Koopman–von Neumann classical mechanics
As its name suggests, the KvN theory is related to work[1][2]: 220 by Bernard Koopman and John von Neumann.
Ergodic theory is a branch of mathematics arising from the study of statistical mechanics.
[3] 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.
The Koopman–von Neumann treatment was further developed over the time by Mário Schenberg in 1952-1953,[6][7] by Angelo Loinger in 1962,[8] by Giacomo Della Riccia and Norbert Wiener in 1966,[9] and by E. C. George Sudarshan himself in 1976.
[10] 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).
Contrast this with quantum mechanics, where observables need not commute, which underlines the uncertainty principle, Kochen–Specker theorem, and Bell inequalities.
[11] The KvN wavefunction is postulated to evolve according to exactly the same Liouville equation as the classical probability density.
times the Hamiltonian vector field considered as a first order differential operator).
[12] [13] 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.
[14] 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.
[14] 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 aka, Newton's laws of motion averaged over ensemble.
With the help of the operator axioms, they can be rewritten as Notice a close resemblance with Ehrenfest theorems in quantum mechanics.
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
This assumption physically means that the classical particle's coordinate and momentum can be measured simultaneously, implying absence of the uncertainty principle.
obeying The need to employ these auxiliary operators arises because all classical observables commute.
evolves according to the Schrödinger-like equation of motion Let us explicitly show that KvN dynamical eq is equivalent to the classical Liouville mechanics.
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.
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
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.
One recovers Newton's laws of motion by applying the method of characteristics to either of these equations.
[26][27][28] The KvN approach is very general, and it has been extended to dissipative systems,[29] relativistic mechanics,[30] and classical field theories.
[14][31][32][33] The KvN approach is fruitful in studies on the quantum-classical correspondence[14][15][34][35][36] as it reveals that the Hilbert space formulation is not exclusively quantum mechanical.
[37] Even Dirac spinors are not exceptionally quantum as they are utilized in the relativistic generalization of the KvN mechanics.
