Arcs and Scaffoldings in the Construction of Relativity and Quantum Theory

By Michel Janssen

Program in the History of Science, Technology, and Medicine, University of Minnesota


Abstract:

It would seem that new theoretical structures in physics, unlike arcs and other architectural structures, could be erected without any scaffolding. After all, that is essentially how such structures as the four-dimensional formalism of special relativity, the curved space-times of general relativity, and the Hilbert space formalism of quantum mechanics are introduced in modern textbooks. Historically, however, these structures, like arcs, were first erected on top of elaborate scaffolding provided by the structures they ultimately replaced. Drawing on more detailed studies of the relevant episodes (undertaken in part with Tony Duncan, Jürgen Renn, and others), I provide thumbnail sketches of a few examples of such arcs and scaffoldings in the history of relativity and quantum theory.

My first example concerns special relativity. Lorentz’s theorem of corresponding states (in modern terms: the Lorentz invariance of Maxwell’s equations) and the electromagnetic mechanics with which Abraham sought to replace Newtonian mechanics formed the scaffolding for Minkowski’s new space-time geometry and Laue’s mechanics of continua in Minkowski space-time, which together offered a new foundation for all of physics and were no longer tied to electrodynamics.

My second example concerns general relativity. Lorentz’s theory of the electromagnetic field provided Einstein with the scaffolding for developing theories of the gravitational field, both his own and the so-called Nordström theory. It was only after this strategy had led him to identify the field equations for these theories that he removed the scaffolding and presented these theories as naturally suggested by the geometry of curved space-time.

My last two examples are taken from the history of quantum theory. Kramers dispersion formula, in which only quantities referring to transitions between orbits occur, provided Heisenberg with the scaffolding for a new theory for all of physics, not just dispersion, formulated entirely in terms of such transition quantities.

Finally, Jordan first realized that in quantum mechanics the usual rules for the composition of probabilities hold for complex probability amplitudes and not for the probabilities themselves, which are the squares of these amplitudes. The formalism Jordan used to implement this insight was rooted in the theory of canonical transformations familiar from classical mechanics. Von Neumann showed that the peculiar behavior of probabilities in quantum mechanics is much more naturally captured in his new Hilbert space formalism and discarded Jordan’s scaffolding.