Koopman–von Neumann classical mechanics

The Koopman–von Neumann mechanics is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively.[1][2][3]

As Koopman and von Neumann demonstrated, a Hilbert space of complex, square integrable wavefunctions can be defined in which classical mechanics can be formulated as an operatorial theory similar to quantum mechanics.

History

Statistical mechanics describes macroscopic systems in terms of statistical ensembles, such as the macroscopic properties of an ideal gas. Ergodic theory is a branch of mathematics arising from the study of statistical mechanics.

Ergodic theory

The origins of Koopman–von Neumann (KvN) theory are tightly connected with the rise of ergodic theory as an independent branch of mathematics, in particular with Boltzmann's ergodic hypothesis.

In 1931 Koopman and André Weil independently observed that the phase space of the classical system can be converted into a Hilbert space by postulating a natural integration rule over the points of the phase space as the definition of the scalar product, and that this transformation allows drawing of interesting conclusions about the evolution of physical observables from Stone's theorem, which had been proved shortly before. This finding inspired von Neumann to apply the novel formalism to the ergodic problem. Already in 1932 he completed the operator reformulation of classical mechanics currently known as Koopman–von Neumann theory. Subsequently, he published several seminal results in modern ergodic theory including the proof of his mean ergodic theorem.

Definition and dynamics

Derivation starting from the Liouville equation

In the approach of Koopman and von Neumann (KvN), dynamics in phase space is described by a (classical) probability density, recovered from an underlying wavefunction – the Koopman–von Neumann wavefunction – as the square of its absolute value (more precisely, as the amplitude multiplied with its own complex conjugate). This stands in analogy to the Born rule in quantum mechanics. In the KvN framework, observables are represented by commuting self-adjoint operators acting on the Hilbert space of KvN wavefunctions. The commutativity physically implies that all observables are simultaneously measurable. Contrast this with quantum mechanics, where observables need not commute, which underlines the uncertainty principle, Kochen–Specker theorem, and Bell inequalities.[4]

The KvN wavefunction is postulated to evolve according to exactly the same Liouville equation as the classical probability density. From this postulate it can be shown that indeed probability density dynamics is recovered.

Dynamics of the probability density (proof)

In classical statistical mechanics, the probability density (with respect to Liouville measure) obeys the Liouville equation[5][6]

${\displaystyle i{\frac {\partial }{\partial t}}\rho (x,p,t)={\hat {L}}\rho (x,p,t)}$

with the self-adjoint Liouvillian

${\displaystyle {\hat {L}}=-i{\frac {\partial H(x,p)}{\partial p}}{\frac {\partial }{\partial x}}+i{\frac {\partial H(x,p)}{\partial x}}{\frac {\partial }{\partial p}},}$

where ${\displaystyle H(x,p)}$ denotes the classical Hamiltonian (i.e. the Liouvillian is ${\displaystyle i}$ times the Hamiltonian vector field considered as a first order differential operator). The same dynamical equation is postulated for the KvN wavefunction

${\displaystyle i{\frac {\partial }{\partial t}}\psi (x,p,t)={\hat {L}}\psi (x,p,t),}$

thus

${\displaystyle {\frac {\partial }{\partial t}}\psi (x,p,t)=\left[-{\frac {\partial H(x,p)}{\partial p}}{\frac {\partial }{\partial x}}+{\frac {\partial H(x,p)}{\partial x}}{\frac {\partial }{\partial p}}\right]\psi (x,p,t),}$

and for its complex conjugate

${\displaystyle {\frac {\partial }{\partial t}}\psi ^{*}(x,p,t)=\left[-{\frac {\partial H(x,p)}{\partial p}}{\frac {\partial }{\partial x}}+{\frac {\partial H(x,p)}{\partial x}}{\frac {\partial }{\partial p}}\right]\psi ^{*}(x,p,t).}$

From

${\displaystyle \rho (x,p,t)=\psi ^{*}(x,p,t)\psi (x,p,t)}$

follows using the product rule that

${\displaystyle {\frac {\partial }{\partial t}}\rho (x,p,t)=\left[-{\frac {\partial H(x,p)}{\partial p}}{\frac {\partial }{\partial x}}+{\frac {\partial H(x,p)}{\partial x}}{\frac {\partial }{\partial p}}\right]\rho (x,p,t)}$

which proves that probability density dynamics can be recovered from the KvN wavefunction.

