) In the context of molecular dynamics, the parameterized wavefunction ψ(x; r) is time-independent (adiabatic approximation) and is given as the fundamental eigenfunction of a specific Hamiltonian operator. Without introducing unnecessary details, it suffices to say that certain approximations are then adopted to solve the dynamics of χ(r, t) numerically. It is important to remark that, while χ and ψ are often referred to as nuclear and electronic wavefunction, respectively, these terms do not correspond to genuine pure states for the nucleus and for the electron. Indeed, as already noticed in [20,33] the molecular density operator ρ(r, x, r ′ , x ′ ) = χ(r)ψ(x; r)χ * (r ′ )ψ * (x ′ ; r ′ ) (29) yields the following expressions for the nuclear and electronic density matrices, respectively: ρ n (r, r ′ ) = χ(r)χ * (r ′ )ˆψ(x; r)ψ * (x; r ′ ) dx , ρ e (r, r ′ ) =ˆ|χ(r)| 2 ψ(x; r)ψ * (x ′ ; r) dr .
(30) Here, the explicit time dependence has been dropped for convenience. Since neither of these two operators is a projection, one concludes that neither the nucleus nor the electron are in a pure state and thus the word 'wavefunction' lacks physical sense in this context. Hence, both the nucleus and the electron are in a mixed quantum state. In particular, the electronic state (second expression above) is represented exactly by the the momentum map (12). This shows that the idea of a generalized mixture emerges naturally in molecular chemistry problems, although its occurrence has not been noticed before.
Nowadays, the Born-Oppenheimer approximation is often replaced by the adoption of more sophisticated methods in order to capture more dynamical features of the electronic motion. Indeed, from the second of (30), we notice that the dynamics of the electron density is entirely slaved to that of the wavefunction χ, which in turn is often approximated by semiclassical methods. More complete models are then necessary in order to capture nonadiabatic effects; that is, to overcome the adiabatic approximation.

Exact factorization and the Berry curvature
Over the last decade, a model due to Gross and collaborators [1,2] has been receiving increasing attention, although its roots are traced back to the works of von Neumann [55] and, in later years, of Hunter [32]. In essence, the parameterized wavefunction is promoted to be timedependent, so that the Born-Oppenheimer approximation (27) is replaced by along with the PNC (28), which now becomes ψ(r, t) 2 =´|ψ(x, t; r)| 2 dx = 1. The dynamical model resulting from the above solution ansatz for the two-body Schrödinger equation is quite involved, although very rich in geometric content as recently presented in [20], where analogies with complex fluid models were also disclosed. A crucial ingredient emerging in the exact factorization model is the dynamical Berry connection (20). Indeed, as outlined in [4,20], this quantity generates a Maxwell-like field thereby producing Lorentz forces in the equations of motion. Thus, the Berry curvature (24) (here, n = 3) plays an essential role in exact factorization dynamics. This is another manifestation of the emergence of momentum maps in molecular chemistry problems: the exact factorization model comprises the dynamics of both momentum maps in (25).
It may be important to remark that the presence of an electric field E (which in this case depends on both wavefunctions χ and ψ) leads to the Faraday-like equation [20] so that d dt The integral of the Berry curvature over a closed surface is then related to topological singularities that form in terms of double-valued expressions of the phase of ψ. In the context of the Born-Oppenheimer approximation, these singularities are related to the so called conical intersections between energy surfaces [6,43], although this aspect will not be covered in this paper. In the case of the exact factorization model, one is left with a picture in which phase singularities may be created by the dynamics and their evolution is an important aspect of the model (unlike the Born-Oppenheimer case, where singularities are fixed in time). The fact that topological singularities are given by the right leg of the momentum map pair (25) is another manifestation of the fundamental role played by momentum maps in mechanical systems.

Clebsch representations
The preceding sections have presented several types of momentum maps which emerge in both quantum and classical dynamical models. While these were already known in the classical setting [42,23,24,31], new momentum maps were found for the case of quantum dynamics. Generally speaking, all these momentum maps are examples of Clebsch representations [16,28,42]. The latter are defined as momentum maps defined on a symplectic manifold endowed with a canonical symplectic form, which is the case for the representation spaces considered so far. The concept of a Clebsch representation may actually lead one to consider special types of solutions for certain Lie-Poisson equations. This fact was first exploited by Clebsch himself in fluid dynamics [16], while the geometric construction underlying Clebsch representations was developed much later [42] in terms of momentum maps generalizing the original formulation of Clebsch canonical variables. In the cases considered before, it is clear that the Clebsch representations are provided by the left legs of the momentum map pairs in (25) and (26).
This picture allows the discovery of other types of momentum map solutions that are defined on different representation spaces carrying a canonical group action. For example, Koopman's wavefunction description of classical dynamics [38] has been attracting increasing attention (see e.g. [10,21,47]) due to its analogies to quantum mechanics. However, other types of Clebsch representations also appeared in the context of density matrix evolution. For example, in 1986 Uhlmann [53] presented an alternative representation of the density matrix in terms of the evolution of linear operators on the quantum Hilbert space. This kind of alternative representations in both quantum and classical mechanics is the subject of the next sections.

