## Abstract

The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical observables are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest’s theorem is shown to be Lie–Poisson for a semidirect-product Lie group, named the *Ehrenfest group*. The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie–Poisson structure associated with another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models that have previously appeared in the chemical physics literature.

## 1. Introduction

The expectation value dynamics for quantum canonical observables *ψ*(** x**)|

^{2}. Then, one is led to look at statistical moments in order to obtain information about macroscopic quantities. Ehrenfest’s equations for expectation value dynamics read

*ρ*=

*ψψ*

^{†}for pure states). Ehrenfest dynamics has recently been considered from a geometric perspective [1] and the present paper uses similar geometric methods to identify its Hamiltonian structure.

The relation (1.1) can also be expressed in terms of the Wigner phase-space function *W*(** q**,

**) associated with**

*p**ρ*[2]. This is conveniently written in Dirac notation, so that in one dimension the ket |

*x*〉 is the eigenvector given by

*x*|:=|

*x*〉

^{†}. Upon generalizing to three dimensions, the Wigner function is associated to

*ρ*by the relations [3]

*Wigner transform of*, while the inverse mapping (second relation above) is called

*ρ**inverse Wigner transform*, or more often

*Weyl transform of*

*W*(

**,**

*q***). These mappings generate a one-to-one correspondence between operators and phase-space functions. The Wigner function is a quasi-probability distribution (i.e. it can be negative) and is sometimes simply called Wigner distribution. Its evolution reads ∂**

*p*_{t}

*W*={{

*H*,

*W*}}, where

*H*(

**,**

*q***) is the Wigner transform of**

*p**Weyl symbol*) and

*f*,

*g*}

_{c}=∂

_{x}

*f*⋅∂

_{p}

*g*−∂

_{x}

*g*⋅∂

_{p}

*f*. The explicit definition of the Moyal bracket operator is rather involved and is omitted here. However, we shall recall that, although it does not identify a Poisson structure (as the Leibniz product rule is not satisfied), the Moyal bracket defines a Lie bracket structure. In this framework, Ehrenfest’s theorem for the expectation values

*d*〈

**〉/**

*ζ**dt*=〈{{

**,**

*ζ**H*}}〉, where

**=(**

*ζ***,**

*q***) is the phase-space coordinate. At this stage, we recall another fundamental property of the Moyal bracket of two phase-space functions: whenever one of these two is a second-degree polynomial, the Moyal bracket drops to the classical Poisson bracket. Consequently, {{**

*p***,**

*ζ**H*}}={

**,**

*ζ**H*}

_{c}and the Wigner–Moyal formulation of Ehrenfest’s theorem reads

_{t}

*ϱ*={

*H*,

*ϱ*}

_{c}), except for the fact that the averages are computed with respect to the quantum Wigner distribution

*W*(whose evolution accounts for quantum non-commutativity).

In the particular case, when *H*(** ζ**)) is a quadratic polynomial, the Wigner equation coincides with the classical Liouville equation and it is solved by a Gaussian distribution on phase space. In this case, an initial Gaussian evolves in time by changing its mean and variance, so that the latter moment tends to zero in the formal limit

**z**:=〈

**〉. Then, one is led to conclude that the expectation values follow canonical Hamiltonian trajectories as long as an initial wavepacket keeps narrow in time, so that the expansion**

*ζ*Indeed, when *Ehrenfest time* and is related to the possible breakdown of classical-quantum correspondence in quantum chaos [7]. Then, a useful way to quantify the classical-quantum differences due to non-commutativity is through the difference between solutions of the classical Liouville and quantum Wigner equations (both initiated in the same state), respectively. In recent years, this has been attempted by comparing the dynamics of quantum and classical statistical moments [8,9], i.e. the moments of the quantum Wigner function and the moments of the classical Liouville distribution, respectively. Still, the relation between expectation value dynamics and canonical Hamiltonian motion remains unclear and this paper aims to shed some light on this point. For example, in [10] (see eqn (6.2) therein), the expectation values **z**=〈** ζ**〉 are shown to obey canonical Hamiltonian dynamics (even for non-Gaussian states) when the energy is expressed in terms of moments 〈(

**−**

*ζ***z**)

^{k}〉. However, this result still lacks a more fundamental description in terms of quantum state dynamics.

