Koopman–von Neumann classical mechanics

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.

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.

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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.

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These axioms allow us to recover the formalism of both classical and quantum mechanics. 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.

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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 ($$) and Liouville equation ($$) 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.