A JARRING DIVIDE cleaves modern physics. On one side lies quantum theory, which portrays subatomic particles as probabilistic waves. On the other lies general relativity, Einstein’s theory that space and time can bend, causing gravity. For 90 years, physicists have sought a reconciliation, a more fundamental description of reality that encompasses both quantum mechanics and gravity. But the quest has run up against thorny paradoxes.
Hints are mounting that at least part of the problem lies with a principle at the center of quantum mechanics, an assumption about how the world works that seems so obvious it’s barely worth stating, much less questioning.
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
Unitarity, as the principle is called, says that something always happens. When particles interact, the probability of all possible outcomes must sum to 100 percent. Unitarity severely limits how atoms and subatomic particles might evolve from moment to moment. It also ensures that change is a two-way street: Any imaginable event at the quantum scale can be undone, at least on paper. These requirements have long guided physicists as they derive valid quantum formulas. “It’s a very restrictive condition, even though it might seem a little bit trivial at first glance,” said Yonatan Kahn, an assistant professor at the University of Illinois.
But what once seemed an essential scaffold may have become a stifling straitjacket preventing physicists from reconciling quantum mechanics and gravity. “Unitarity in quantum gravity is a very open question,” said Bianca Dittrich, a theorist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada.
The main problem is that the universe is expanding. This expansion is well described by general relativity. But it means that the future of the cosmos looks totally different from its past, while unitarity demands a tidy symmetry between past and future on the quantum level. “There is a tension there, and it’s something quite puzzling if you think about it,” said Steve Giddings, a quantum gravity theorist at the University of California, Santa Barbara.
Concern over this conflict has been in the air for years. But recently, two quantum gravity theorists may have found a way to loosen unitarity’s buckles to better fit our growing cosmos. Andrew Strominger and Jordan Cotler of Harvard University argue that a more relaxed principle called isometry can accommodate an expanding universe while still satisfying the stringent requirements that first made unitary a guiding light.
“You don’t need unitarity,” said Strominger. “Unitarity is too strong of a condition.”
While many physicists are receptive to the isometry proposal—some have even come to similar conclusions independently—opinions vary as to whether the update is too radical or not radical enough.
A Fixed Sum
In everyday life, events can’t help but play out in a unitary way. A coin toss, for instance, has a 100 percent chance of coming up heads or tails.
But a century ago, the pioneers of quantum mechanics made a surprising discovery—one that elevated unitarity from common sense to a hallowed principle. The surprise was that, mathematically, the quantum world operates not by probabilities but by more complicated numbers known as amplitudes. An amplitude is essentially the degree to which a particle is in a certain state; it can be a positive, negative or imaginary number. To calculate the probability of actually observing a particle in a certain state, physicists square the amplitude (or, if the amplitude is an imaginary number, they square its absolute value), which gets rid of the imaginary and negative bits and produces a positive probability. Unitarity says the sum of these probabilities (really, the squares of all the amplitudes) must equal 1.
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It’s this twist—the squaring of hidden amplitudes to calculate the outcomes we actually see—that gives unitarity teeth. As a particle’s state changes (as it flies through a magnetic field, say, or collides with another particle), its amplitudes change too. In working out how a particle is allowed to evolve or interact, physicists use the fact that amplitudes never change in a way that disrupts the fixed sum of their squares. In the 1920s, for instance, this unitarity requirement guided the British physicist Paul Dirac to discover an equation that implied the existence of antimatter. “I was not interested in considering any theory which would not fit in with my darling,” Dirac wrote, referring to unitarity.
Physicists keep probabilities and amplitudes in line by tracking how the quantum state of a particle moves around in Hilbert space—an abstract space representing all possible states available to the particle. The particle’s amplitudes correspond to its coordinates in Hilbert space, and physicists capture changes to the particle with mathematical objects called matrices, which transform its coordinates. Unitarity dictates that a physically allowed change must correspond to a special “unitary” matrix that rotates the particle’s state in Hilbert space without changing that the sum of the squares of its coordinates equals 1.
It’s a mathematical fact with philosophical consequences: If you know the specific unitary matrix corresponding to some change over time, any quantum state can be swiveled into the future or unswiveled into the past. It will always land on another viable state in the Hilbert space, which never grows or shrinks. “The past completely determines the future, and the future completely determines the past,” said Cotler. “It’s related to the statement that information is neither created nor destroyed.”
And yet, this bedrock assumption seems to conflict with the universe that surrounds us.
A Cosmic Clash
Galaxies are flying ever farther apart. While our expanding universe is a perfectly valid solution to the equations of general relativity, physicists have increasingly realized that its growth spells trouble for quantum mechanics, by presenting particles with an expanding smorgasbord of options for where to be and how to behave. As space grows, how can the Hilbert space of possibilities not grow with it? “It’s definitely true that there are more degrees of freedom in the universe now than in the early universe,” said Nima Arkani-Hamed, a theoretical physicist at the Institute for Advanced Study in Princeton, New Jersey.
