EPJ H Highlight - Crediting the real pioneers of classical wavefunctions

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Published on 20 January 2026

Study corrects a long-standing misconception surrounding the origins of classical wavefunctions in Hilbert space

A Hilbert space is an abstract space of finitely or infinitely many dimensions, and its mathematical properties make it incredibly useful for quantum and classical theories alike. In the 1930s, Bernard Koopman and John von Neumann found a way to formulate classical observable quantities in terms of ingredients in Hilbert spaces. Decades later, conceptually distinct methods emerged for formulating classical states as wavefunctions in Hilbert spaces – methods that have since become central to modern theories. In recent years, however, credit for both approaches has incorrectly gone entirely to Koopman and von Neumann.

In a new investigation published in EPJ H: Historical Perspectives on Contemporary Physics, Jacob Barandes at Harvard University shows how the Koopman-von Neumann formulation and the method of classical wavefunctions, despite both using Hilbert spaces, are each based on different underlying principles – with the latter possibly emerging through calculations carried out by Brazilian physicist Mario Schönberg in the 1950s. Barandes’s work corrects a long-standing misconception surrounding a cornerstone of modern classical theory, and could finally give the physicists really responsible the credit they deserve.

Hilbert spaces provide a robust framework for representing classical and quantum states as vectors – which can be treated mathematically just like vectors in real space. For classical systems, a Hilbert space enables physicists to represent probability distributions over positions and momenta as wavefunctions – which evolve over time according to equations which mathematically resemble quantum equations.

By revisiting the papers which led to this approach, Barandes shows that it was never explored by Koopman or von Neumann – despite the pair still being widely credited with it in recent years. Rather, the method of classical wavefunctions emerged through several independent studies in the 1950s – perhaps first by Schönberg, followed by others including Angelo Loinger, Giacomo Della Riccia, Norbert Wiener, and E. C. George Sudarshan. In clarifying the history of the approach, Barandes ultimately hopes that the study may finally provide credit long due to the physicists who really developed it.