View source for Koopman–von Neumann classical mechanics

{{Classical mechanics|cTopic=Formulations}}
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.<ref name=Koopman1931>{{Cite journal | last1 = Koopman | first1 = B. O. | title = Hamiltonian Systems and Transformations in Hilbert Space | doi = 10.1073/pnas.17.5.315 | journal = Proceedings of the National Academy of Sciences | volume = 17 | issue = 5 | pages = 315–318 | year = 1931 | pmid =  16577368| pmc = 1076052| bibcode = 1931PNAS...17..315K }}</ref><ref name=Neumann1932>{{cite journal | doi = 10.2307/1968537 | last1 = von Neumann | first1 = J.  | year = 1932 | title = Zur Operatorenmethode In Der Klassischen Mechanik | journal = Annals of Mathematics | volume = 33 | issue = 3 | pages = 587–642 | jstor = 1968537}}</ref><ref name=Neumann1932a>{{cite journal | doi = 10.2307/1968225 | last1 = von Neumann | first1 = J.  | year = 1932 | title = Zusatze Zur Arbeit "Zur Operatorenmethode..." | journal = Annals of Mathematics | volume = 33 | issue = 4 | pages = 789–791 | jstor = 1968225}}</ref>

As Koopman and von Neumann demonstrated, a [[Hilbert space]] of [[Complex number|complex]], [[Square-integrable function|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 ensemble (mathematical physics)|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{{when|date=October 2016}} of [[ergodic theory]] as an independent branch of mathematics, in particular with [[Ludwig Boltzmann|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 on one-parameter unitary groups|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 quantum mechanics currently known as Koopman–von Neumann theory. Subsequently, he published several seminal results in modern ergodic theory including the proof of his [[Ergodic theory#Mean ergodic theorem|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]].<ref name=Landau1987>{{Cite journal | last1 = Landau | first1 = L. J. | title = On the violation of Bell's inequality in quantum theory | doi = 10.1016/0375-9601(87)90075-2 | journal = Physics Letters A | volume = 120 | issue = 2 | pages = 54–56| year = 1987 | pmid =  | pmc = |bibcode = 1987PhLA..120...54L }}</ref>

The KvN wavefunction is postulated to evolve according to exactly the same [[Liouville's theorem (Hamiltonian)|Liouville equation]] as the classical probability density. From this postulate it can be shown that indeed probability density dynamics is recovered.

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<div class="NavHead" style="background-color:#CCCCFF; text-align:left; font-size:larger;">Dynamics of the probability density (proof)</div>
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In classical statistical mechanics, the probability density obeys the Liouville equation<ref name=DaniloMauro2002/><ref name=Mauro2002/>
:<math>i\frac{\partial}{\partial t} \rho (x, p, t) = \hat{L} \rho(x, p, t)</math>
with the self-adjoint Liouvillian
:<math>\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},</math>
where <math> H(x,p) </math> denotes the [[Hamiltonian mechanics|classical Hamiltonian]].
The same dynamical equation is postulated for the KvN wavefunction
:<math>i\frac{\partial}{\partial t} \psi (x, p, t) = \hat{L} \psi (x, p, t),</math>
thus
:<math>\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),</math>
and for its complex conjugate
:<math>\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).</math>
From
:<math>\rho(x, p, t) = \psi^*(x, p, t) \psi(x, p, t)</math>
follows using the [[Product rule#Higher partial derivatives|product rule]] that
:<math>\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)</math>
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.
<ref name=DaniloMauro2002>
{{Cite arXiv | last=Mauro | first=D. | title=Topics in Koopman–von Neumann Theory | year=2002 | eprint=quant-ph/0301172 }} PhD thesis, Università degli Studi di Trieste.
</ref>
<ref name=Mauro2002>{{Cite journal | last1 = Mauro | first1 = D. | title = On Koopman–Von Neumann Waves | doi = 10.1142/S0217751X02009680 | journal = International Journal of Modern Physics A | volume = 17 | issue = 9 | pages = 1301–1325 | year = 2002 | pmid =  | pmc = |arxiv = quant-ph/0105112 |bibcode = 2002IJMPA..17.1301M | citeseerx = 10.1.1.252.9355 }}</ref>
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===Derivation starting from operator axioms===
Conversely, it is possible to start from operator postulates, similar to the [[Operator (physics)#Operators in quantum mechanics|Hilbert space axioms of quantum mechanics]], and derive the equation of motion by specifying how expectation values evolve.<ref name=Bondar2012>{{Cite journal | last1 = Bondar | first1 = D. | last2 = Cabrera | first2 = R. | last3 = Lompay | first3 = R. | last4 = Ivanov | first4 = M. | last5 = Rabitz | first5 = H. | title = Operational Dynamic Modeling Transcending Quantum and Classical Mechanics | doi = 10.1103/PhysRevLett.109.190403 | journal = Physical Review Letters | volume = 109 | issue = 19 | year = 2012 | pmid =  23215365| pmc = |arxiv = 1105.4014 |bibcode = 2012PhRvL.109s0403B | pages=190403}}</ref>