Without entering further the difficult questions concerning the interpretation of Ehrenfest’s equations, this paper unfolds their geometric properties and expresses the quantum evolution in the phase-space frame co-moving with the expectation values **z**(*t*). More particularly, if one writes the total energy *W*(** ζ**,

*t*) and the expectation values

**z**(that is

**z**. Interestingly, equation (1.3) is identical in form to the equation of motion of a classical particle in the Ehrenfest mean-field model of mixed classical-quantum dynamics [11,12]. To avoid confusion, it is important to point out that (1.3) does not mean that the expectation values follow classical trajectories, as they would be obtained by the classical limit of a quantum system. Indeed, while the canonical structure implies Hamiltonian trajectories of classical type, these trajectories do

*not*coincide with those of the classical physical system, which in turn would be obtained upon replacing

*h*(

**z**). This fact avoids the possibility of contradiction in equation (1.3), which is to be coupled to the evolution of the relative quantum distribution

In this paper, we shall focus on the geometric nature of the classical-quantum coupling that emerges from expectation value dynamics. The first goal is to present (in §2) a new formulation of Ehrenfest’s theorem in terms of a classical-quantum Poisson bracket that couples the classical canonical bracket to the Poisson structure underlying quantum dynamics. This classical-quantum bracket naturally incorporates classical and quantum mechanics as special cases. However, as it was briefly mentioned earlier, this work is not meant to provide a new interpretation of the classical-quantum correspondence. Rather, one of the targets of this paper is to express (in §3) the quantum dynamics in the phase-space frame co-moving with the expectation values, so that Ehrenfest equations possess a standard canonical form independently of the quantum state (not necessarily Gaussian). This canonical structure was found in [10] in the context of quantum cosmology: in this case, equation (1.3) is accompanied by the evolution of the moments ** ζ**〉 and 〈

*ζ***〉. These quantities enjoy an (equivariant) momentum map structure that confers them a Lie–Poisson bracket [14,15,16]. Then, this bracket is analysed in detail and related to the role of metaplectic transformations for the particular case of Gaussian wavepackets. Eventually, this construction is applied to provide energy-conserving Gaussian closure models. More particularly, the characterization of Gaussian state dynamics enables providing energy-conserving variants of previous Gaussian moment models [17,18] in which the role of energy conservation has previously posed some issues [19]. As a general comment, we emphasize that this paper does not deal with the convergence issues that may emerge in quantum mechanics. In particular, probability densities are assumed to decay sufficiently fast so that expectation values converge at all times.**

*ζ*## 2. Hamiltonian structure of Ehrenfest’s theorem

We start this section with a mathematical result that lies at the basis of equation (1.3): expectation values are momentum maps for the standard representation of the Heisenberg group *G*. For a symplectic vector space (*V*,*ω*) carrying a (symplectic) *G*-representation, the momentum map *J*(*z*) is given by 2〈*J*(*z*),*ξ*〉:=*ω*(*ξ*_{V}(*z*),*z*), for all *z*∈*V* and all *G*, *ξ*_{V}(*z*) denotes the infinitesimal generator of the *G*-representation and 〈⋅,⋅〉 is the natural duality pairing on *G*-representation on the quantum state space is the map *ξ*(*ψ*) is the infinitesimal generator of the group action *Φ*_{g}(*ψ*) (with *g*∈*G*). When *G* is the whole group of unitary transformations, the corresponding momentum map is given by ** z**=(

**,**

*q***), this action is given by**

*p*_{c}(up to a multiplicative normalization factor ∥

*ψ*∥

^{2}). Note that the momentum map

**J**(

*ψ*) induces an equivariant momentum map

*ρ*) denotes the standard trace of trace–class operators): in this case, the infinitesimal action of the Heisenberg group reads

