Koopman–von Neumann classical mechanics
The Koopman–von Neumann theory is a description of classical mechanics as an operatorial theory similar to quantum mechanics, based on a Hilbert space of complex, square-integrable functions representing classical observables on phase spaces. As its name suggests, the KvN theory was developed by Bernard Koopman and John von Neumann.
The KvN theory is mathematically related to the conceptually distinct method of classical wavefunctions originally introduced by Mario Schönberg and others. These two methods will collectively be called Hilbert-space formulations of classical mechanics in what follows.
History
Statistical Mechanics and Ergodic Theory
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 the Koopman–von Neumann theory are tightly connected with the rise of ergodic theory as an independent branch of mathematics, in particular with Ludwig Boltzmann's ergodic hypothesis.
In 1931, Koopman 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 in 1932. Subsequently, he published several seminal results in modern ergodic theory, including the proof of his mean ergodic theorem.
Schönberg's Method of Classical Wavefunctions
Independently, several researchers developed a framework for representing classical probability distributions on phase spaces with complex-valued wavefunctions. This method of classical wavefunctions is conceptually distinct from the KvN theory, which was based on treating functions representing classical observables on phase spaces as the elementary vectors of Hilbert spaces.The method of classical wavefunctions was developed by Mário Schönberg in 1952-1953, Angelo Loinger in 1962, Giacomo Della Riccia and Norbert Wiener in 1966, and E. C. George Sudarshan in 1976. In particular, Loinger established an isomorphism between the Hilbert spaces of classical wavefunctions and KvN Hilbert spaces, yielding an overall Hilbert-space framework for classical mechanics.
In 2001, Danilo Mauro made important advances to the line of work tracing back to Schönberg but Mauro attributed the concepts to Koopman and von Neumann. Since that time the method of classical wave functions became conflated with the KvN theory, despite being conceptually different theories with different origins.
Definition and dynamics
Derivation starting from the Liouville equation
In the method of classical wavefunctions, as part of Hilbert-space classical mechanics, dynamics in phase space is described by a probability density, recovered from an underlying wavefunction – the classical wavefunction – as the square of its absolute value. This stands in analogy to the Born rule in quantum mechanics. In Hilbert-space classical mechanics, observables are represented by commuting self-adjoint operators acting on the Hilbert space of classical 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 classical 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 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, the expectation value of an observable is obtained in the manner as the expectation value in quantum mechanics, the probabilities of measuring certain values of some observables are calculated by the Born rule, and 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 classical 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.
Measurements
In Hilbert-space classical mechanics, the classical wavefunction takes the form of a superposition of eigenstates, and measurement collapses the classical wavefunction to the eigenstate which is associated the measurement result, in analogy to the wave function collapse of quantum mechanics.However, Mauro showed that, in his approach, non-selective measurements leave the classical wavefunction unchanged.
Hilbert-space classical mechanics vs Liouville mechanics
The dynamical equation for Hilbert-space classical mechanics and the 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 Hilbert-space classical mechanics and Liouville mechanics lies in weighting individual trajectories: Arbitrary weights, underlying the classical wave function, can be utilized in Hilbert-space classical mechanics, while only positive weights, representing the probability density, are permitted in the Liouville mechanics.Quantum analogy
Being explicitly based on the Hilbert space language, the Hilbert-space approach to classical mechanics adopts many techniques from quantum mechanics, for example, perturbation and diagram techniques as well as functional integral methods. Hilbert-space classical mechanics is very general, and it has been extended to dissipative systems, relativistic mechanics, and classical field theories.The Hilbert-space classical mechanics is fruitful in studies on the quantum-classical correspondence as it reveals that the use of Hilbert spaces is not exclusively quantum mechanical. Even Dirac spinors are not exceptionally quantum as they are utilized in the relativistic generalization of Hilbert-space classical mechanics. Similarly as the more well-known phase space formulation of quantum mechanics, Hilbert-space classical mechanics 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 classical 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 Hilbert-space classical mechanics.