Remark

The last step of this derivation relies on the classical Liouville operator containing only first-order derivatives in the coordinate and momentum; this is not the case in quantum mechanics where the Schrödinger equation contains second-order derivatives. [5] [6]

Derivation starting from operator axioms

Conversely, it is possible to start from operator postulates, similar to the Hilbert space axioms of quantum mechanics, and derive the equation of motion by specifying how expectation values evolve.[7]

The relevant axioms are that as in quantum mechanics (i) the states of a system are represented by normalized vectors of a complex Hilbert space, and the observables are given by self-adjoint operators acting on that space, (ii) the expectation value of an observable is obtained in the manner as the expectation value in quantum mechanics, (iii) the probabilities of measuring certain values of some observables are calculated by the Born rule, and (iv) the state space of a composite system is the tensor product of the subsystem's spaces.

Mathematical form of the operator axioms

The above axioms (i) to (iv), with the inner product written in the bra–ket notation, are

(i) ${\displaystyle \langle \psi (t)|\psi (t)\rangle =1}$,

(ii) The expectation value of an observable ${\displaystyle {\hat {A}}}$ at time ${\displaystyle t}$ is ${\displaystyle \langle A(t)\rangle =\langle \Psi (t)|{\hat {A}}|\Psi (t)\rangle .}$

(iii) The probability that a measurement of an observable ${\displaystyle {\hat {A}}}$ at time ${\displaystyle t}$ yields ${\displaystyle A}$ is ${\displaystyle \left|\langle A|\Psi (t)\rangle \right|^{2}}$, where ${\displaystyle {\hat {A}}|A\rangle =A|A\rangle }$. (This axiom is an analogue of the Born rule in quantum mechanics.[8])

(iv) (see Tensor product of Hilbert spaces).

These axioms allow us to recover the formalism of both classical and quantum mechanics.[7] Specifically, under the assumption that the classical position and momentum operators commute, the Liouville equation for the KvN wavefunction is recovered from averaged Newton's laws of motion. However, if the coordinate and momentum obey the canonical commutation relation, the Schrödinger equation of quantum mechanics is obtained.

Classical mechanics from the operator axioms (derivation)

We begin from the following equations for expectation values of the coordinate x and momentum p

${\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ,\qquad {\frac {d}{dt}}\langle p\rangle =\langle -U'(x)\rangle ,}$

aka, Newton's laws of motion averaged over ensemble. With the help of the operator axioms, they can be rewritten as

${\displaystyle {\begin{aligned}m{\frac {d}{dt}}\langle \Psi (t)|{\hat {x}}|\Psi (t)\rangle &=\langle \Psi (t)|{\hat {p}}|\Psi (t)\rangle ,\\{\frac {d}{dt}}\langle \Psi (t)|{\hat {p}}|\Psi (t)\rangle &=\langle \Psi (t)|-U'({\hat {x}})|\Psi (t)\rangle .\end{aligned}}}$

Notice a close resemblance with Ehrenfest theorems in quantum mechanics. Applications of the product rule leads to

${\displaystyle {\begin{aligned}\langle d\Psi /dt|{\hat {x}}|\Psi \rangle +\langle \Psi |{\hat {x}}|d\Psi /dt\rangle &=\langle \Psi |{\hat {p}}/m|\Psi \rangle ,\\\langle d\Psi /dt|{\hat {p}}|\Psi \rangle +\langle \Psi |{\hat {p}}|d\Psi /dt\rangle &=\langle \Psi |-U'({\hat {x}})|\Psi \rangle ,\end{aligned}}}$

into which we substitute a consequence of Stone's theorem ${\displaystyle i|d\Psi (t)/dt\rangle ={\hat {L}}|\Psi (t)\rangle }$ and obtain

${\displaystyle {\begin{aligned}im\langle \Psi (t)|[{\hat {L}},{\hat {x}}]|\Psi (t)\rangle &=\langle \Psi (t)|{\hat {p}}|\Psi (t)\rangle ,\\i\langle \Psi (t)|[{\hat {L}},{\hat {p}}]|\Psi (t)\rangle &=-\langle \Psi (t)|U'({\hat {x}})|\Psi (t)\rangle .\end{aligned}}}$

Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for the unknown ${\displaystyle {\hat {L}}}$ is derived

${\displaystyle im[{\hat {L}},{\hat {x}}]={\hat {p}},\qquad i[{\hat {L}},{\hat {p}}]=-U'({\hat {x}}).}$

(commutator eqs for L)

Assume that the coordinate and momentum commute ${\displaystyle [{\hat {x}},{\hat {p}}]=0}$. This assumption physically means that the classical particle's coordinate and momentum can be measured simultaneously, implying absence of the uncertainty principle.

