Operator algebras and the substructure of space and time

A 1930s-era breakthrough is helping physicists understand how quantum threads could
weave together into a holographic space-time fabric.

Motivated by the mysteries of black holes, theorists are advancing their ability to describe space-time as a hologram.

Señor Salme for Quanta Magazine

Introduction

John von Neumann came about as close as humanly possible to embodying the Platonic ideal of a genius. Conversant in ancient Greek by age 6, the Hungarian made significant mathematical advances in his teens. Then, as an adult, he invented game theory and helped design the atomic bomb and the modern computer.

Along the way, as a young man in 1932, von Neumann rewrote the rules of quantum mechanics, formulating the strange new theory of particles and their fluctuating, probabilistic behavior in the mathematical language used today. Then he went further. He developed a framework known as “operator algebras” to describe quantum systems in a more powerful but more abstract way. Unlike his earlier work on quantum theory, this framework was hard to understand and did not catch on widely in theoretical physics. It was literally a century ahead of its time.

Over the past few years, however, more physicists have been dusting off von Neumann’s ideas. His operator algebras are now helping them see their way around the most mysterious quantum system yet: the substructure of space and time.

Even before von Neumann did his work, Albert Einstein’s theories of relativity merged space and time into a four-dimensional fabric known as “space-time.” Einstein showed that the force of gravity is generated by curves in this fabric. But physicists know that the fabric can’t be the whole story. Dying stars puncture it, creating intensely warped regions called black holes where the equations of general relativity break down. And even in calmer parts of space-time, when you zoom in to the smallest scales, quantum fluctuations seem to shred it apart.

Many theoretical physicists therefore believe that space-time will go the way of water, metals, and so many other substances before it; what seems like a smooth and simple medium will turn out to be made of a complicated collection of primitive quantum entities. For decades, theorists have wondered about those entities and how the space-time fabric emerges from them.

These physicists are now gaining a deeper understanding of space-time’s quantum weave. They’re developing new ways of predicting what happens in extreme regions where space-time as we know it unravels, as well as identifying the conditions that normally allow it to hang together. At the heart of the progress has been von Neumann’s abstruse research.

“People have been kind of scared of it,” said Antony Speranza, a physicist at the University of Amsterdam. But “it does seem to give you these algebraic tools for seeing that a space-time is emerging.”

The Emergence of Emergence
In the spring I took the train to Princeton, New Jersey, and walked to the pastoral campus of the Institute for Advanced Study. This was where von Neumann developed the math of operator algebras, and where Albert Einstein lived out his days, after both immigrated to the United States and joined the institute’s first generation of professors. It remains a major hub of fundamental research. My first stop was the chalkboard-lined office of Juan Maldacena, one of the most respected theorists working today.

In 1997, the Argentinian physicist caught the first glimpse of the most famous example of how space-time can emerge — an enigmatic relationship known as the AdS/CFT correspondence. “It gives you an explicit model of emergent space-time,” Maldacena told me.

The correspondence amounts to a striking quantum conspiracy.

Juan Maldacena, a physicist at the Institute for Advanced Study, discovered a way of recasting a black hole in an exotic space-time as a collection of quantum ripples.

Sasha Maslov for Quanta Magazine

To get a sense of how it works, imagine you have a two-dimensional sheet of metal wrapped into a sphere, like a hollow aluminum ball (it remains 2D in the sense that you can locate any point on it with a longitude and a latitude). The sheet hosts quantum particles, which can be thought of as tiny ripples in media known as quantum fields. These fields and their ripples obey complicated but well-tested mathematical rules, known as quantum field theory. In this case, the ripples follow a symmetric theory known as a conformal field theory or CFT.

The big surprise, which Maldacena and others have now explored in thousands of papers, is that this two-dimensional surface is mathematically equivalent, or “dual,” to the three-dimensional volume it encloses, called the bulk. The duality gives rise to an entire toy universe. Certain collections of ripples on the 2D boundary might represent a 3D star in the bulk, for instance, and others a bulk planet.

The bulk universe differs from ours in that its space has an intrinsically negative energy, making it “anti-de Sitter” or AdS space. But other than that, it looks a lot like our home; it’s a malleable space-time fabric whose curves produce gravity. The AdS/CFT correspondence opens up the tantalizing possibility that physicists could do an end run around physics they don’t understand (quantum gravity in the bulk) by using only physics they do understand (quantum field theory).

M.C. Escher’s Circle Limit IV woodcut depicts a geometry in which an infinite expanse of angels and demons fits into a bounded region. Anti-de Sitter space has the same geometry.

M.C. Escher

“It’s saying gravity is not something separate from regular quantum theory,” said Josephine Suh, a physicist at the Korea Advanced Institute of Science and Technology. “It’s saying that gravity is just a different description of a quantum theory.”