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 operator]]s acting on that space, (ii) the expectation value of an observable is obtained in the manner as the [[Expectation value (quantum mechanics)#Formalism in quantum mechanics|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.

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<div class="NavHead" style="background-color:#CCCCFF; text-align:left; font-size:larger;">{{anchor|sec_formulation}}Mathematical form of the operator axioms</div>
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The above axioms (i) to (iv), with the [[inner product]] written in the [[bra–ket notation]], are
:(i) <math> \langle \psi(t) | \psi(t) \rangle = 1</math>,
:(ii) The expectation value of an observable <math>\hat{A}</math> at time <math>t</math> is <math>\langle A (t)\rangle =  \langle \Psi (t)| \hat{A} | \Psi(t) \rangle.</math>
:(iii)  The probability that a measurement of an observable <math>\hat{A}</math> at time <math>t</math> yields <math>A</math> is <math> \left|\langle A | \Psi(t)\rangle \right|^2 </math>, where <math> \hat{A} |A\rangle = A |A \rangle </math>. (This axiom is an analogue of the [[Born rule]] in quantum mechanics.<ref name=Brumer2006>{{Cite journal | last1 = Brumer | first1 = P. | last2 = Gong | first2 = J. | doi = 10.1103/PhysRevA.73.052109 | title = Born rule in quantum and classical mechanics | journal = Physical Review A | volume = 73 | issue = 5 | pages = 052109 | year = 2006 | pmid =  | pmc = |arxiv = quant-ph/0604178 |bibcode = 2006PhRvA..73e2109B | hdl = 1807/16870 }}</ref>)
:(iv) (see [[Tensor product of Hilbert spaces#Definition|Tensor product of Hilbert spaces]]).
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These axioms allow us to recover the formalism of both classical and quantum mechanics.<ref name=Bondar2012/> Specifically, under the assumption that the classical position and momentum operators [[Commutative property|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.

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<div class="NavHead" style="background-color:#CCCCFF; text-align:left; font-size:larger;">{{anchor|sec_KvN_from_Newton}}Classical mechanics from the [[#sec_formulation|operator axioms]] (derivation)</div>
<div class="NavContent" style="text-align:left;">
We begin from the following equations for expectation values of the coordinate ''x'' and momentum ''p''
:<math>
m\frac{d}{dt} \langle x \rangle = \langle p \rangle, \qquad \frac{d}{dt} \langle p \rangle =\langle -U'(x) \rangle,
</math>
aka, [[Newton's laws of motion]] averaged over ensemble. With the help of the [[#sec_formulation|operator axioms]], they can be rewritten as
:<math>
\begin{align}
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{align}
</math>
Notice a close resemblance with [[Ehrenfest theorem]]s in quantum mechanics. Applications of the [[product rule]] leads to
:<math>
\begin{align}
\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{align}
</math>
into which we substitute a consequence of [[Stone's theorem on one-parameter unitary groups|Stone's theorem]] <math> i | d\Psi(t)/dt \rangle = \hat{L} | \Psi(t) \rangle </math> and obtain
:<math>
\begin{align}
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{align}
</math>
Since these identities must be valid for any initial state, the averaging can be dropped and the  system of commutator equations for the unknown <math>\hat{L}</math> is derived
{{NumBlk|:|<math>
	im  [\hat{L}, \hat{x}] =  \hat{p} , \qquad i [\hat{L}, \hat{p}] = -U'(\hat{x}).
</math>|{{EquationRef|commutator eqs for L}}}}
'''Assume that the coordinate and momentum commute <math>[ \hat{x}, \hat{p} ] = 0</math>.''' This assumption physically means that the classical particle's coordinate and momentum can be measured simultaneously, implying absence of the [[uncertainty principle]].