The solution ${\displaystyle {\hat {L}}}$ cannot be simply of the form ${\displaystyle {\hat {L}}=L({\hat {x}},{\hat {p}})}$ because it would imply the contractions ${\displaystyle im[L({\hat {x}},{\hat {p}}),{\hat {x}}]=0={\hat {p}}}$ and ${\displaystyle i[L({\hat {x}},{\hat {p}}),{\hat {p}}]=0=-U'({\hat {x}})}$. Therefore, we must utilize additional operators ${\displaystyle {\hat {\lambda }}_{x}}$ and ${\displaystyle {\hat {\lambda }}_{p}}$ obeying

${\displaystyle [{\hat {x}},{\hat {\lambda }}_{x}]=[{\hat {p}},{\hat {\lambda }}_{p}]=i,\quad [{\hat {x}},{\hat {p}}]=[{\hat {x}},{\hat {\lambda }}_{p}]=[{\hat {p}},{\hat {\lambda }}_{x}]=[{\hat {\lambda }}_{x},{\hat {\lambda }}_{p}]=0.}$

(KvN algebra)

The need to employ these auxiliary operators arises because all classical observables commute. Now we seek ${\displaystyle {\hat {L}}}$ in the form ${\displaystyle {\hat {L}}=L({\hat {x}},{\hat {\lambda }}_{x},{\hat {p}},{\hat {\lambda }}_{p})}$. Utilizing KvN algebra, the commutator eqs for L can be converted into the following differential equations

[7][9]

${\displaystyle mL'_{\lambda _{x}}(x,\lambda _{x},p,\lambda _{p})=p,\qquad L'_{\lambda _{p}}(x,\lambda _{x},p,\lambda _{p})=-U'(x).}$

Whence, we conclude that the classical KvN wave function ${\displaystyle |\Psi (t)\rangle }$ evolves according to the Schrödinger-like equation of motion

${\displaystyle i{\frac {d}{dt}}|\Psi (t)\rangle ={\hat {L}}|\Psi (t)\rangle ,\qquad {\hat {L}}={\frac {\hat {p}}{m}}{\hat {\lambda }}_{x}-U'({\hat {x}}){\hat {\lambda }}_{p}.}$

(KvN dynamical eq)

Let us explicitly show that KvN dynamical eq is equivalent to the classical Liouville mechanics.

Since ${\displaystyle {\hat {x}}}$ and ${\displaystyle {\hat {p}}}$ commute, they share the common eigenvectors

${\displaystyle {\hat {x}}|x,p\rangle =x|x,p\rangle ,\quad {\hat {p}}|x,p\rangle =p|x,p\rangle ,\quad A({\hat {x}},{\hat {p}})|x,p\rangle =A(x,p)|x,p\rangle ,}$

(xp eigenvec)

with the resolution of the identity ${\displaystyle 1=\int dxdp\,|x,p\rangle \langle x,p|.}$ Then, one obtains from equation (KvN algebra)

${\displaystyle \langle x,p|{\hat {\lambda }}_{x}|\Psi \rangle =-i{\frac {\partial }{\partial x}}\langle x,p|\Psi \rangle ,\qquad \langle x,p|{\hat {\lambda }}_{p}|\Psi \rangle =-i{\frac {\partial }{\partial p}}\langle x,p|\Psi \rangle .}$

Projecting equation (KvN dynamical eq) onto ${\displaystyle \langle x,p|}$, we get the equation of motion for the KvN wave function in the xp-representation

${\displaystyle \left[{\frac {\partial }{\partial t}}+{\frac {p}{m}}{\frac {\partial }{\partial x}}-U'(x){\frac {\partial }{\partial p}}\right]\langle x,p|\Psi (t)\rangle =0.}$

(KvN dynamical eq in xp)

The quantity ${\displaystyle \langle x,\,p|\Psi (t)\rangle }$ is the probability amplitude for a classical particle to be at point ${\displaystyle x}$ with momentum ${\displaystyle p}$ at time ${\displaystyle t}$. According to the axioms above, the probability density is given by ${\displaystyle \rho (x,p;t)=\left|\langle x,p|\Psi (t)\rangle \right|^{2}}$. Utilizing the identity