Uhlmann's quantum density operator
Within the context of holonomy in quantum dynamics, in 1986 Uhlmann [53] wrote the density operator in terms of an abstract linear operator W ∈ L(V, H ) from some vector space V (finite-or infinite-dimensional) to the quantum Hilbert space. More specifically, the density operator was written as It is clear that if V is trivial, then W reduces to a wavefunction ψ ∈ H . Otherwise, the density matrix (34) does not identify a pure state unless W † W = 1, that is ρ 2 = ρ. One of the purposes of this section is to show that (34) determines a Clebsch representation L(V, H ) → u(H ) * . This proof needs only two ingredients on L(V, H ): a canonical symplectic form and a Hamiltonian action of U(H ). The first is simply given by while the action U(H ) is given by Since the infinitesimal generator reads W → ξW , with ξ ∈ u(H ), we compute thereby leading to the momentum map W → −i ρ. Now, since the space L(V, H ) is canonical, W satisfies canonical Hamiltonian motion so that replacing (34) in the quantum Liouville equation yields the following Schrödinger-type equation on L(V, H ): Notice that, in the special case V = H , the operator W is a linear operator on the quantum Hilbert space H (that is W ∈ L(H )). In this particular case, the unitary group U(H ) carries the alternative representation W → UW U † , whose infinitesimal generator reads In this particular setting, the corresponding momentum map reads Notice, however, that this momentum map does not produce a Clebsch representation for the density operator ρ, since Tr[W, W † ] = 0 (upon assuming that the trace is indeed defined). Still, this last construction can be adopted to provide a generalized Clebsch representation for ρ that is defined on the Cartesian product H × L(H ). Indeed, upon importing the natural product symplectic form on H × L(H ), the Hamiltonian action produces the momentum map (ψ, W ) → −i ρ, with Here, Tr ρ = 1 and ρ > 0 are both preserved by the unitary evolution ρ = Uρ 0 U † , which in turn preserves also the purity of the state since ρ 2 − ρ = U(ρ 2 0 − ρ 0 )U. Then substitution of the above expression in the quantum Liouville equation yields the uncoupled equations It is not known whether this type of momentum map solutions of the quantum Liouville equation (1) may have any physical meaning. It is certainly true that the density operator in quantum mechanics is only defined up to a commutator and this observation might be used to formulate generalized theories of quantum mechanics. However, these are beyond the scope of this paper.

Wavefunctions in classical mechanics
In the classical setting, a Clebsch representation for the Liouville equation has been known since the early 80's [45] and it is essentially an immediate generalization of the Clebsch representation for the vorticity of planar incompressible flows. If (S, D) ∈ T * C ∞ (R 6 , S 1 ), then a Clebsch representation momentum map T * C ∞ (R 6 , S 1 ) → Den(R 6 ) is given as where we recall that {·, ·} denotes the canonical Poisson bracket. This momentum map is associated to the cotangent lift of the natural right action of Diff Ham (R 6 ) × R on C ∞ (R 6 , S 1 ), that is given by the pullback S → η * S with (η, κ) ∈ Diff Ham (R 6 ) × R. The next section shows how this construction applies to the Koopman-von Naumann formulation of classical mechanics [38,10].

Koopman-von Neumann classical mechanics
A similar structure as in (39) can also be found by considering the symplectic Hilbert space L 2 (R 6 ) with the symplectic form in (2), that is Here, we have introduced the notation z = (q, p) ∈ R 6 . In this case, the pullback action gives the momentum map Notice that the polar form ψ = √ De iS/ returns exactly the expression in (39). Then, we notice that replacing the Clebsch representation in the Liouville equation (4) yields the evolution equation for ψ, which can be written in the Schrödinger-like form i ∂ t ψ = L H ψ , The self-adjoint operator L H is called the Liouvillian and the ψ−equation in (43) is the Koopman-von Neumann (KvN) equation of classical mechanics [38,10]. However, we notice that the Clebsch representation (42) is not compatible with the normalization conditioń f = 1 and thus it is not a genuine representation of classical mechanics. However, since |ψ| 2 satisfies the Liouville equation, the KvN construction adopts the identification in place of (42). We note in passing that the quantity |ψ| 2 is itself another momentum map for the action ψ(z) → e −iθ(z)/ ψ(z) of local phases θ(z) ∈ C ∞ (R 6 , S 1 ) on the Hilbert space L 2 (R 6 ).

