Wikipedia details Von Neumann expanded Koopman’s 1931 Hilbert-Space work

In 1931, Bernard Koopman published a paper in Proceedings of the National Academy of Sciences laying out a novel method for identifying functions representing observables on a classical system’s phase space as vectors in a Hilbert space; however, the full story of this now-ubiquitous technique, and its connection to quantum theory, reveals a complex history of attribution. John von Neumann followed up in 1932 with further development published in German Annals of Mathematics, papers that were never translated into English.

As Jacob Barandes details in a new analysis, the method of “classical” wave functions often linked to Koopman and von Neumann was, in fact, independently developed starting with Mario Schönberg, with later key contributions from Angelo Loinger, Giacomo Della Riccia, Norbert Wiener, and E. C. George Sudarshan. Indeed, as Jordan and Sudarshan noted in 1961, the resulting Hilbert space “corresponds not to the space of state vectors in quantum mechanics but to the Hilbert space of operators on the state vectors.” A mathematical framework initially developed to bridge classical and quantum mechanics has, for decades, been subject to significant misattribution. While often referred to as the “Koopman–von Neumann (KvN) formulation,” the widely accepted understanding of this approach obscures the contributions of several key researchers who built upon the original work. Barandes details in recent research that Koopman’s initial formulation did not involve classical wave functions linked to probability distributions via the Born rule; rather, his Hilbert spaces “consisted of functions representing classical observables.” This erroneous association, exemplified by a 2021 conference and a 2022 Journal of Physics A special issue both titled “Koopman Methods in Classical and Classical-Quantum Mechanics,” which referenced “Koopman–von Neumann wave functions,” persists even in contemporary literature, including a 2025 Wikipedia entry stating that the approach.

Classical Wave Functions: Origins Beyond Koopman-von Neumann Following renewed interest in applying Hilbert-space methods to classical physics, a historical clarification is emerging regarding the origins of “classical” wave functions. While often linked to the work of Bernard Koopman and John von Neumann, the technique was not their invention, but rather developed independently by several researchers in subsequent decades. Angelo Loinger, Giacomo Della Riccia, Norbert Wiener, and E. C. George Sudarshan made key contributions. Koopman’s 1931 paper, “Hamiltonian Systems and Transformations in Hilbert Space,” laid out a novel method for representing classical observables as vectors within a Hilbert space, a concept further expanded by von Neumann in 1932. These researchers did not propose classical wave functions analogous to their quantum counterparts. As Barandes details in recent work, the subsequent development of using complex-valued functions in place of classical probability distributions, linked by the modulus-square operation, is where the divergence occurs. Danilo Mauro’s 2002 paper, “On Koopman–von Neumann Waves,” replaced the observables-as-vectors method described above with an important but different technique, inadvertently cementing the misattribution. Barandes explains that this method of classical wave functions was due to other researchers who came decades after Koopman and von Neumann’s papers. The pursuit of classical analogues to quantum wave functions, now a tool in fields ranging from plasma physics to weather forecasting, has a surprisingly obscured history. While the “Koopman–von Neumann (KvN) formulation” is frequently invoked when discussing these “classical” wave functions, attributing their origins to Bernard Koopman and John von Neumann is inaccurate. The shift towards classical wave functions, where complex-valued functions replace probability distributions and are linked via the modulus-square operation, emerged decades later. The widespread association of methods with classical wave functions is a historical misstep, obscuring the true origins of this mathematical technique. John von Neumann’s subsequent papers in 1932, appearing in German in Annals of Mathematics, further developed Koopman’s approach. Both Koopman and von Neumann’s papers were never translated into English. As of 2025, the Wikipedia entry for “Koopman–von Neumann Classical Mechanics” even begins by defining the framework, highlighting the extent of this enduring misattribution. Von Neumann’s two papers were never translated into English. Mauro’s work demonstrated that the two resulting Hilbert spaces, one arising from observables-as-vectors and the other from classical wave functions, are mathematically equivalent, a consequence of the GNS construction (Gelfand, Naimark 1943; Segal 1947). Despite this equivalence, the conceptual origins are distinct, a point Mauro emphasized, drawing a parallel to the separate development of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics in quantum theory. The apparent connection between two distinct approaches to representing classical physics within Hilbert spaces, the Koopman-von Neumann (KvN) formulation and the method of “classical” wave functions, has long been a source of historical confusion. While both utilize the mathematical framework of Hilbert spaces, their origins and underlying principles differ significantly, a distinction often blurred in contemporary discussions. Recent work clarifies that the mathematical equivalence between these seemingly disparate methods stems directly from the GNS construction, a result in operator algebra established by Gelfand, Naimark in 1943, and Segal in 1947. This construction, as the author details, takes a C*-algebra representing observables and combines it with a positive linear functional representing a quantum state, ultimately defining an inner product that underpins the Hilbert space. Despite this underlying mathematical link, the author emphasizes that Koopman and von Neumann did not originate the concept of using classical wave functions to represent probability distributions. Researchers at Harvard University are meticulously re-examining the historical foundations of classical and quantum mechanics, with a particular focus on the often-blurred lines between Hilbert space formulations. This work challenges a long-held assumption regarding the development of “classical” wave functions, revealing a more complex history than previously understood, and often conflates two distinct approaches. 👉 More information🗞 The History of Hilbert-Space Formulations of Classical Physics✍️ Jacob A. Barandes🧠 ArXiv: https://arxiv.org/abs/2607.17408 Stay currentSee today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals. Tags: