Solutions of Koopman-von Neumann equations, their superpositions, orthogonality and uncertainties

Abstract: The Koopman-von Neumann (KvN) formulation brings classical mechanics to Hilbert space, but many techniques familiar from quantum mechanics remain missing. One would hope to solve eigenvalue problems, obtain orthonormal eigenstates of Hermitian operators and ascribe meaning to a coherent superposition of states, among other things. Here we consider the general KvN equation for a classical probability amplitude and show that its so-called gauge freedom allows the separation of variables. The amenability to Hilbert-space methods of the resulting KvN solutions is investigated. We construct superpositions from differently-gauged Liouvillian eigenstates, and find an orthonormal set among them. We find that some separable solutions describe the canonical ensemble with temperature related to the separation constant. Classical uncertainty relations arise naturally in the KvN formalism. We discuss one between the dynamical time and the Liouvillian in terms of the statistical description of classical systems.

• The paper demonstrates that classical mechanics can be reformulated in Hilbert space by deriving separable KvN solutions that yield canonical ensemble distributions.
• The paper develops a gauge freedom framework that unifies different KvN solutions into a common Hilbert space structure, enabling robust orthonormal state expansions.
• The paper reveals classical uncertainty relations and epistemic state collapse, offering a fresh perspective on the classical–quantum boundary.

Koopman-von Neumann Equations: Solutions, Superpositions, Orthonormality, and Uncertainties

Overview and Motivation

This paper addresses foundational and technical aspects of the Koopman-von Neumann (KvN) formulation, which extends classical mechanics into Hilbert space via probability amplitudes that evolve under a Schrödinger-like equation—a formalism that parallels quantum theory but remains intrinsically classical. The central focus is the exploration of general solution structures for the KvN equation, their gauge freedom, superposability, orthogonality, and the emergence of classical uncertainty relations.

The motivation lies in enabling classical mechanics to fully utilize Hilbert space techniques—specifically, the expansion in terms of orthonormal eigenstates, the separation of variables, and a systematic understanding of state superpositions. The work investigates the depth to which these quantum-inspired tools can be meaningfully applied, examines their statistical origin, and clarifies how phenomena such as uncertainty and state collapse manifest in a strictly classical but epistemic context.

KvN Formalism and Gauge Freedom

At the heart of KvN mechanics is the replacement of the classical phase space probability density $\rho(q,p,t)$ (obeying the Liouville equation) with a complex probability amplitude $\chi(q,p,t)$ such that $|\chi|^2 = \rho$. The evolution of $\chi$ is dictated by the generalized KvN (gauged-KvN) equation: $$i\hbar \partial_t \chi = \hat{H} \chi, \quad \hat{H} = i\hbar\{H, \cdot\} + \alpha(q,p,t)$$ where $\alpha(q,p,t)$ is a real, arbitrary gauge function. The gauge freedom—rooted in the ability to adjust $\chi$ up to a phase without affecting $\rho$—enables the tailoring of solutions and the application of powerful Hilbert-space methodologies.

This freedom is essential for inducing separability in variables where the canonical Liouville or traditional KvN equations are insufficient, and it underpins all subsequent technical constructions in the paper.

Separable Solutions and Emergence of the Canonical Ensemble

Leveraging the full gauge freedom, the authors demonstrate that for time-independent, separable Hamiltonians, it is possible to choose $\alpha(q,p)$ such that the KvN equation admits separable solutions in phase space variables. Specifically, for additive Hamiltonians of the form $H(q,p) = K(p) + U(q)$, they show:

• Time-separable solutions correspond to stationary states and, upon squaring, yield equilibrium probability distributions.
• Full separability in $q$ and $p$ leads to solutions whose squared modulus is precisely the canonical ensemble distribution:

$$|\chi|^2 = \frac{1}{Z(\beta, \Gamma)} \exp(-\beta H)$$

where the separation constant $\beta$ plays the role of inverse temperature.

This construction directly ties the gauge structure of the KvN equation to statistical mechanical ensembles and highlights that the statistical temperature emerges as a parameter encoding an observer’s knowledge, rather than dynamical properties per se.

