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+--- Thread: Article In Hilbert space, all things are quantumly possible (/thread-21195.html)
In Hilbert space, all things are quantumly possible - C C - Aug 27, 2026
https://www.quantamagazine.org/in-hilbert-space-all-things-are-quantumly-possible-20260826/
EXCERPTS: 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.
Here’s how von Neumann’s first commandment of quantum physics came to be, and how to understand it.
[...] The more possible futures an object has, the bigger its Hilbert space. A coinlike particle with two possible futures is a “qubit,” the computational building block of quantum computers. It has a two-dimensional Hilbert space. Our three-color traffic light has a three-dimensional Hilbert space. But that’s just the beginning. A freely floating particle could be found in any location in the universe, so its Hilbert space must span an infinite number of dimensions.
This size — whether it’s two dimensions or an infinite number — is the only fundamental feature of a Hilbert space, according to von Neumann’s rules. The axes are arbitrary and imagined by us; they aren’t intrinsic to the space.
[...] This freedom to carve up Hilbert space as we see fit is what allowed Heisenberg and Schrödinger to come up with two distinct versions of the same theory. [...] John von Neumann developed an underlying structure for quantum mechanics that involves arrows moving inside an abstract Hilbert space...
[...] A century after the birth of quantum mechanics, the theory has left physicists in an awkward position. We live in a world where objects change position as they move through three dimensions of physical space. But our most fundamental theory takes place somewhere else, in von Neumann’s vast realm of possibilities. What does that imply about the reality of our world, or that of Hilbert space?
To Sean Carroll , a philosopher and physicist at Johns Hopkins University, the message is clear. If quantum mechanics is the fundamental theory of nature, then Hilbert space should be considered the fundamental theater of reality, he argued in a 2022 paper. One of his lines of research seeks to distill our familiar world from the disorienting Hilbert space that encapsulates all the ways the universe could possibly be.
Other physicists take a more pragmatic stance. Jonathan Sorce, a physicist at Princeton University, says that Hilbert space is a handy mathematical construction that is remarkably useful for describing many quantum systems — but not all of them. He belongs to a community of researchers searching for a mathematical construction that can describe the fabric of space and time as a quantum object. Such a theory is a prerequisite for answering big questions such as what goes on at the heart of a black hole.
Physicists asking these questions have recently focused on an even more abstract space that seems especially well suited for their purposes. This kind of space is made up of the things you could do to a Hilbert space, such as slicing it up in different ways or rotating one slice into another. In this arena, they have found that black holes seem a bit less mysterious. This sort of über-space is known today as a von Neumann algebra... (MORE - missing detail)