Koopman–von Neumann classical mechanics

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.

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 statistical mechanics arising from the study of statistical mechanics.

The origins of Koopman–von Neumann (KvN) 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 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 quantum 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.

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