The Koopman–von Neumann mechanics is a description of classical mechanics in terms of

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

, introduced by

Bernard Koopman Bernard Osgood Koopman (January 19, 1900 – August 18, 1981) was a French-born American mathematician, known for his work in ergodic theory, the foundations of probability, statistical theory and operations research. Education and work Af ...

and

John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...

in 1931 and 1932, respectively. As Koopman and von Neumann demonstrated, a

of complex, square-integrable wavefunctions can be defined in which classical mechanics can be formulated as an operatorial theory similar to

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

History

Statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...

describes macroscopic systems in terms of statistical ensembles, such as the macroscopic properties of an

ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...

. Ergodic theory is a branch of mathematics arising from the study of statistical mechanics.

Ergodic theory

The origins of Koopman–von Neumann (KvN) theory are tightly connected with the rise of

ergodic theory Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expr ...

as an independent branch of mathematics, in particular with Boltzmann's

ergodic hypothesis In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., t ...

. In 1931 Koopman and

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...

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 In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...

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 In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...

). This stands in analogy to the

Born rule The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of findi ...

in quantum mechanics. In the KvN framework, observables are represented by commuting self-adjoint operators acting on the

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 In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...

Kochen–Specker theorem In quantum mechanics, the Kochen–Specker (KS) theorem, also known as the Bell–Kochen–Specker theorem, is a "no-go" theorem proved by John S. Bell in 1966 and by Simon B. Kochen and Ernst Specker in 1967. It places certain constraints on the ...

, and Bell inequalities. The KvN wavefunction is postulated to evolve according to exactly the same

Liouville equation :''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelf ...

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 operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...

s 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

, and (iv) the state space of a composite system is the

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...

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 Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...

, the Liouville equation for the KvN wavefunction is recovered from averaged

Newton's laws of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in mo ...

. However, if the coordinate and momentum obey the

canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p_ ...

, the

Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...

of quantum mechanics is obtained.

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

() are first-order linear partial differential equations. One recovers

by applying the

method of characteristics In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partia ...

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 Perturbation or perturb may refer to: * Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly * Perturbation (geology), changes in the nature of alluvial deposits over time * Perturbat ...

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 In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...

, 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.Bracken, A. J. (2003). "Quantum mechanics as an approximation to classical mechanics in Hilbert space", ''Journal of Physics A: Mathematical and General'', 36(23), L329. Even

Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain co ...

s 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 The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...

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 In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanic ...

and

Aharonov–Bohm effect The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confine ...

are explicitly demonstrated in the KvN framework.