The solution <math>\hat{L}</math> cannot be simply of the form <math>\hat{L} = L(\hat{x}, \hat{p})</math> because it would imply the contractions <math> im  [L(\hat{x}, \hat{p}), \hat{x}] =  0 = \hat{p} </math> and <math> i [L(\hat{x}, \hat{p}), \hat{p}] = 0 = -U'(\hat{x}) </math>. Therefore, we must utilize additional operators <math>\hat{\lambda}_x</math> and <math>\hat{\lambda}_p</math> obeying
{{NumBlk|:|<math>
	[ \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.
</math>|{{EquationRef|KvN algebra}}}}
The need to employ these auxiliary operators arises because all classical observables commute. Now we seek <math>\hat{L}</math> in the form <math>\hat{L} = L(\hat{x}, \hat{\lambda}_x, \hat{p}, \hat{\lambda}_p)</math>. Utilizing {{EquationNote|KvN algebra}}, the {{EquationNote|commutator eqs for L}} can be converted into the following differential equations
:<ref name=Bondar2012/><ref name=Transtrum2005>{{Cite journal | last1 = Transtrum | first1 = M. K. | last2 = Van Huele | first2 = J. F. O. S. | doi = 10.1063/1.1924703 | title = Commutation relations for functions of operators | journal = Journal of Mathematical Physics | volume = 46 | issue = 6 | pages = 063510 | year = 2005 | pmid =  | pmc = |bibcode = 2005JMP....46f3510T }}</ref>
:<math>
m L'_{\lambda_x} (x, \lambda_x, p, \lambda_p) = p, \qquad L'_{\lambda_p} (x, \lambda_x, p, \lambda_p) = -U'(x).
</math>
Whence, we conclude that the classical KvN wave function <math>|\Psi(t)\rangle</math> evolves according to the  [[Schrödinger equation|Schrödinger-like]] equation of motion
{{NumBlk|:|<math>
	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.
</math>|{{EquationRef|KvN dynamical eq}}}}

Let us explicitly show that {{EquationNote|KvN dynamical eq}} is equivalent to the [[Liouville's theorem (Hamiltonian)#Liouville equations|classical Liouville mechanics]].

Since <math>\hat{x}</math> and <math>\hat{p}</math> commute, they share the common [[eigenvectors]]
{{NumBlk|:|<math>
\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,
</math>|{{EquationRef|xp eigenvec}}}}
with the [[Self-adjoint operator#Resolution of the identity|resolution of the identity]]
<math>
1 = \int dx dp \, |x,p\rangle \langle x,p|.
</math>
Then, one obtains from equation ({{EquationNote|KvN algebra}})
:<math>
\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.
</math>
Projecting equation ({{EquationNote|KvN dynamical eq}}) onto <math>\langle x,p|</math>, we get the equation of motion for the KvN wave function in the xp-representation
{{NumBlk|:|<math>
\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.
</math>|{{EquationRef|KvN dynamical eq in xp}}}}
The quantity <math>\langle x,\, p |\Psi(t) \rangle</math> is the probability amplitude for a classical particle to be at point <math>x</math> with momentum <math>p</math> at time <math>t</math>. According to the [[#sec_formulation|axioms above]], the probability density is given by
<math>\rho(x,p;t) = \left| \langle x, p |\Psi(t) \rangle \right|^2</math>. Utilizing the identity
:<math>
\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)^*
</math>
as well as ({{EquationNote|KvN dynamical eq in xp}}), we recover the classical Liouville equation
{{NumBlk|:|<math>
\left[ \frac{\partial }{\partial t} + \frac{p}{m} \frac{\partial}{\partial x} -  U'(x) \frac{\partial}{\partial p}  \right]  \rho(x,p;t) = 0.
</math>|{{EquationRef|Liouville eq}}}}