${\displaystyle {\frac {\partial }{\partial t}}\rho (x,p;t)=\langle \Psi (t)|x,p\rangle {\frac {\partial }{\partial t}}\langle x,p|\Psi (t)\rangle +\langle x,p|\Psi (t)\rangle \left({\frac {\partial }{\partial t}}\langle x,p|\Psi (t)\rangle \right)^{*}}$

as well as (KvN dynamical eq in xp), we recover the classical Liouville equation

${\displaystyle \left[{\frac {\partial }{\partial t}}+{\frac {p}{m}}{\frac {\partial }{\partial x}}-U'(x){\frac {\partial }{\partial p}}\right]\rho (x,p;t)=0.}$

(Liouville eq)

Moreover, according to the operator axioms and (xp eigenvec),

${\displaystyle {\begin{aligned}\langle A\rangle &=\langle \Psi (t)|A({\hat {x}},{\hat {p}})|\Psi (t)\rangle =\int dxdp\,\langle \Psi (t)|x,p\rangle A(x,p)\langle x,p|\Psi (t)\rangle \\&=\int dxdp\,A(x,p)\langle \Psi (t)|x,p\rangle \langle x,p|\Psi (t)\rangle =\int dxdp\,A(x,p)\rho (x,p;t).\end{aligned}}}$

Therefore, the rule for calculating averages of observable ${\displaystyle A(x,p)}$ in classical statistical mechanics has been recovered from the operator axioms with the additional assumption ${\displaystyle [{\hat {x}},{\hat {p}}]=0}$. As a result, the phase of a classical wave function does not contribute to observable averages. Contrary to quantum mechanics, the phase of a KvN wave function is physically irrelevant. Hence, nonexistence of the double-slit experiment[6][10][11] as well as Aharonov–Bohm effect[12] is established in the KvN mechanics.

Projecting KvN dynamical eq onto the common eigenvector of the operators ${\displaystyle {\hat {x}}}$ and ${\displaystyle {\hat {\lambda }}_{p}}$ (i.e., ${\displaystyle x\lambda _{p}}$-representation), one obtains classical mechanics in the doubled configuration space,[13] whose generalization leads [13] [14] [15] [16] [17] to the phase space formulation of quantum mechanics.

Quantum mechanics from the operator axioms (derivation)

As in the derivation of classical mechanics, we begin from the following equations for averages of coordinate x and momentum p

${\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ,\qquad {\frac {d}{dt}}\langle p\rangle =\langle -U'(x)\rangle .}$

With the help of the operator axioms, they can be rewritten as

${\displaystyle {\begin{aligned}m{\frac {d}{dt}}\langle \Psi (t)|{\hat {x}}|\Psi (t)\rangle &=\langle \Psi (t)|{\hat {p}}|\Psi (t)\rangle ,\\{\frac {d}{dt}}\langle \Psi (t)|{\hat {p}}|\Psi (t)\rangle &=\langle \Psi (t)|-U'({\hat {x}})|\Psi (t)\rangle .\end{aligned}}}$

These are the Ehrenfest theorems in quantum mechanics. Applications of the product rule leads to

${\displaystyle {\begin{aligned}\langle d\Psi /dt|{\hat {x}}|\Psi \rangle +\langle \Psi |{\hat {x}}|d\Psi /dt\rangle &=\langle \Psi |{\hat {p}}/m|\Psi \rangle ,\\\langle d\Psi /dt|{\hat {p}}|\Psi \rangle +\langle \Psi |{\hat {p}}|d\Psi /dt\rangle &=\langle \Psi |-U'({\hat {x}})|\Psi \rangle ,\end{aligned}}}$

into which we substitute a consequence of Stone's theorem

${\displaystyle i\hbar |d\Psi (t)/dt\rangle ={\hat {H}}|\Psi (t)\rangle ,}$

where ${\displaystyle \hbar }$ was introduced as a normalization constant to balance dimensionality. Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for the unknown quantum generator of motion ${\displaystyle {\hat {H}}}$ are derived

${\displaystyle im[{\hat {H}},{\hat {x}}]=\hbar {\hat {p}},\qquad i[{\hat {H}},{\hat {p}}]=-\hbar U'({\hat {x}}).}$

