In Hilbert Space, All Things Are Quantumly Possible

At the heart of quantum mechanics lie a few sacred rules for how to use the theory. First and foremost is, roughly, that thou shalt not think about ordinary objects presently whizzing through ordinary space. Rather, quantum mechanics predicts — in exquisite detail — all the possible ways that an object might turn out to be in the future.

Exploring those possible futures requires tracking an entirely different mathematical object — an arrow known as a vector, one oriented in an expansive, alien domain. These arrows aren’t pointing at locations. They’re “really pointing in a direction in a possibility space.” This possibility space is called Hilbert space, and it acts as the primary arena for quantum physics.

The early quantum pioneers didn’t realize — at first — that the arcane math that strikingly captured the conduct of atoms had left the real world behind. It took a visionary mathematical physicist, John von Neumann, to recognize and define the quantum world as a Hilbert space. Once he did, exploring the ins and outs of Hilbert space would lead physicists to a deeper, more unified understanding of quantum physics.

Here’s how von Neumann’s first commandment of quantum physics came to be, and how to understand it. Von Neumann’s commandments, or axioms, were his way of making sense of the two distinct forms of quantum mechanics developed back-to-back in the 1920s. First came Werner Heisenberg’s “matrix mechanics” in 1925.

It used inscrutable tables and, in later formulations, interminable towers of numbers to calculate the odds that an electron circling an atom would jump to a higher or lower orbit. The next year, Erwin Schrödinger introduced his “wave mechanics.” It used waves to track, for instance, the probability of a particle being found at a certain location in space. While the pictures evoked by these two physicists looked completely distinct, they yielded identical predictions.

Heisenberg and Schrödinger had come up with two radically different incarnations of one theory. But what was that theory? The question fascinated David Hilbert, a renowned mathematician who had devoted much of his life to rebuilding physics on a sturdy foundation of crisp axioms.