“I’ve felt for many years [that] it was the elephant in the room,” said Strominger.
Andrew Strominger, left, and Jordan Cotler of Harvard University have collaborated on an effort to replace unitarity in quantum physics with an alternative rule called isometry. PHOTOGRAPH: MIGUEL MONTRERO
Giddings sharpens the issue with a paradoxical thought experiment set in a universe that’s both unitary and expanding. Imagine taking the current state of the universe, said Giddings, and adding “one innocuous photon”—perhaps lodged in newly created space halfway between here and the Andromeda galaxy. Unitarity insists that we must be able to calculate what this universe looked like in the past, unswiveling its quantum state as much as we wish.
But rewinding the state of the universe plus an extra photon creates a glitch. Going into the past, the universe gets smaller, and the wavelength of photons will shrink too. In our real universe, this isn’t a problem: A photon shrinks only until the moment of its creation through some subatomic process; the reversal of that process will make it disappear. But the extra photon wasn’t created by that special process, so instead of disappearing when you turn back time, its wavelength will eventually get impossibly small, concentrating its energy so greatly that the photon collapses into a black hole. This creates a paradox, absurdly implying that—in this fictional, expanding universe—microscopic black holes convert into photons. The thought experiment suggests that a naïve mashup of unitarity and cosmic expansion doesn’t work.
Dittrich thinks unitarity smells fishy on more general grounds. Quantum mechanics treats time as absolute, but general relativity messes with the ticking of clocks, complicating the notion of change from one moment to the next. “I personally never relied so much on unitarity,” she said.
The question is: What sort of alternative framework could accommodate both cosmic expansion and the rigid mathematics of quantum theory?
Unitarity 2.0
Last year, Strominger struck up a collaboration with Cotler, who splits his time between quantum gravity research and quantum information theory—the study of information stored in quantum states. The duo realized that there is a well-studied scheme in quantum information theory that resembles the expanding universe: quantum error correction, a scheme where a small message made from quantum states is redundantly encoded inside a bigger system. Perhaps, they thought, the contents of the young universe are similarly stitched into the modern cosmos’s swollen form.
“In hindsight, the obvious answer is this is exactly what people doing quantum encoding have been doing,” Strominger said.
In a paper earlier this year, the two homed in on a class of transformations that quantum error-correcting codes belong to, known as isometries. An isometric change resembles a unitary one with added flexibility.
Bianca Dittrich, of the Perimeter Institute for Theoretical Physics, struck upon isometry a decade ago while formulating a toy quantum theory of space-time. PHOTOGRAPH: GABRIELA SECARA/PERIMETER INSTITUTE
Think of an electron that can occupy two possible locations. Its Hilbert space consists of all possible combinations of amplitudes in the two locations. These possibilities can be imagined as the points on a circle—every point has some value in both the horizontal and vertical directions. Unitary changes rotate states around the circle but do not expand or shrink the set of possibilities.
To visualize an isometric change, though, let the universe of this electron swell just enough to allow a third position. The electron’s Hilbert space grows, but in a special way: It gains another dimension. The circle becomes a sphere, on which the particle’s quantum state can swivel around to accommodate mixtures of all three locations. The distance between any two states on the circle holds steady under the change—another requirement of unitarity. In short, the options increase, but without unphysical consequences.
“Working with isometries is sort of a generalization” of unitarity, said Giddings. “It keeps some of the essence.”
Our universe would have a Hilbert space with a huge number of dimensions that proliferate continuously as real space expands. As a simpler proof of concept, Strominger and Cotler studied the expansion of a toy universe consisting of a line ending in a receding mirror. They calculated the probability that the universe would grow from one length to another.
For such calculations, quantum practitioners often use the Schrödinger equation, which predicts how a quantum system evolves in time. But changes dictated by the Schrödinger equation are perfectly reversible; its “literal purpose in life is to enforce unitarity,” Arkani-Hamed said. So instead, Strominger and Cotler used an alternative version of quantum mechanics dreamed up by Richard Feynman, called the path integral. This method, which involves tallying up all the paths a quantum system can take from some starting point to an endpoint, has no trouble accommodating the creation of new states (which appear as branching paths leading to multiple endpoints). In the end, Strominger and Cotler’s path integral spit out a matrix encapsulating the growth of the toy cosmos, and it was indeed an isometric matrix rather than a unitary one.
“If you want to describe an expanding universe, the Schrödinger equation as it stands just won’t work,” Cotler said. “But in the Feynman formulation, it keeps on working on its own volition.” Cotler concludes that this alternative way of doing quantum mechanics based on isometry “will be more useful to us in understanding an expanding universe.”