Cross-Gauge Superpositions and Hilbert Space Structure

The authors address the nontrivial issue that superpositions of KvN solutions, each in a different gauge, are not automatically solutions of any KvN-like equation. To resolve this, they construct a formalism of cross-gauge superposition: individual solutions are gauge transformed to a common "superposition gauge" via

$$\chi_j \to \chi_j^{(s)} = \chi_j e^{i\varphi_j/\hbar},\quad \alpha_j \to \alpha^{(s)} = \alpha_j - D\varphi_j$$

so that their superpositions evolve under a single unified KvN equation. In this way, the Hilbert space of solutions supports orthonormal expansion in terms of eigenstates with well-characterized time-evolution and statistical interpretation.

Orthonormal Bases and Hermiticity

Through explicit computation, the paper establishes:

• The existence of complete orthonormal systems formed from Liouvillian (or tilde-Hamiltonian) eigenstates of the form

$\chi_n = f_n(H) e^{i \epsilon_n (\tau - t)/\hbar}$

where $\tau$ is canonically conjugate to $H$, and $f_n$ is fixed by the initial distribution.

• The Hermiticity of the (gauged) Liouvillian operator in the Hilbert space of square-integrable functions, contingent upon precise boundary conditions (e.g., finite intervals in $\tau$ for bounded systems like the oscillator).
• An explicit construction of (typically discrete) spectra of orthogonal states even in entirely classical settings.

Expansion coefficients for arbitrary initial states are derived, and completeness relations are explicitly verified in nontrivial cases.

Illustration: The Harmonic Oscillator

The technical machinery is employed in detail for the 1D harmonic oscillator:

• The canonical coordinates $(\tau, H)$ are found, with $\tau$ as the dynamical time variable.
• Orthonormal eigenstates are constructed, and specific expansions are detailed for both microcanonical and shifted canonical initial distributions.
• The normalization and completeness of the expansions are explicitly proved using analytic and integral estimates.

This example exemplifies the general method and confirms the practical use of Hilbert-space tools for classical time-evolution in the KvN formalism.

Uncertainty Relations and Epistemic Collapse

A notable conceptual development is the emergence of classical uncertainty relations in the KvN framework: $\Delta\tau\,\Delta\hat{H} \geq \frac{\hbar}{2}$ where $\tau$ (the internal dynamical time) and $\hat{H}$ (the Liouvillian generator of time evolution) form a noncommuting canonical pair. The implication is that equilibrium states (Liouvillian eigenstates) enforce maximal ignorance of $\tau$; conversely, precise knowledge of $\tau$ precludes definite equilibrium.

This is framed as a strictly epistemic uncertainty: every phase space point is ontically well-defined, but the KvN formalism organizes knowledge in such a way that epistemic restrictions are encoded by these uncertainty relations. The paper emphasizes that KvN uncertainty is a classical analogue of quantum uncertainty, rooted entirely in the statistical mechanical (information-theoretic) structure, not in any quantum indeterminacy.

Similarly, the collapse of the classical KvN probability amplitude upon measurement is interpreted as knowledge update, avoiding quantum measurement paradoxes.

Implications, Theoretical and Practical

The results have several implications:

• Conceptual Unification: The work strengthens the position that Hilbert space methods, noncommutativity, and superposition are not uniquely quantum, but can serve classical statistical mechanics under the KvN representation, clarifying sharp boundaries between quantum and classical statistical descriptions.
• Statistical Foundations: The emergence of canonical ensembles, uncertainty relations, and state collapse as statistical/informational phenomena in the KvN formalism supports the view that many apparent quantum structures have classical epistemic origins.
• Technical Utility: The demonstrated capability to construct orthonormal bases and to use Hilbert space tools for solving classical evolution, including time-dependent distributions and state preparations, renders the KvN formalism more practically relevant for hybrid quantum-classical problems and for foundational studies.
• Future Prospects: There are strong suggestions that phenomena such as interference, entanglement analogues, and nonlocal correlations—studied in epistemically restricted toy theories—might be systematically investigated in the KvN framework, potentially yielding new insights into the classical-quantum interface and the role of epistemic constraints.

Conclusion

This paper provides a rigorous foundation for the deployment of Hilbert space concepts—superposition, orthonormal eigenstates, gauge transformations, and uncertainty relations—within purely classical mechanics via the KvN formalism. It elucidates the profound statistical mechanical origin of these structures and reinforces the view that many traditionally "quantum" features arise naturally in classical epistemic descriptions when couched in Hilbert space language. The technical apparatus developed herein opens new directions for the analysis of the classical-quantum boundary, the information-theoretic structure of classical physics, and the practical application of KvN techniques to complex systems.