In Hilbert Space, All Things Are Quantumly Possible
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Physics Black Holes Evolution Home In Hilbert Space, All Things Are Quantumly Possible Comment Save Article Read Later Share Facebook Copied! Copy link Email Pocket Reddit Ycombinator Comment Comments Save Article Read Later Read Later quantum physics In Hilbert Space, All Things Are Quantumly Possible By Charlie Wood August 26, 2026
explainers history of science physics quantum physics All topics A t 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. “It’s a much more abstract space than that,” said Lucien Hardy , a physicist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada. 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. He got von Neumann thinking about the problem in the mid-1920s. In 1927 the 23-year-old prodigy — building on insights from Paul Dirac — solved it in a single-author trilogy of papers .
The mathematician David Hilbert sought a mathematical structure that would unify the different forms of quantum mechanics.
“The ideas specifically were von Neumann’s, but the inspiration — why do you axiomatize and what for — this is something that he took from Hilbert,” said Leo Corry , a historian of mathematics.
These papers laid out the rules for quantum mechanics, carefully defining the theory’s central objects and how they behaved. Von Neumann showed that Heisenberg’s towers and Schrödinger’s waves were reflections of the same entity, just as 0.5 and ½ indicate the same point on the number line. They both represented the main character in quantum mechanics: the quantum state.
Everything has a state. A coin can read heads or tails. A grandfather clock’s bob can take on any number of positions as it swings. Most of physics amounts to capturing an object’s state and predicting how it will change.
Von Neumann’s quantum rules complicate the notion of a state. Before you observe a quantum object, it does not have a fixed set of properties, such as a specific position. Instead it has a combination of possible properties unique to quantum mechanics — a “quantum superposition.” A superposition combines, for instance, all possible places the particle might end up being. Those possibilities can be precise and informative; perhaps there is a 99% chance you’ll find your particle to your left and a 1% chance you’ll find it to your right. You know how to place your bets, but you can’t know for sure if you’ve won until you check.
Von Neumann rendered the quantum state as a mathematical arrow called a vector. This arrow points in some direction through a space capturing all the possible futures of any quantum object — a Hilbert space.
Imagine a quantum traffic light with three possible states — red, yellow, or green. Its arrow exists in a three-dimensional Hilbert space, where the three axes represent the three possible future colors. Until the moment the light is observed, it doesn’t have a color, but rather a mixture of possible colors. So its arrow points into the space’s central region. The more closely the arrow aligns with, say, the red axis, the more likely the light is to shine red.
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The state of any object, from an electron to a galaxy, can be captured by such a vector, pointing in some direction through such a Hilbert space. This is von Neumann’s first rule of quantum mechanics.
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