Moreover, according to the [[#sec_formulation|operator axioms]] and ({{EquationNote|xp eigenvec}}),
:<math>
\begin{align}
\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{align}
</math>
Therefore, the rule for calculating averages of observable <math> A(x,p) </math> in classical statistical mechanics has been recovered from the [[#sec_formulation|operator axioms]] with the additional assumption <math>[ \hat{x}, \hat{p} ] = 0</math>. 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]]<ref name=Mauro2002/><ref name=Gozzi2003>{{Cite journal | last1 = Gozzi | first1 = E. | last2 = Mauro | first2 = D. | doi = 10.1142/S0217751X04017872 | title = On Koopman–Von Neumann Waves Ii | journal = International Journal of Modern Physics A | volume = 19 | issue = 9 | pages = 1475 | year = 2004 | pmid =  | pmc = |arxiv = quant-ph/0306029 |bibcode = 2004IJMPA..19.1475G | citeseerx = 10.1.1.252.1596 }}</ref><ref name=Gozzi2010>{{Cite journal | last1 = Gozzi | first1 = E. | last2 = Pagani | first2 = C. | doi = 10.1103/PhysRevLett.105.150604 | title = Universal Local Symmetries and Nonsuperposition in Classical Mechanics | journal = Physical Review Letters | volume = 105 | issue = 15 | year = 2010 | pmid =  21230883| pmc = |arxiv = 1006.3029 |bibcode = 2010PhRvL.105o0604G | page=150604}}</ref> as well as [[Aharonov–Bohm effect]]<ref name=Gozzi2002>{{Cite journal | last1 = Gozzi | first1 = E. | last2 = Mauro | first2 = D. | doi = 10.1006/aphy.2001.6206 | title = Minimal Coupling in Koopman–von Neumann Theory | journal = Annals of Physics | volume = 296 | issue = 2 | pages = 152–186 | year = 2002 | pmid =  | pmc = |arxiv = quant-ph/0105113 |bibcode = 2002AnPhy.296..152G | citeseerx = 10.1.1.252.9506 }}</ref> is established in the KvN mechanics.