The momentum map property of quantum expectation values suggests looking for the Poisson bracket structure of Ehrenfest’s equations. We recall that the latter have to be accompanied by the evolution of the quantum state, this being given by a wavefunction, a density matrix or its Wigner function. In order to find the Poisson bracket for the expectation values, we start with the following Poisson bracket [22] for the quantum Liouville equation *f* and *g* are formally defined as function(al)s on the space of Hermitian operators. This Poisson bracket returns the quantum Liouville equation as *δh*/*δρ* is a Hermitian operator and we identify vector spaces of linear operators with their dual spaces by using the pairing 〈*A*,*B*〉=*Re*〈*A*|*B*〉, where 〈*A*|*B*〉=Tr(*A*^{†}*B*) is the natural inner product. Since Ehrenfest dynamics advances both expectation values (denoted by *ρ*), we allow to consider functionals of the type *f* to depend on *ρ* both explicitly and through the expectation value *ρ*) and in (2.4) we have set Tr(*ρ*)=1, which is the standard normalization of the density matrix. Without this particular choice, the change of variables *f*,*g*}_{±}(*μ*)=±〈*μ*,[*δf*/*δμ*,*δg*/*δμ*]〉 (where *Ehrenfest algebra* and its underlying Lie group *Ehrenfest group* (here, *Ehrenfest bracket*: this is the first example of a classical-quantum bracket that couples the canonical Poisson bracket underlying classical motion to the Lie–Poisson bracket (2.2) underlying quantum Liouville dynamics. However, note that this Poisson bracket does not model the correlation effects occurring in the interaction of quantum and classical particles. Indeed, Poisson bracket structures modelling the backreaction of a quantum particle on a classical particle have been sought for decades and are still unknown despite several efforts [24,25,26,27,28,29]. Rather, the Ehrenfest bracket governs the classical-quantum coupling (middle term in (2.4)) between expectation value dynamics (first classical term in (2.4)) and quantum state evolution (last term in (2.4)) for the same physical system.

At this point, in order to write down explicit equations of motion, one has to find the expression of the total energy *ρ*, so that it can still be expressed as an expectation value. For example, the kinetic energy in one spatial dimension can be rewritten by using the relation **z** (and not explicitly on *ρ*). For example, in the simplest case of a free particle in one spatial dimension one writes **z** and *ρ* as independent variables (e.g. one has

Equations (2.5) and (2.6) were obtained in [23], upon postulating a specific variational principle, based on analogies with the variational structure of the Ehrenfest mean-field model of mixed classical-quantum dynamics. Then, the action principle postulated in [23] is justified here in terms of its corresponding Hamiltonian structure. More importantly, we have shown how these equations are totally equivalent to the quantum Liouville equation. Indeed, these equations were derived from the Poisson bracket (2.2) for the density matrix evolution, without any sort of assumption or approximation: the only step involved was rewriting the total energy as

Depending on the specific form of the classical-quantum Hamiltonian, equations (2.5) and (2.6) allow for two different limits. Indeed, while Ehrenfest’s theorem is obtained in the case

While equations (2.5) and (2.6) recover Ehrenfest’s theorem as a special case, it is important to remark that they carry a redundancy, in the sense that equation (2.5) is simply the expectation value equation associated with (2.6) (if the latter is interpreted as a nonlinear non-local equation). This redundancy is not new, since it already occurs in Ehrenfest’s original equations. However, the redundancy of equation (2.5) can be eliminated by expressing the quantum dynamics in the frame of the expectation values. As we shall see, this operation separates the expectation values from the fluctuations arising from quantum uncertainty. While doing this is not generally trivial when using wavefunctions or density matrices, it becomes rather straightforward when using Wigner’s phase-space description. We remark that changes of frames in the configuration space for quantum dynamics were studied in the past [30,31], while changing the phase-space frame comes here as a new concept. This is the topic of the following section.

