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arXiv:1705.07427 · Quantum Studies Mathematics and Foundations

From Koopman–von Neumann theory to quantum theory

U. Klein

Published May 31, 2017• 25 citations
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Abstract

Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function $$S^{(K)}(q,p,t)$$ whose physical meaning is unknown. We show that a different S(q, p, t), with well-defined physical meaning, may be introduced without destroying the attractive “quantum-like” mathematical features of the KvN theory. This new S(q, p, t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrödinger’s equation if functions on phase space are reduced to functions on configuration space. This new kind of “quantization” does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing $$\frac{\partial }{\partial p}$$ by 0 and p by $$\frac{\hbar }{\imath } \frac{\partial }{\partial q}$$ , thus providing an explanation for the common quantization rules.

DOI
10.1007/s40509-017-0113-2

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Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function S(K)(q,p,t)S^{(K)}(q,p,t)S(K)(q,p,t) whose physical meaning is unknown. We show that a different S(q, p, t), with well-defined physical meaning, may be introduced without destroying the attractive “quantum-like” mathematical features of the KvN theory. This new S(q, p, t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrödinger’s equation if functions on phase space are reduced to functions on configuration space. This new kind of “quantization” does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing ∂∂p\frac{\partial }{\partial p}∂p∂​ by 0 and p by ℏı∂∂q\frac{\hbar }{\imath } \frac{\partial }{\partial q}ℏ​∂q∂​ , thus providing an explanation for the common quantization rules.
Abstract We investigate operator algebraic origins of the classical Koopman–von Neumann wave function ψKvN\psi _{KvN}ψKvN​ ψ KvN as well as the quantum-mechanical one ψQM\psi _{QM}ψQM​ ψ QM . We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, ψKvN\psi _{KvN}ψKvN​ ψ KvN and ψQM\psi _{QM}ψQM​ ψ QM are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle E{{\mathcal {E}}}E E follows from this. ψKvN\psi _{KvN}ψKvN​ ψ KvN and ψQM\psi _{QM}ψQM​ ψ QM are shown to be sections in E{{\mathcal {E}}}E E . The difference between ψKvN\psi _{KvN}ψKvN​ ψ KvN and ψQM\psi _{QM}ψQM​ ψ QM originates from a quantization map interpreted as “twisting” of sections over E{{\mathcal {E}}}E E . We also show that the Schrödinger equation is obtained from the Koopman–von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.
The Koopman–von Neumann (KvN) formulation recasts deterministic classical mechanics in a Hilbert-space operator framework analogous to quantum mechanics. In this approach, the Liouville equation is expressed as a Schrödinger-type evolution equation generated by a Hermitian operator, yielding unitary propagation for phase-space-volume-preserving dynamics. We first clarify the relationship among the Koopman, Perron–Frobenius, and Liouville operators through an explicit example. Wilczek's construction shows that the KvN formalism is not only restricted to canonical Hamiltonian mechanics but this can be generalized any first-order classical dynamical system. Wilczek shows how a general autonomous first-order system can be generated by a Hermitian temporal-evolution operator after the introduction of auxiliary conjugate variables. We develop the KvN description of systems driven by Hamiltonian curl forces–position-dependent, non-conservative forces that are non-dissipative and preserve phase-space volume. Finally, we extend the KvN formalism to non-Hamiltonian systems using the Jacobi Last Multiplier and examine its connection to the Boltzmann H-theorem.
A semiclassical approximation is derived by using a family of wavepackets to map arbitrary wavefunctions into phase space. If the Hamiltonian can be approximated as linear over each individual wavepacket, as often done when presenting Ehrenfest's theorem, the resulting approximation is a linear first-order partial differential equation on phase space, which will be referred to as the Schrödinger-Ehrenfest or SE equation. This advectively transports wavefunctions along classical trajectories, so that as a trajectory is followed in time the amplitude remains constant while the phase changes by the action divided by ℏ\hbarℏ. The wavefunction's squared-magnitude is a plausible phase space density and obeys Liouville's equation for the classical time evolution. This is a derivation of the Koopman-von~Neumann (KvN) formulation of classical mechanics, which previously was postulated but not derived. With the time-independent SE equation, continuity of the wavefunction requires the change of phase around any closed path in the torus covered by a classical trajectory to be an integer multiple of 2π2π2π, giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.
The so-called Koopman-von Neumann (KvN) formulation brings classical mechanics to Hilbert space, but many techniques familiar from quantum mechanics remain missing. We use the gauge freedom of the KvN equation to derive separable solutions of the KvN equation, and by extension the classical Liouville equation. The canonical ensemble solution of statistical mechanics is shown to arise from separable solutions. We discuss the Hermiticity of the Liouvillian operator, and treat a case where its spectrum is discrete. Finally, we analyze an uncertainty relation between the so-called dynamical time and the Liouvillian, and argue that such a relation is natural and can be understood entirely within a classical framework.
The Koopman-von Neumann (KvN) theory is one where the dynamical momentum is not canonically conjugate to position, i.e., position and momentum are deconjugated. From this point of view, we show that the KvN theory arises from quantum mechanics, extracting classical equations of motion from quantum ones. However, preserving the canonical structure of the theory requires introducing ``auxiliary'' canonical conjugates to position and momentum. We show that using the KvN formulation to study the interaction between quantum and classical systems forces the auxiliary variables to take on a physical role. While giving rise to classical behaviour, the KvN theory might be more than classical.
We show that quantum theory (QT) is a substructure of classical probabilistic physics. The central quantity of the classical theory is Hamilton’s function, which determines canonical equations, a corresponding flow, and a Liouville equation for a probability density. We extend this theory in two respects: (1) The same structure is defined for arbitrary observables. Thus, we have all of the above entities generated not only by Hamilton’s function but also by every observable. (2) We introduce for each observable a phase space function representing the classical action. This is a redundant quantity in a classical context but indispensable for the transition to QT. The basic equations of the resulting theory take a “quantum-like” form, which allows for a simple derivation of QT by means of a projection to configuration space reported previously [Quantum Stud Math Found 5:219–227, 2018]. We obtain the most important relations of QT, namely the form of operators, Schrödinger’s equation, eigenvalue equations, commutation relations, expectation values, and Born’s rule. Implications for the interpretation of QT are discussed, as well as an alternative projection method allowing for a derivation of spin.