Projecting {{EquationNote|KvN dynamical eq}} onto the common eigenvector of the operators <math>\hat{x}</math> and <math>\hat{\lambda}_p</math> (i.e., <math>x\lambda_p</math>-representation), one obtains classical mechanics in the doubled configuration space,<ref name=Blokhintsev1977>{{Cite journal | last1 = Blokhintsev | first1 = D. I. | title = Classical statistical physics and quantum mechanics | doi = 10.1070/PU1977v020n08ABEH005457 | journal = Soviet Physics Uspekhi | volume = 20 | issue = 8 | pages = 683–690 | year = 1977 | pmid =  | pmc = | bibcode = 1977SvPhU..20..683B }}</ref> whose generalization leads
<ref name=Blokhintsev1977/>
<ref name=Blokhintsev1940a>{{Cite journal|last=[[Blokhintsev]] | first=D.I. | journal = J. Phys. U.S.S.R. |issue=1 |pages = 71–74 | title = The Gibbs Quantum Ensemble and its Connection with the Classical Ensemble | volume = 2 |year = 1940 }}
</ref>
<ref name=Blokhintsev1940>{{Cite journal| last1= [[Blokhintsev]] | first1= D.I. | last2=Nemirovsky | first2= P | journal = J. Phys. U.S.S.R. | issue=3 | pages = 191–194 | title = Connection of the Quantum Ensemble with the Gibbs Classical Ensemble. II | volume = 3 | year = 1940 }}
</ref>
<ref name=Blokhintsev1941>{{Cite journal| last1=[[Blokhintsev]] | first1= D.I. | last2=Dadyshevsky | first2=Ya. B. | journal = Zh. Eksp. Teor. Fiz. | issue=2–3 | pages = 222–225 | title = On Separation of a System into Quantum and Classical Parts | volume = 11 |year = 1941 }}
</ref>
<ref name=blokhintsev2010philosophy>{{Cite book|last=[[Blokhintsev]] | first=D.I.|publisher = Springer|isbn=9789048183357|title = The Philosophy of Quantum Mechanics |year = 2010}}
</ref>
to the [[Phase space formulation|phase space formulation of quantum mechanics]].
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<div class="NavHead" style="background-color:#CCCCFF; text-align:left; font-size:larger;">{{anchor|sec_quantum_mech}}Quantum mechanics from the [[#sec_formulation|operator axioms]] (derivation)</div>
<div class="NavContent" style="text-align:left;">
As in the [[#sec_KvN_from_Newton|derivation of classical mechanics]], we begin from the following equations for averages of coordinate ''x'' and momentum ''p''
:<math>
m\frac{d}{dt} \langle x \rangle = \langle p \rangle, \qquad \frac{d}{dt} \langle p \rangle =\langle -U'(x) \rangle.
</math>
With the help of the [[#sec_formulation|operator axioms]], they can be rewritten as
:<math>
\begin{align}
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{align}
</math>
These are the [[Ehrenfest theorem]]s in quantum mechanics. Applications of the [[product rule]] leads to
:<math>
\begin{align}
\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{align}
</math>
into which we substitute a consequence of [[Stone's theorem on one-parameter unitary groups|Stone's theorem]]
:<math>
i\hbar | d \Psi(t)/dt \rangle = \hat{H} | \Psi(t) \rangle,
</math>
where <math>\hbar</math> 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 <math>\hat{H}</math> are derived
:<math>
im [\hat{H}, \hat{x}] = \hbar \hat{p}, \qquad i [\hat{H}, \hat{p}] = -\hbar U'(\hat{x}).
</math>
Contrary to the case of [[#sec_KvN_from_Newton|classical mechanics]], we '''assume that observables of the coordinate and momentum obey the [[canonical commutation relation]]''' <math>[ \hat{x}, \hat{p} ] = i\hbar</math>. Setting <math>\hat{H} = H(\hat{x}, \hat{p})</math>, the commutator equations can be converted into the differential equations
<ref name=Bondar2012/><ref name=Transtrum2005/>
:<math>
m H'_p (x,p) = p, \qquad H'_x (x,p) = U'(x),
</math>
whose solution is the familiar [[Hamiltonian (quantum mechanics)|quantum Hamiltonian]]
:<math>
\hat{H} = \frac{\hat{p}^2}{2m} + U(\hat{x}).
</math>
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 [[#sec_KvN_from_Newton|derivation of classical KvN mechanics]] shows that the difference between quantum and classical mechanics essentially boils down to the value of the commutator <math>[ \hat{x}, \hat{p} ]</math>.
</div>
</div>

===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.<ref name=DaniloMauro2002/>

==KvN vs Liouville mechanics==

The KvN dynamical equation ({{EquationNote|KvN dynamical eq in xp}}) and [[Liouville's theorem (Hamiltonian)#Liouville equations|Liouville equation]] ({{EquationNote|Liouville eq}}) are [[first-order partial differential equation|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 [[#fig_KvN_vs_Liouville|this scheme]]).