## 3. Quantum dynamics in the frame of expectation values

As anticipated above, the splitting between quantum averages and fluctuations can be conveniently performed by writing the quantum dynamics in the frame of the expectation values. For this purpose, it is convenient to work in the Wigner phase-space formalism. Then, the equations (2.5) and (2.6) become
*H*_{CQ}(**z**,** ζ**) is the Weyl symbol of the classical-quantum operator

*f*,

*g*}

_{ζ}, while the Moyal bracket operates only on the phase-space quantum coordinates (so that {{

*f*,

*g*}}={{

*f*,

*g*}}

_{ζ}). Note the abuse of notation, as

*f*and

*g*are very different in (3.1) and in (3.2) (cf. (2.2) and (2.4)). Then, changing variables to

**z**. On the other hand, as already found in [10], the first equation in (3.4) implies that expectation values evolve along canonical Hamiltonian trajectories with Hamiltonian

Once quantum dynamics is expressed in the frame of expectation values, one can continue to compute moments **z**(*t*). For example, in terms of the density matrix

Note that, unlike the moment approach, the present method does not require the Weyl symbol of the original Hamiltonian to be analytic. As an illustrative example in one dimension, here we consider a unit mass subject to a step potential, so that the phase-space Hamiltonian is *H*(** ζ**)=

*p*

^{2}/2+

*μ*

*Θ*(

*q*), where

*μ*is a physical parameter,

*Θ*denotes Heaviside’s step function, and we have used the notation

**=(**

*ζ**q*,

*p*). Upon introducing the deviation coordinate

**z**(

*t*)=(

*q*(

*t*),

*p*(

*t*)). Then, the total energy

**z**(

*t*). In this framework, the latter evolve according to the first equation in (3.4), that is

*q*. While explicit solutions can be constructed by taking the Wigner transform of the usual wavefunction of a unit mass subject to a step potential, the detailed study of the nonlinear non-local evolution of

Note, the whole treatment proceeds analogously for classical Liouville dynamics upon replacing Moyal brackets with Poisson brackets. Once more, this means that the essential difference between classical and quantum statistical effects lies in the non-commutative terms (higher order in *γ* in phase-space moving with the Hamiltonian vector field

In the next section, we focus on Gaussian quantum states, thereby restricting to consider only second-order moments. More particularly, we shall characterize the Hamiltonian structure of Gaussian state dynamics in terms of expectation values **z** and covariance matrix

## 4. Hamiltonian structure of Gaussian quantum states

Once the dynamics of Ehrenfest expectation values has been completely characterized in terms of Poisson brackets, one may consider higher order moments. The moment hierarchy does not close in the general case, although it is well known that it does for quadratic Hamiltonians. In the latter case, the moment algebra acquires an interesting structure, which is the subject of the present section. Before entering this matter, we emphasize that quadratic Hamiltonians restrict to consider linear oscillator motion and so they are uninteresting for practical purposes. An interesting situation, however, occurs when the total energy 〈*H*〉 is computed with respect to a Gaussian state, so that higher moments are expressed in terms of the first two. This is the Gaussian moment closure for nonlinear quantum Hamiltonians.

Gaussian quantum states (a.k.a. squeezed states in quantum optics) have been widely studied over the decades in many different contexts, mostly quantum optics and physical chemistry. Recently, applications of Gaussian states in quantum information have also been proposed (see e.g. [36]). Generally speaking, a Gaussian quantum state is identified with a Gaussian Wigner function of the form
*N* is a normalizing factor, **z**=〈** ζ**〉 is the mean and

_{t}

*G*={{

*H*,

*G*}} for Gaussian states. We emphasize that the expression (4.1) incorporates the Wigner transform of Gaussian wavepackets [39,40] as a special case.