Maldacena’s “holographic duality” linked CFTs on a lower-dimensional boundary with AdS space-time in the bulk. But his work did not specify exactly which patterns of quantum ripples on the boundary would represent, say, a star in the bulk, and which would pinch space-time into a black hole. So over the following decades, researchers developed increasingly sophisticated ways to do so. These methods, which involve powerful mathematics called tensor networks and quantum error-correcting codes, amount to, very roughly, tapping out patterns of ripples in the boundary sphere that correspond to measurements at specific locations in the bulk.

No one knows if the space-time fabric of our real universe is holographic. One convenient feature of negative-energy AdS space is that it has a spatial boundary for those quantum ripples to live on; our universe does not. But the AdS/CFT correspondence provides a toy model for exploring this kind of space-time emergence.

“AdS/CFT is an insane suggestion that should be stupid,” said Geoff Penington, a physicist at the University of California, Berkeley who studies holography. “But then you try all these things, and it all ends up being consistent.”

Geoff Penington studies holography at the University of California, Berkeley. He helped devise a new way to compare the entropy of two black holes.

Lee Sandberg, Institute for Advanced Study

But holography can’t yet tell physicists what they most want to know: What happens deep inside a black hole at the point known as a singularity, where Einstein’s equations fail and the smooth space-time fabric breaks down? What strange occurrences would an astronaut — or a sensor — observe as they approached this singularity? Theorists know how to tap out the boundary ripples for a measurement outside a black hole, but they still don’t know the rhythm corresponding to sending a probe into the hole and retrieving its reading. These are esoteric questions today, but many holographers aspire to someday program such ripples into future quantum computers and simulate the breakdown of Einstein’s space-time fabric.

“If you want to simulate a black hole on a quantum computer in 60 years, what question are you going to ask?” said Jonathan Sorce, a physicist at the Massachusetts Institute of Technology. “I can’t even tell you what calculation to do.”

To find out, physicists have been trying to figure out what that genius von Neumann was up to nearly a century ago.

Perfect Space-Time, Infinite Entanglement
In 2020, Hong Liu, a physicist at MIT, was puzzling over just this problem. The blind spot deep inside a black hole tortured him. He specifically wondered what set of boundary ripples would simulate the flow of time inside a black hole — the ticking of a clock on board a spaceship that had flown in.

“This time is very mysterious,” Liu told me on a visit to his office, where a leaning tower of yellow legal pads on his desk threatened to undergo a pedestrian form of gravitational collapse. “How can you use this boundary to describe time going inside the [black hole] horizon?”

To investigate, Liu and his student Sam Leutheusser conjured up a black hole in the purest space-time they could imagine. In holography, the more rippling fields there are on the boundary, the more closely the bulk resembles Einstein’s fabric — smooth and continuous. Real space-time (like everything else in nature) should experience quantum fluctuations, which blur the notions of “here” and “there.” Understanding smooth, idealized space-time first can serve as a sort of warmup problem to understanding the real, quantum mechanically fluctuating fabric described by a quantum theory of gravity.

Hong Liu, a professor at the Massachusetts Institute of Technology, recently argued that a smooth space-time must be described by a particular type of algebra.

Hong Liu

Liu and Leutheusser wondered exactly what would change as the boundary fields became infinitely numerous — corresponding to the dying gasp of the last quantum fluctuation of the bulk space-time. “What kind of mathematical and physical structure is needed for the emergence of all this space-time?” Liu asked.

But more fields meant more problems. The ripples (that is, the particles) in these fields can come to depend on one other through an intrinsically quantum relationship known as entanglement. When two particles are strongly entangled, measuring their orientations will reveal them to always point in opposite directions. Similarly, how a field ripples at a certain point can depend on a faraway ripple in some other field.

Since Liu and Leutheusser wanted to describe a perfectly smooth black hole in a flawless space-time, they needed to have an infinite number of quantum fields on the boundary. But this created problems. Any region of the boundary would have an infinite amount of entanglement, since the quantum ripples in that patch would be entangled with the infinitude of ripples outside it. Because of this, familiar holographic tools became useless. To understand the transition from fluctuating to smooth space-time, the duo needed to get a handle on this new infinity.

“You want to really find some kind of intrinsic way to describe this infinite amount of entanglement,” Liu said. “Surprisingly, it turns out some work of von Neumann from the early 1930s is the perfect tool for this.”

The Importance of Uncertainty
By 1932, the 29-year-old von Neumann had reinvented the mathematical language of the nascent quantum mechanics. The verbs that glued his new grammar together were physical actions — measuring a particle’s position, say, or moving it, or flipping it upside down. By listing these operations and the rules for combining them to make new operations, one could capture every physical facet of any quantum system, from a hydrogen atom to a solar system.

These lists are known as operator algebras. They amount to a detailed accounting of everything that could happen inside a given region, when you know nothing about the rest of the universe outside it.

John von Neumann, who was born in Hungary in 1903, launched or revolutionized multiple fields including quantum mechanics, game theory, computer science and information theory. He developed a universal language for all quantum systems that’s now being applied to the quantum features of space-time.

Alan W. Richards, Emilio Segre Visual Archives

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.