[[File:KvN vs Liouville mechanics.pdf|thumb|350px|center|{{anchor|fig_KvN_vs_Liouville}}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. ([[#animation_KvN_for_Morse|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 theory (quantum mechanics)|perturbation]] and [[Feynman diagrams|diagram techniques]]<ref name=Liboff2003>{{Cite book| last=Liboff | first=R. L. | title=Kinetic theory: classical, quantum, and relativistic descriptions | publisher=Springer | year=2003 |isbn=9780387955513}}</ref> as well as [[Functional integration|functional integral methods]]<ref name=Gozzi1988>{{Cite journal | last1 = Gozzi | first1 = E. | title = Hidden BRS invariance in classical mechanics | doi = 10.1016/0370-2693(88)90611-9 | journal = Physics Letters B | volume = 201 | issue = 4 | pages = 525–528 | year = 1988 | pmid =  | pmc = |bibcode = 1988PhLB..201..525G | url = http://cds.cern.ch/record/199310 }}</ref><ref name=Gozzi1989>{{Cite journal | last1 = Gozzi | first1 = E. | last2 = Reuter | first2 = M. | last3 = Thacker | first3 = W. | doi = 10.1103/PhysRevD.40.3363 | title = Hidden BRS invariance in classical mechanics. II | journal = Physical Review D | volume = 40 | issue = 10 | pages = 3363 | year = 1989 | pmid =  | pmc = |bibcode = 1989PhRvD..40.3363G | url = http://cds.cern.ch/record/199310 }}</ref><ref name=Blasone2005>{{Cite journal | last1 = Blasone | first1 = M. | last2 = Jizba | first2 = P. | last3 = Kleinert | first3 = H. | title = Path-integral approach to 't Hooft's derivation of quantum physics from classical physics | doi = 10.1103/PhysRevA.71.052507 | journal = Physical Review A | volume = 71 | issue = 5 | pages = 052507 | year = 2005 | pmid =  | pmc = |arxiv = quant-ph/0409021 |bibcode = 2005PhRvA..71e2507B }}</ref>. The KvN approach is very general, and it has been extended to [[dissipative systems]],<ref name=Chruscinski2006>{{Cite journal | last1 = Chruściński | first1 = D. | title = Koopman's approach to dissipation | doi = 10.1016/S0034-4877(06)80023-6 | journal = Reports on Mathematical Physics | volume = 57 | issue = 3 | pages = 319–332 | year = 2006 | pmid =  | pmc = | bibcode=2006RpMP...57..319C}}</ref> [[relativistic mechanics]],<ref name=Cabrera2012/> and [[classical field theories]]<ref name=Bondar2012/><ref name=Carta2006>{{Cite journal | last1 = Carta | first1 = P. | last2 = Gozzi | first2 = E. | last3 = Mauro | first3 = D. | doi = 10.1002/andp.200510177 | title = Koopman–von Neumann formulation of classical Yang–Mills theories: I | journal = Annalen der Physik | volume = 15 | issue = 3 | pages = 177–215 | year = 2006 | pmid =  | pmc = |arxiv = hep-th/0508244 |bibcode = 2006AnP...518..177C }}</ref><ref name=Gozzi2011>{{Cite journal | last1 = Gozzi | first1 = E. | last2 = Penco | first2 = R. | doi = 10.1016/j.aop.2010.11.018 | title = Three approaches to classical thermal field theory | journal = Annals of Physics | volume = 326 | issue = 4 | pages = 876–910 | year = 2011 | pmid =  | pmc = |arxiv = 1008.5135 |bibcode = 2011AnPhy.326..876G }}</ref><ref name=Cattaruzza2011>{{Cite journal | last1 = Cattaruzza | first1 = E. | last2 = Gozzi | first2 = E. | last3 = Francisco Neto | first3 = A. | doi = 10.1016/j.aop.2011.05.009 | title = Diagrammar in classical scalar field theory | journal = Annals of Physics | volume = 326 | issue = 9 | pages = 2377–2430 | year = 2011 | pmid =  | pmc = |arxiv = 1010.0818 |bibcode = 2011AnPhy.326.2377C | citeseerx = 10.1.1.750.8350 }}</ref>.

