To explore quantum phenomena, we must leave the familiar world and enter the abstract realm of Hilbert space.
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. “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.
