What Is Hilbert Space? The 1927 Math That Still Defines Quantum Mechanics

In the 1920s, physicists developed two very different-looking descriptions of quantum mechanics. Werner Heisenberg's 'matrix mechanics', introduced in 1925, used tables of numbers to calculate the odds that an electron circling an atom would jump to a higher or lower orbit. The following year, Erwin Schrodinger's 'wave mechanics' used waves to track the probability of finding a particle at a given location. The two pictures looked nothing alike, but they produced identical predictions.

The mathematician David Hilbert, who had spent much of his career trying to rebuild physics on precise mathematical axioms, set the young John von Neumann the challenge of explaining why. In 1927, building on the work of Paul Dirac, the 23-year-old von Neumann solved it in a trilogy of single-author papers, showing that Heisenberg's tables and Schrodinger's waves were two different representations of the same underlying object: the quantum state.

Von Neumann represented that state as a mathematical arrow, or vector, pointing through an abstract space of every possible outcome, which came to be called Hilbert space. Before a quantum object is measured, it does not have a fixed property such as a specific position; instead it exists in a mix of possibilities called a superposition, with the vector's direction encoding the odds of each possible outcome.

Historian of mathematics Leo Corry says the idea of building quantum theory on strict axioms came from Hilbert, even though the specific mathematics was von Neumann's own. Von Neumann's framework also describes how the vector behaves. It turns smoothly and predictably through Hilbert space as a quantum object interacts with its surroundings. The moment it is measured, though, it snaps instantly and unpredictably onto a single definite outcome.