Koopman-van Hove classical mechanics
Here, we are left with a picture in which the KvN equation is Hamiltonian with symplectic form (40) and Hamiltonian functional h(ψ) = i ´H {ψ * , ψ}. The latter differs from the physical total energy, which instead would read´H|ψ| 2 by following the KvN prescription f = |ψ| 2 . This apparent inconsistency, was recently overcome in [21] by considering an alternative action of Diff Ham (R 6 ) × R on classical wavefunctions ψ ∈ L 2 (R 6 ). As reported in van Hove's thesis, this action is given by where A is the symplectic potential (for example, A = −p · dq) such that the canonical symplectic form on R 6 is given as ω can = dA. In turn, as shown in [21], this action produces the Clebsch representation momentum map While comprising both momentum maps appearing previously in this section, this representation has the advantage that´f =´|ψ| 2 = 1, for a suitably normalized wavefunction. In turn, replacing (46) in the Liouville equation (4) yields a modified version of the KvN equation (43) previously appeared in [12,52], that is Here, is the Lagrangian function, as it arises from the phase term in the group action (45). Notice that this group action produces the infinitesimal generator i L H , which in turn satisfies [L H , L K ] = i L {H,K} . The equation (47) is called here Koopman-van Hove (KvH) equation and the selfadjoint operator L H is called prequantum operator in prequantization theory [27]. As mentioned earlier, the KvH equation (47) already appeared in [12,52], although the relation (46) between the classical wavefunction ψ(z) and the Liouville density function was discovered only recently in [21]. The main relation between the KvN and KvH constructions is that KvH reproduces the KvN equation for the modulus D = |ψ| 2 , while it also carries the evolution for the phase. Indeed, the polar form ψ = √ De iS/ yields the relations [36] ∂ t S = {H, S} + L , ∂ t D = {H, D} .
The (unobservable) classical phase is then a fundamental ingredient of KvH theory, which therefore can be regarded as a completion of the KvN construction. The dynamics of the phase can be written in terms of A as follows. Using £ X H A = −dL (here, £ X H denotes the Lie derivative along X H ) leads to (∂ t + £ X H )(dS + A) = 0 , which produces the relation η * (dS + A) = const. Then, upon setting the constant to be A itself, we have the usual relation [41] dS = η * A − A. This is simply another manifestation of the evolution equation for the classical wavefunction ψ = e i −1´z 0 (η * A−A) η * ψ 0 (up to a global phase factor), as it emerges by formally integrating the KvH equation (47).
The KvH construction was recently used in [21] to formulate a classical-quantum wave equation for the Hamiltonian dynamics of hybrid classical-quantum systems. Such a formulation has been an open question for over 40 years, since Sudarshan's first proposal [51] of using KvN for modeling hybrid systems. The fact that the Clebsch representation (46) has finally led to a consistent Hamiltonian theory for classical-quantum dynamics is among the best successes of momentum map methods.

Conclusions
This paper has disclosed various types of momentum maps underlying both quantum and classical dynamics. While most of them were already known in the classical setting, new momentum map features were presented for mixed quantum states and it was shown how they emerge in dynamical models for molecular dynamics. As an example, we showed how the celebrated Berry curvature determines the right leg of a momentum map pair, whose left leg identifies the electronic density matrix in the Born-Oppenheimer approximation. It would be interesting to know whether the momentum map pair underlying quantum mixed states enjoys a dual pair property, similar to the classical case.
In the second part of the paper, we showed how new momentum maps produce different representations of both the quantum density matrix and the classical probability density. Indeed, Uhlmann's density matrix was recovered as a special example of this construction and it would be interesting to know whether its possible generalizations could be of any physical significance. In the classical case, the Koopman-von Neumann construction was completed to include the dynamics of the classical phase, thereby leading to the Koopman-von Neumann theory. In the latter case, KvH theory is being currently used for designing hybrid classical-quantum models.
The momentum maps which appeared in this paper are fundamental objects in both quantum and classical mechanics, since they are produced by the actions of the most general groups determining the equations of motion in each case (e.g. the unitary group and the strict contact transformations). For example, many other momentum maps can be reproduced from those in this paper by appropriate projections arising from the action of suitable subgroups. It is expected that the momentum maps presented here will open the way to the development of geometric tools for new models in quantum physics and chemistry. An example is provided by the recent work [20] on exact factorization models.