If the linear form ** w** and the quadratic form

*S*defining the quadratic Hamiltonian

*H*=

**⋅**

*ζ**S*

**+**

*ζ***⋅**

*w***are functions (possibly nonlinear) of the first and second order moments 〈**

*ζ***〉 and 〈**

*ζ*

*ζ***〉, then a Gaussian initial state will remain a Gaussian under time evolution by changing its mean and variance. In more generality, a nonlinear (analytic) Hamiltonian will produce a total energy**

*ζ**G*(

**), depending only on the first two moments. For convenience, we shall denote**

*ζ**X*=〈

*ζ***〉/2, so that**

*ζ**h*(

**z**,

*G*)=

*H*(

**z**,

*X*). The corresponding Poisson structure is easily found by using the chain rule relation

*H*(

**z**,

*X*):

The Poisson bracket (4.3) has appeared earlier in the literature [41,42] in the context of classical Liouville (Vlasov) equations. This is no surprise, as second order moments are associated with quadratic phase-space polynomials, which do not involve the higher-order non-commutative terms in the Moyal bracket (Gaussian quantum states undergo classical Liouville-type evolution). Under the change of variables *Φ*_{S}(** z**,

*φ*)=(

*S*

**,**

*z**φ*), where

*j*=1,2,3 and Tr denotes the matrix trace. For example, setting

*j*=1 and expanding yields (up to multiplicative factors)

These results come as no surprise. The relation between the Jacobi group and Gaussian states has been known for decades in the theory of coherent states [46,47], under the statement that wavepackets evolve under the action of the semidirect product ** z**,

*φ*) is an element of the Heisenberg group. This action is given by the pullback of the Wigner function

*W*(

**) by the phase-space transformation**

*ζ*The emergence of a Lie–Poisson bracket for the first and second moments is also not surprising. Indeed, this is due to the fact that the moment triple (〈1〉,**z**,*X*) is itself an equivariant momentum map for the action (4.6); see section III.C of [41]. (Here, we have formally denoted **z**,*X*) and this identification enables the description of Gaussian state dynamics in terms of coadjoint orbits.

## 5. Gaussian moment models and energy conservation

Note that one can rewrite the above dynamics in terms of the covariance matrix *Σ*=2*X*−**z****z**. This is easily done by restricting the bracket (3.3) to functions of the type *H*(**z**,*Σ*). This process yields the direct sum bracket
*h*=*h*(**z**,*Σ*):

Equations (5.2) can be directly applied to modify certain moment models that have previously appeared in the chemical physics literature [17,18,19]. This class of models suffers from lack of energy conservation in the general case [19], with possible consequent drawbacks on the time evolution properties. In references [17,18,19] and related papers on the same topic, a class of moment models was developed by adopting a Gaussian moment closure on the equations of motion for the expectation values

## 6. Conclusion and perspectives

Based on the Hamiltonian Poisson bracket approach, this paper has unfolded the geometric properties of Ehrenfest’s expectation value dynamics. More particularly, the search for the Hamiltonian structure of Ehrenfest’s theorem has produced a new classical-quantum Poisson structure that incorporates classical and quantum dynamics as special cases. The corresponding equations are Lie–Poisson for the Ehrenfest group

This paper has shown that the use of Wigner functions and the properties of the Moyal bracket are particularly advantageous for studying expectation values. Then, combining Poisson brackets with momentum map structures unfolds the geometry underlying quantum dynamics. For example, momentum map structures may also appear in quantum hydrodynamics, where local averages (e.g.

## Authors' contributions

E.B.L. focused mainly on §§4 and 5, while C.T. worked on the remaining parts of the paper.

## Competing interests

The authors have no competing interests to declare.

## Funding

Financial support by the Leverhulme Trust Research Project grant no. 2014-112, the London Mathematical Society grant no. 31320 (Applied Geometric Mechanics Network) and the EPSRC grant no. EP/K503186/1 is also acknowledged.

## Acknowledgements

The authors are indebted to Tomoki Ohsawa, Alessandro Torrielli and the anonymous referees for providing extensive and valuable feedback on these results. In addition, the authors are grateful to Dorje Brody, François Gay-Balmaz, Darryl Holm, David Meier, Juan-Pablo Ortega and Paul Skerritt for several discussions on this and related topics. This work was partially carried out at the Bernoulli Center of the Swiss Federal Institute of Technology in Lausanne: C.T. acknowledges hospitality during the program ‘Geometric Mechanics, Variational and Stochastic Methods’.

- Received November 8, 2015.
- Accepted April 8, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.