Formulation of classical mechanics in terms of Hilbert spaces
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The Koopman–von Neumann (KvN) theory is a description of classical mechanics as an operatorial theory similar to quantum mechanics, based on a Hilbert space of complex, square-integrable wavefunctions. As its name suggests, the KvN theory is loosely related to work by Bernard Koopman and John von Neumann in 1931 and 1932, respectively. As explained in this entry, however, the historical origins of the theory and its name are complicated.
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 the Koopman–von Neumann 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. According to this formulation, functions representing physical observables become vectors, with an inner product defined in terms of a natural integration rule over the system's probability density on phase space. This reformulation makes it possible to draw 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. Subsequently, he published several seminal results in modern ergodic theory, including the proof of his mean ergodic theorem.
Historical Misattribution.
The Koopman-von Neumann theory is often used today to refer to a reformulation of classical mechanics in which a classical system's probability density on phase space is expressed in terms of an underlying wavefunction, meaning that the vectors of the classical Hilbert space are wavefunctions, rather than physical observables.
This approach did not originate with Koopman or von Neumann, for whom the classical Hilbert space consisted of physical observables, rather than wavefunctions. Indeed, as noted in 1961 by Thomas F. Jordan and E. C. George Sudarshan:It was shown by Koopman how the dynamical transformations of classical mechanics, considered as measure preserving transformations of the phase space, induce unitary transformations on the Hilbert space of functions which are square integrable with respect to a density function over the phase space. This Hilbert space formulation of classical mechanics was further developed by von Neumann. It is to be noted that this Hilbert space corresponds not to the space of state vectors in quantum mechanics but to the Hilbert space of operators on the state vectors (with the trace of the product of two operators being chosen as the scalar product).The practice of expressing classical probability distributions on phase space in terms of underlying wavefunctions goes back at least to the 1952–1953 work of Mário Schenberg on statistical mechanics. This method was independently developed several more times, by Angelo Loinger in 1962, by Giacomo Della Riccia and Norbert Wiener in 1966, and by E. C. George Sudarshan himself in 1976.
The name "Koopman-von Neumann theory" for representing classical systems based on Hilbert spaces made up of classical wavefunctions is therefore an example of Stigler's law of eponymy. This misattribution appears to have first shown up in a paper by Danilo Mauro in 2002.
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.
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.
 ;templatestyles src="Math_proof/styles.css" / ;Dynamics of the probability density (proof)
In classical statistical mechanics, the probability density (with respect to Liouville measure) obeys the Liouville equation
formula_1
with the self-adjoint Liouvillian 
formula_2
where formula_3 denotes the classical Hamiltonian (i.e. the Liouvillian is formula_4 times the Hamiltonian vector field considered as a first order differential operator).
The same dynamical equation is postulated for the KvN wavefunction
formula_5
thus
formula_6
and for its complex conjugate
formula_7
From
formula_8
follows using the product rule that
formula_9
which proves that probability density dynamics can be recovered from the KvN wavefunction.
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.
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.
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.
 ;templatestyles src="Math_proof/styles.css" / ;Mathematical form of the operator axioms
The above axioms (i) to (iv), with the inner product written in the bra–ket notation, are
 ;ol type="i" ;
 ;li ; formula_10
aka, Newton's laws of motion averaged over ensemble. With the help of the operator axioms, they can be rewritten as
formula_11 and formula_12 commute, they share the common eigenvectors
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with the resolution of the identity
formula_13
Then, one obtains from equation (KvN algebra)
formula_14
Projecting equation (KvN dynamical eq) onto formula_15, we get the equation of motion for the KvN wave function in the xp-representation
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The quantity formula_16 is the probability amplitude for a classical particle to be at point formula_17 with momentum formula_18 at time formula_19. According to the axioms above, the probability density is given by
formula_20. Utilizing the identity
formula_21
as well as (KvN dynamical eq in xp), we recover the classical Liouville equation
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Moreover, according to the operator axioms and (xp eigenvec),
formula_22
Therefore, the rule for calculating averages of observable formula_23 in classical statistical mechanics has been recovered from the operator axioms with the additional assumption formula_24. 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 as well as Aharonov–Bohm effect is established in the KvN mechanics.
Projecting KvN dynamical eq onto the common eigenvector of the operators formula_25 and formula_26 (i.e., formula_27-representation), one obtains classical mechanics in the doubled configuration space, whose generalization leads
to the phase space formulation of quantum mechanics.
 ;templatestyles src="Math_proof/styles.css" / ;Derivation of quantum mechanics from the operator axioms
As in the derivation of classical mechanics, we begin from the following equations for averages of coordinate "x" and momentum "p"
formula_28
With the help of the operator axioms, they can be rewritten as
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m\frac{d}{dt} \langle \Psi(t)
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.
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).
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 as well as functional integral methods. The KvN approach is very general, and it has been extended to dissipative systems, relativistic mechanics, and classical field theories.
The KvN approach is fruitful in studies on the quantum-classical correspondence as it reveals that the Hilbert space formulation is not exclusively quantum mechanical. Even Dirac spinors are not exceptionally quantum as they are utilized in the relativistic generalization of the KvN mechanics. 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. However, a mathematical resemblance to quantum mechanics does not imply the presence of hallmark quantum effects. In particular, impossibility of double-slit experiment and Aharonov–Bohm effect are explicitly demonstrated in the KvN framework.
References.
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