Koopman–von Neumann classical mechanics
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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.
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.
The origins of Koopman–von Neumann (KvN) theory are tightly connected with the rise[when?] 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.
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.
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.
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.
We begin from the following equations for expectation values of the coordinate x and momentum p
into which we substitute a consequence of Stone's theorem and obtain
Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for the unknown is derived
(commutator eqs for L)
Assume that the coordinate and momentum commute . This assumption physically means that the classical particle's coordinate and momentum can be measured simultaneously, implying absence of the uncertainty principle.
The solution cannot be simply of the form because it would imply the contractions and . Therefore, we must utilize additional operators and obeying
The need to employ these auxiliary operators arises because all classical observables commute. Now we seek in the form . Utilizing KvN algebra, the commutator eqs for L can be converted into the following differential equations
Whence, we conclude that the classical KvN wave function evolves according to the Schrödinger-like equation of motion
(KvN dynamical eq)
Since and commute, they share the common eigenvectors
Projecting equation (KvN dynamical eq) onto , we get the equation of motion for the KvN wave function in the xp-representation
(KvN dynamical eq in xp)
The quantity is the probability amplitude for a classical particle to be at point with momentum at time . According to the axioms above, the probability density is given by . Utilizing the identity
as well as (KvN dynamical eq in xp), we recover the classical Liouville equation
Therefore, the rule for calculating averages of observable in classical statistical mechanics has been recovered from the operator axioms with the additional assumption . 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 and (i.e., -representation), one obtains classical mechanics in the doubled configuration space, whose generalization leads      to the phase space formulation of quantum mechanics.
As in the derivation of classical mechanics, we begin from the following equations for averages of coordinate x and momentum p
With the help of the operator axioms, they can be rewritten as
into which we substitute a consequence of Stone's theorem
where 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 are derived
Contrary to the case of classical mechanics, we assume that observables of the coordinate and momentum obey the canonical commutation relation . Setting , the commutator equations can be converted into the differential equations 
whose solution is the familiar quantum Hamiltonian
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 .
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).
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.
The time evolution of the classical KvN wave function for the Morse potential: . Black dots are classical particles following Newton's law of motion. The solid lines represent the level set of the Hamiltonian . This video illustrates the fundamental difference between KvN and Liouville mechanics.
Quantum counterpart of the classical KvN propagation on the left: The Wigner function time evolution of the Morse potential in atomic units (a.u.). The solid lines represent the level set of the underlying Hamiltonian. Note that the same initial condition used for this quantum propagation as well as for the KvN propagation on the left.
- Classical mechanics
- Statistical mechanics
- Liouville's theorem
- Quantum mechanics
- Phase space formulation of quantum mechanics
- Wigner quasiprobability distribution
- Dynamical systems
- Ergodic theory
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