Contrary to the case of classical mechanics, we assume that observables of the coordinate and momentum obey the canonical commutation relation ${\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar }$. Setting ${\displaystyle {\hat {H}}=H({\hat {x}},{\hat {p}})}$, the commutator equations can be converted into the differential equations [7][9]

${\displaystyle mH'_{p}(x,p)=p,\qquad H'_{x}(x,p)=U'(x),}$

whose solution is the familiar quantum Hamiltonian

${\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+U({\hat {x}}).}$

Whence, the Schrödinger equation was derived from the Ehrenfest theorems by assuming the canonical commutation relation between the coordinate and momentum. This derivation as well as the derivation of classical KvN mechanics shows that the difference between quantum and classical mechanics essentially boils down to the value of the commutator ${\displaystyle [{\hat {x}},{\hat {p}}]}$.

Measurements

In the Hilbert space and operator formulation of classical mechanics, the Koopman von Neumann–wavefunction takes the form of a superposition of eigenstates, and measurement collapses the KvN wavefunction to the eigenstate which is associated the measurement result, in analogy to the wave function collapse of quantum mechanics.

However, it can be shown that for Koopman–von Neumann classical mechanics non-selective measurements leave the KvN wavefunction unchanged.[5]

KvN vs Liouville mechanics

The KvN dynamical equation (KvN dynamical eq in xp) and Liouville equation (Liouville eq) are first-order linear partial differential equations. One recovers Newton's laws of motion by applying the method of characteristics to either of these equations. Hence, the key difference between KvN and Liouville mechanics lies in weighting individual trajectories: Arbitrary weights, underlying the classical wave function, can be utilized in the KvN mechanics, while only positive weights, representing the probability density, are permitted in the Liouville mechanics (see this scheme).

The essential distinction between KvN and Liouville mechanics lies in weighting (coloring) individual trajectories: Any weights can be utilized in KvN mechanics, while only positive weights are allowed in Liouville mechanics. Particles move along Newtonian trajectories in both cases. (Regarding a dynamical example, see below.)

Quantum analogy

Being explicitly based on the Hilbert space language, the KvN classical mechanics adopts many techniques from quantum mechanics, for example, perturbation and diagram techniques[18] as well as functional integral methods[19][20][21]. The KvN approach is very general, and it has been extended to dissipative systems,[22] relativistic mechanics,[23] and classical field theories[7][24][25][26].

The KvN approach is fruitful in studies on the quantum-classical correspondence[7][8][27][28][29] as it reveals that the Hilbert space formulation is not exclusively quantum mechanical.[30] Even Dirac spinors are not exceptionally quantum as they are utilized in the relativistic generalization of the KvN mechanics.[23] Similarly as the more well-known phase space formulation of quantum mechanics, the KvN approach can be understood as an attempt to bring classical and quantum mechanics into a common mathematical framework. In fact, the time evolution of the Wigner function approaches, in the classical limit, the time evolution of the KvN wavefunction of a classical particle.[23][31] However, a mathematical resemblance to quantum mechanics does not imply the presence of hallmark quantum effects. In particular, impossibility of double-slit experiment[6][10][11] and Aharonov–Bohm effect[12] are explicitly demonstrated in the KvN framework.

KvN propagation vs Wigner propagation

• The time evolution of the classical KvN wave function for the Morse potential: ${\displaystyle U(x)=20(1-e^{-0.16x})^{2}}$. Black dots are classical particles following Newton's law of motion. The solid lines represent the level set of the Hamiltonian ${\displaystyle H(x,p)=p^{2}/2+U(x)}$. This video illustrates the fundamental difference between KvN and Liouville mechanics.

See also

• Classical mechanics
• Statistical mechanics
• Liouville's theorem
• Quantum mechanics
• Phase space formulation of quantum mechanics
• Wigner quasiprobability distribution
• Dynamical systems
• Ergodic theory

