What Properties Define Hilbert Space?
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.
Consider an electron. It has one state, one arrow, pointing in a vast Hilbert space. Its Hilbert space spans all possible measurements — energy, position, momentum, etc. If you are curious where the electron might be, you can mark the space with the axes that represent possible positions. If you are wondering where the particle might be going, you apply a different set of axes, those representing possible momenta. No matter which measurement you intend to make, the underlying Hilbert space remains the same.
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. Heisenberg’s picture essentially put in axes and let them rotate around the vector, while Schrödinger’s picture did the opposite: It put in a fixed set of axes and let the vector rotate relative to them. They were two completely different mathematical perspectives on the same arrows, in the same Hilbert spaces.
In the first of his 1927 papers, von Neumann laid out two mathematical criteria that defined such a space. First, it had to be “complete.” It couldn’t be missing any regions or points. And second, you had to be able to calculate the alignment between a state and an axis, which you can visualize by imagining a light shining straight down onto an arrow so it casts a shadow on an axis. (The longer the shadow, the more aligned with that axis the arrow is.) The space had to allow for this operation, known as an inner product. Any space with these two features, no matter its size or origin, was a Hilbert space.
“This was a major step in creating what we call Hilbert space quantum mechanics,” said Miklós Rédei, a philosopher of physics at the London School of Economics. “It’s a beautiful example of how mathematical generalization or abstraction takes place.”
Von Neumann referred to these abstract spaces as Hilbert spaces because his mentor Hilbert had been the first mathematician to explore specific spaces with infinite dimensions in the early 1900s. Hilbert’s work had relied on those spaces being complete and having an inner product, but he didn’t think of them as examples of a more general class of spaces until his protégé grouped them together. The older mathematician may have been surprised to find his name attached to this new mathematical structure. “Dr. von Neumann, I am really curious to know what these Hilbert spaces are, after all,” Hilbert reportedly asked during a 1929 lecture.
Is Hilbert Space Real or Just an Abstraction?
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. Von Neumann himself helped develop it as a potential remedy for some logical inconsistencies with Hilbert space that troubled him. “I would like to make a confession which may seem immoral: I do not believe in Hilbert space anymore,” he wrote in a 1935 letter while exploring the virtues of algebras.
Sorce, for his part, doesn’t share von Neumann’s desire for one space to rule them all. He’s content to use whichever mathematical construction best suits the quantum object he’s studying. Often it’s a Hilbert space. Sometimes it’s a von Neumann algebra. And occasionally it might even be one of the many other spaces mathematicians have cooked up over the last century.
“There’s a whole zoo of these things,” he said.
