A simple system, like a computer bit that can be either zero or one, might have two dimensions.
Most quantum systems are much more complex. Take a single hydrogen atom. Its electron can reach higher and higher orbits as you give it more energy. In this case, the number of possible states is unlimited, and so its Hilbert space is infinite-dimensional. Most real quantum systems have this feature.
Edgar Shaghoulian, a physicist at the University of California, Santa Cruz, noticed a connection of the strange behavior to topological field theories.
Physicists therefore expect a whole universe to have an infinite number of states too. But when Maldacena applied the island formula to a closed universe, he found instead that it had a Hilbert space with just one dimension. There was no information to be found. The whole universe and everything in it could be in only one quantum state. It lacked even the complexity of a single bit.
This conclusion struck physicists as paradoxical, given that we too could conceivably live in a closed universe. And we clearly see far more than a single state around us.
“On my desk there are an infinite number of states,” said Edgar Shaghoulian, a physicist at the University of California, Santa Cruz.
But as physicists continued to study different types of closed universes, they kept seeing the same pattern. While the IAS group considered black holes, Maxfield and his collaborator Donald Marolf looked at hypothetical quantum bubbles of space-time called baby universes. They found the same stark simplicity. Increasingly, it appeared that the barrenness of closed universes was a universal trend.
“Eventually we believed it,” Zhao said.
Complexity Returns
The situation presents a paradox: Calculations consistently imply that any closed universe has only one possible state. But our universe, which may very well be closed, seems infinitely more complex. So what’s going on?
In a 2023 essay, Shaghoulian noted that physicists had seen this strange behavior before in theories called topological field theories. Mathematicians use these theories to chart the shape, or topology, of geometric spaces. Topological field theories can also have one-dimensional Hilbert spaces. But if you split up the geometric space into multiple zones, you can describe the space in many different ways. To keep track of all the new possibilities, you need a bigger Hilbert space.
“The rules of the game change,” Shaghoulian said.
Shaghoulian proposed that there might be a similar way to split up a closed universe: Bring in an observer.