References

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3. von Neumann, J. (1932). "Zusatze Zur Arbeit "Zur Operatorenmethode..."". Annals of Mathematics. 33 (4): 789–791. doi:10.2307/1968225. JSTOR 1968225.
4. Landau, L. J. (1987). "On the violation of Bell's inequality in quantum theory". Physics Letters A. 120 (2): 54–56. Bibcode:1987PhLA..120...54L. doi:10.1016/0375-9601(87)90075-2.
5. Mauro, D. (2002). "Topics in Koopman–von Neumann Theory". arXiv:quant-ph/0301172. PhD thesis, Università degli Studi di Trieste.
6. Mauro, D. (2002). "On Koopman–Von Neumann Waves". International Journal of Modern Physics A. 17 (9): 1301–1325. arXiv:quant-ph/0105112. Bibcode:2002IJMPA..17.1301M. CiteSeerX 10.1.1.252.9355. doi:10.1142/S0217751X02009680.
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12. Gozzi, E.; Mauro, D. (2002). "Minimal Coupling in Koopman–von Neumann Theory". Annals of Physics. 296 (2): 152–186. arXiv:quant-ph/0105113. Bibcode:2002AnPhy.296..152G. CiteSeerX 10.1.1.252.9506. doi:10.1006/aphy.2001.6206.
13. Blokhintsev, D. I. (1977). "Classical statistical physics and quantum mechanics". Soviet Physics Uspekhi. 20 (8): 683–690. Bibcode:1977SvPhU..20..683B. doi:10.1070/PU1977v020n08ABEH005457.
14. Blokhintsev, D.I. (1940). "The Gibbs Quantum Ensemble and its Connection with the Classical Ensemble". J. Phys. U.S.S.R. 2 (1): 71–74.
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18. Liboff, R. L. (2003). Kinetic theory: classical, quantum, and relativistic descriptions. Springer. ISBN 9780387955513.
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22. Chruściński, D. (2006). "Koopman's approach to dissipation". Reports on Mathematical Physics. 57 (3): 319–332. Bibcode:2006RpMP...57..319C. doi:10.1016/S0034-4877(06)80023-6.
23. Renan Cabrera; Bondar; Rabitz (2011). "Relativistic Wigner function and consistent classical limit for spin 1/2 particles". arXiv:1107.5139 [quant-ph].
24. Carta, P.; Gozzi, E.; Mauro, D. (2006). "Koopman–von Neumann formulation of classical Yang–Mills theories: I". Annalen der Physik. 15 (3): 177–215. arXiv:hep-th/0508244. Bibcode:2006AnP...518..177C. doi:10.1002/andp.200510177.
25. Gozzi, E.; Penco, R. (2011). "Three approaches to classical thermal field theory". Annals of Physics. 326 (4): 876–910. arXiv:1008.5135. Bibcode:2011AnPhy.326..876G. doi:10.1016/j.aop.2010.11.018.
26. Cattaruzza, E.; Gozzi, E.; Francisco Neto, A. (2011). "Diagrammar in classical scalar field theory". Annals of Physics. 326 (9): 2377–2430. arXiv:1010.0818. Bibcode:2011AnPhy.326.2377C. CiteSeerX 10.1.1.750.8350. doi:10.1016/j.aop.2011.05.009.
27. Wilkie, J.; Brumer, P. (1997). "Quantum-classical correspondence via Liouville dynamics. I. Integrable systems and the chaotic spectral decomposition". Physical Review A. 55 (1): 27–42. arXiv:chao-dyn/9608013. Bibcode:1997PhRvA..55...27W. doi:10.1103/PhysRevA.55.27. hdl:1807/16867.
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30. Bracken, A. J. (2003). "Quantum mechanics as an approximation to classical mechanics in Hilbert space", Journal of Physics A: Mathematical and General, 36(23), L329.
31. Bondar; Renan Cabrera; Zhdanov; Rabitz (2013). "Wigner Function's Negativity Demystified". Physical Review A. 88 (5): 263. arXiv:1202.3628. Bibcode:2013PhRvA..88e2108B. doi:10.1103/PhysRevA.88.052108.

Further reading

• Mauro, D. (2002). "Topics in Koopman–von Neumann Theory". arXiv:quant-ph/0301172. PhD thesis, Università degli Studi di Trieste.
• H.R. Jauslin, D. Sugny, Dynamics of mixed classical-quantum systems, geometric quantization and coherent states, Lecture Note Series, IMS, NUS, Review Vol., August 13, 2009
• The Legacy of John von Neumann (Proceedings of Symposia in Pure Mathematics, vol 50), edited by James Glimm, John Impagliazzo, Isadore Singer. — Amata Graphics, 2006. — ISBN 0821842196
• U. Klein, From Koopman–von Neumann theory to quantum theory, Quantum Stud.: Math. Found. (2018) 5:219–227.