The KvN approach is fruitful in studies on the [[Classical limit|quantum-classical correspondence]]<ref name=Bondar2012/><ref name=Brumer2006/><ref name=Wilkie1997>{{Cite journal | last1 = Wilkie | first1 = J. | last2 = Brumer | first2 = P. | doi = 10.1103/PhysRevA.55.27 | title = Quantum-classical correspondence via Liouville dynamics. I. Integrable systems and the chaotic spectral decomposition | journal = Physical Review A | volume = 55 | issue = 1 | pages = 27–42 | year = 1997 | pmid =  | pmc = |bibcode = 1997PhRvA..55...27W | arxiv = chao-dyn/9608013 | hdl = 1807/16867 }}</ref><ref name=Wilkie1997a>{{Cite journal | last1 = Wilkie | first1 = J. | last2 = Brumer | first2 = P. | doi = 10.1103/PhysRevA.55.43 | title = Quantum-classical correspondence via Liouville dynamics. II. Correspondence for chaotic Hamiltonian systems | journal = Physical Review A | volume = 55 | issue = 1 | pages = 43–61 | year = 1997 | pmid =  | pmc = |bibcode = 1997PhRvA..55...43W | arxiv = chao-dyn/9608014 | hdl = 1807/16874 }}</ref><ref name=Abrikosovjr2005>{{Cite journal | last1 = Abrikosov | first1 = A. A. | last2 = Gozzi | first2 = E. | last3 = Mauro | first3 = D. | title = Geometric dequantization | doi = 10.1016/j.aop.2004.12.001 | journal = Annals of Physics | volume = 317 | issue = 1 | pages = 24–71 | year = 2005 | pmid =  | pmc = |arxiv = quant-ph/0406028 |bibcode = 2005AnPhy.317...24A }}</ref> as it reveals that the Hilbert space formulation is not exclusively quantum mechanical.<ref>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.</ref> Even [[Dirac spinor]]s are not exceptionally quantum as they are utilized in the relativistic generalization of the KvN mechanics.<ref name=Cabrera2012/> 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 quasiprobability distribution|Wigner function]] approaches, in the classical limit, the time evolution of the KvN wavefunction of a classical particle.<ref name=Cabrera2012>{{cite arXiv|eprint=1107.5139|author1=Renan Cabrera|author2=Bondar|author3=Rabitz|title=Relativistic Wigner function and consistent classical limit for spin 1/2 particles|class=quant-ph|year=2011}}</ref><ref name=Bondar2013>{{Cite journal|arxiv=1202.3628|author1=Bondar|author2=Renan Cabrera|author3=Zhdanov|author4=Rabitz|title=Wigner Function's Negativity Demystified|year=2013|doi=10.1103/PhysRevA.88.052108|volume=88|issue=5|pages=263|journal=Physical Review A|bibcode=2013PhRvA..88e2108B}}</ref> However, a mathematical resemblance to quantum mechanics does not imply the presence of hallmark quantum effects. In particular, impossibility of [[double-slit experiment]]<ref name=Mauro2002/><ref name=Gozzi2003/><ref name=Gozzi2010/> and [[Aharonov–Bohm effect]]<ref name=Gozzi2002/> are explicitly demonstrated in the KvN framework.

{{Gallery
|title=KvN propagation vs Wigner propagation
|align=center
|lines=10
|width=320
|height=320
|File:KvN evolution for Morse potential.ogv|{{anchor|animation_KvN_for_Morse}}The time evolution of the classical KvN wave function for the [[Morse potential]]: <math>U (x) = 20 ( 1 - e^{-0.16x} )^2 </math>. Black dots are classical particles following [[Newton's law of motion]]. The solid lines represent the [[level set]] of the [[Hamiltonian mechanics|Hamiltonian]] <math>H(x,p) = p^2 / 2 + U(x) </math>. This video illustrates [[#fig_KvN_vs_Liouville|the fundamental difference between KvN and Liouville mechanics]].
|File:Wigner function propagation for morse potential.ogv|Quantum counterpart of the classical KvN propagation on the left: The [[Wigner quasiprobability distribution|Wigner function]] time evolution of the [[Morse potential]] in [[atomic units]] (a.u.). The solid lines represent the [[level set]] of the underlying [[Hamiltonian mechanics|Hamiltonian]]. Note that the same initial condition used for this quantum propagation as well as for the KvN propagation on the left.
}}

==See also==

* [[Classical mechanics]]
* [[Statistical mechanics]]
* [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]
* [[Quantum mechanics]]
* [[Phase space formulation|Phase space formulation of quantum mechanics]]
* [[Wigner quasiprobability distribution]]
* [[Dynamical systems]]
* [[Ergodic theory]]

==References==
{{reflist|30em}}

==Further reading==
* {{Cite arXiv | last=Mauro | first=D. | title=Topics in Koopman–von Neumann Theory | year=2002 | eprint=quant-ph/0301172 }} PhD thesis, Università degli Studi di Trieste.
* H.R. Jauslin, D. Sugny, [http://icb.u-bourgogne.fr/omr/dqnl/sugny/files/MHQP-LN-Jauslin.pdf 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.[https://link.springer.com/content/pdf/10.1007%2Fs40509-017-0113-2.pdf]



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