If the Universe Is a Hologram, This Long-Forgotten Math Could Decode It

Von Neumann and a collaborator, Francis Murray, eventually identified three types of operator algebras. Each one applies to a different kind of physical system. The systems are classified by two physical quantities: entanglement and a property called entropy.

Physicists first discovered entropy while studying steam engines in the 1800s. They later came to understand it as a measure of uncertainty. You might know the temperature of a gas, for instance, but you’ll remain uncertain of the specific locations of all its molecules. Entropy counts how many possible states of the molecules’ positions and trajectories there could be. Similarly, in quantum systems, entropy is also a measure of your ignorance. It tells you how much information you can’t access because of the entanglement between your quantum system and the world outside.

Von Neumann algebras specify what kind of entanglement a system has, and accordingly, how well you can come to know it.

Type I algebras are the simplest. They describe systems with a finite number of parts, which can be completely disentangled from the rest of the universe. So if the parts of the system do become entangled with the outside, you can tell precisely how much they have done so. Their entropy — your ignorance — is limited. You can always calculate exactly what it is. Sorce likens such algebras to a beaker with the water level representing the entropy. You can see the bottom, so you know the height of the water.

Type II algebras are trickier. They describe systems that have an infinite number of parts, all inextricably entangled with the outside. Absolute entropy is infinite — and therefore meaningless. But the system has some uniformity that gives you a reference point. The parts might all be as entangled with the outside as they can possibly be, for instance. Then, if you disentangle five particles, you know that entanglement has decreased by five units. The absolute amount of uncertainty is unknowable, but you’re a little less uncertain than before; five units less, to be precise. You can’t see the beaker’s bottom, but you can see when the water level rises or falls.

The final type, type III, is the worst: It describes a system with infinite parts, infinite entanglement with the outside, and no uniform pattern in the entanglement to help you get oriented. Not even changes in entropy are knowable. The beaker’s bottom is too distant to see, and so is the water level above you.

“Type III is flipping horrible, and no one wants to deal with them,” Penington said (using stronger language than “flipping”).

When von Neumann and Murray first encountered type III algebras, they found them too alien to understand. The nature of these algebras would remain mysterious for more than three decades until Alain Connes, a French mathematician, managed to define them in 1973. The feat won Connes the Fields Medal, math’s highest honor. He determined that what set type III algebras apart was related to a fearsomely technical property called modular flow.

Very roughly speaking, modular flow resembles the flow of time — but it’s more abstract. It’s a physical process that takes a system at a particular temperature and keeps it at that temperature. A room-temperature cup of tea naturally experiences modular flow (and normal physical time) because it stays at room temperature. But for a steaming-hot cup of tea, modular flow is the sequence of operations needed to keep it eternally hot. That’s not something that would ever happen naturally, since it requires constantly fiddling with all the tea’s atoms, but it’s a process that can be specified mathematically. Connes realized that a type III algebra describes a system so entangled with its surroundings that the system’s modular flow also becomes inseparable from what’s going on outside.

Mathematicians — and a few intrepid physicists — would continue to study von Neumann algebras and their modular flows. But only in the last few years have quantum gravity researchers come to appreciate their power.

Alien Algebra

When Liu and Leutheusser were trying to understand what happens inside a black hole, they situated it in a perfectly smooth bulk space-time. They knew that fluctuating, quantum space-time corresponded to a finite number of entangled fields on the boundary and a type I theory. But as they added fields to the boundary to ensure that space-time became smooth, they saw that the algebra changed from type I to type III. In other words, the more fields there were, and the more entanglement, the closer space-time behaved to its idealized, classical version.

They then used the hopelessly entangled modular flow of the type III algebra to sneak a peek inside the black hole lurking in their bulk. Starting with a simple pattern of boundary ripples that they knew simulated a measurement device outside the black hole, they argued that a certain procedure involving a type III modular flow would bring the device inside the hole, where it could measure the flow of time. This process achieved Liu’s goal of determining what intricate pattern of boundary ripples was equivalent to a ticking clock inside a holographic black hole.

“These new structures give you emergent time,” Liu said.

They weren’t the only physicists rediscovering von Neumann algebras. Other groups were also using modular flow to understand black holes. A 2017 proposal, for instance, took a measurement device inside a black hole and scrambled it in such a way that it ended up outside. And in 2020 researchers imagined firing a small black hole into a larger one and using the little black hole’s modular flow to get it back out.

Sorce, who worked on another modular flow procedure this spring, says these algorithms are all pushing toward a single goal: understanding how quantum particles would behave near a singularity. The singularity would live in AdS space rather than in a realistic universe, but most holographers expect that all space-time fabrics should fray in similar ways. (Physicists outside the holography community question that assumption.) “If you could understand singularities in AdS space at the quantum level, you would be very happy in declaring victory in understanding them in our universe,” Sorce said.

Liu and Leutheusser put a spotlight on what had been something of a backwater of mathematical physics. “Before Hong’s paper,” said Elliott Gesteau, a mathematical physicist at the California Institute of Technology, “it was kind of like a dream. There was a hunch that this must be important, but it wasn’t clear how to make this intuition precise.”