A Mirage of Possibilities
Relaxing unitarity could resolve the glitches in the thought experiment that has troubled Giddings and others. It would do so through a conceptual change to how we think about the relationship between the past and the future, and which states of the universe are really possible.
ILLUSTRATION: MERRILL SHERMAN/QUANTA MAGAZINE
To see why isometry solves the problem, Cotler describes a toy universe, one born in one of two possible initial states, 0 or 1 (a two-dimensional Hilbert space). He makes up an isometric rule to govern this universe’s expansion: At every successive moment, each 0 becomes 01, and each 1 becomes 10. If the universe starts at 0, its first three moments will see it grow as follows: 0 → 01 → 0110 → 01101001 (an 8D Hilbert space). If it starts at 1, it will become 10010110. The string captures everything about this universe—all its particles’ positions, for instance. A considerably longer string made from superpositions of 0s and 1s presumably describes the real universe.
At any given time, the toy universe has two possible states: one arising from 0 and another arising from 1. The initial one-digit configuration has been “encoded” in a larger, eight-digit state. That evolution resembles a unitary one, in that there are two possibilities at the beginning and two at the end. But the isometric evolution provides a more capable framework for describing the expanding universe. Crucially, it does so without creating the freedom to add, say, an extra photon between here and Andromeda, which would spell trouble when you turn back the clock. Imagine, for instance, that the universe is in the 01101001 state. Flip the first 0 to a 1—representing a minor, local tweak, such as the extra photon—and you’ll get a state that looks fine on paper (11101001), with a seemingly valid set of coordinates in the larger Hilbert space. But knowing the specific isometric rule, you can see that such a state has no parent state. This imaginary universe could never have arisen.
“There are some configurations of the future that don’t correspond to anything in the past,” Cotler said. “There’s nothing in the past that would evolve into them.”
Giddings has proposed a similar principle for ruling out paradoxical states he encountered while studying black holes last year. He calls it “history matters,” and it holds that a given state of the universe is only physically possible if it can evolve backward without generating contradictions. “This has been kind of a lingering puzzle,” he said. Strominger and Cotler “are taking that puzzle and using it to try to motivate possibly a new way of thinking about things.”
Giddings thinks the approach deserves further development. So does Dittrich, who came to similar realizations about isometry a decade ago while attempting to formulate a toy quantum theory of space-time with her collaborator Philipp Höhn. One hope is that such work could eventually lead to the specific isometric rule that might govern our universe—a rather more complicated prescription than “0 goes to 01.” A true cosmological isometry, Cotler speculates, could be verified by calculating which specific patterns in the distribution of the matter in the sky are possible and which aren’t, and then testing those predictions against observational data. “If you look closer at it, you’ll find this but not this,” he said. “That could be really useful.”
To Isometry and Beyond
While such experimental evidence could accrue in the future, in the near term, evidence for isometry is more likely to come from theoretical studies and thought experiments showing that it helps combine the malleability of space-time with the amplitudes of quantum theory.
One thought experiment where unitarity looks creaky involves black holes, intense concentrations of matter that warp space-time into a dead end. Stephen Hawking calculated in 1974 that black holes evaporate over time, erasing the quantum state of anything that fell in—a seemingly blatant unitarity violation known as the black hole information paradox. If black holes have Hilbert spaces that mature isometrically, as Cotler and Strominger hypothesize, physicists may face a somewhat different puzzle than they thought. “I don’t think there can be a solution that doesn’t take this into account,” Strominger said.
Another prize would be a detailed quantum theory that described not just how the cosmos grows, but where everything came from in the first place. “We have no universe, and all of a sudden we have a universe,” Arkani-Hamed said. “What the hell kind of unitary evolution is that?”
For his part, however, Arkani-Hamed doubts that swapping in isometry for unitarity goes far enough. He is one of the leaders of a research program that is trying to break free of many fundamental assumptions in quantum theory and general relativity, not just unitarity.
Whatever theory comes next, he suspects, will take a completely novel form, just as quantum mechanics was a clean break from Isaac Newton’s laws of motion. As an illustrative example of what a new form might look like, he points to a research program stemming from a 2014 discovery he made together with Jaroslav Trnka, his student at the time. They showed that when certain particles collide, the amplitude of each possible outcome equals the volume of a geometric object, dubbed the amplituhedron. Calculating the object’s volume is much easier than using standard methods for computing the amplitudes, which laboriously reconstruct all the ways a particle collision might play out, moment by moment.
Intriguingly, while the amplituhedron gives answers that obey unitarity, the principle isn’t used to construct the shape itself. Neither are any assumptions about how particles move in space and time. The success of this purely geometric formulation of particle physics raises the possibility of a fresh perspective on reality, one free from the cherished principles that currently conflict. Researchers have gradually been generalizing the approach to explore related geometric shapes pertaining to different particles and quantum theories.
“[It] may be a different way to organize unitarity,” Cotler said, “and perhaps it has the seeds to transcend it.”
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