The S-Matrix Is the Oracle Physicists Turn To in Times of Crisis
Even without knowing exactly what goes on during a particle collision, the “S-matrix” lets physicists assess the possible outcomes.
Nico Roper/Quanta Magazine
Introduction
In 1943, the German physicist Werner Heisenberg distracted himself from World War II by pondering a crisis in quantum theory. Predictions about how particles should behave were occasionally giving nonsensical, infinite results. These infinities led Heisenberg to distrust the way quantum physics was depicting reality, and to expect that a revolutionary new theory would eventually overthrow particle physics and fix the problem. But even with no such theory at hand, he realized, progress could still continue. The key was to focus on unassailable facts that would survive no matter what new theory might arise in the future.
Those facts, Heisenberg decided, were observations — specifically, the outcomes of particle collisions. When two particles collide, they may experience many quantum transformations before the final products emerge. Heisenberg ignored the mystifying dynamical events in the middle, and instead kept tabs only on the initial and final particles. He collected the possible outcomes in a table called a scattering matrix, or S-matrix for short. No matter how strange the ultimate theory of particle physics turned out to be, it must predict the correct S-matrix. So by studying the rules and patterns of this matrix, Heisenberg guaranteed that his work would stand the test of time.
Heisenberg’s austere perspective would wax and wane, fading as physicists gained confidence in quantum theory and surging when they faced new mysteries. Now, particle physicists are again seeking a revolutionary new theory of reality. To find it, they have turned back to the only facts they can count on: the entries in the S-matrix.
An Inventory of the Possible
The S-matrix wasn’t Heisenberg’s invention: Paul Dirac first explored the concept, and John Wheeler, who coined the terms “quantum foam” and “wormhole,” came up with the name. Heisenberg took the idea further by treating it not just as a tool, but as a perspective: What would a collision look like from the end of time, at the edge of the universe?
In the 1940s, Werner Heisenberg, a pioneer of quantum mechanics, expected that a revolutionary new theory would soon replace particle physics. He developed the S-matrix to prepare for its arrival.
Alpha Historica/Alamy
Consider the Higgs boson — the latest fundamental particle to be discovered, in 2012 — which lives for less than a billionth of a billionth of a second and can travel less than a millionth of a millionth of a meter. Compared to those tiny scales, the physicists operating the particle colliders that momentarily give rise to Higgs bosons exist for long, long times at distances far, far away. This far-off perspective is captured by the S-matrix.
To see how the S-matrix works, imagine that, instead of making particles collide, you throw dice. With one die, you have six possible outcomes and a 1/6 probability of rolling each one. With two dice, you have 12 possible outcomes with differing probabilities.
Now chart the probabilities in a table. Each entry in the table displays the probability of rolling a certain number with a given number of dice. The table is called a matrix.
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What that matrix does for dice, the S-matrix does for particles. Each row is a choice of initial particles, like the number of dice. You might start with a Higgs boson traveling alone, or with an electron and a positron. Each column is an output: The Higgs might morph into a pair of W bosons, or (more likely) a pair of quarks. Each entry is the probability of observing a given output.
(In quantum theory, each entry is a complex number, a value that may include the “imaginary” square root of −1. You use the absolute value to get familiar probabilities.)
If you could calculate a table of probabilities like this, Heisenberg reasoned, you could check a theory against experiments. As the next century would show, though, you could do more. By studying the S-matrix, Heisenberg and his intellectual descendants would study, in general, how physics ought to be.
Logical Predictions
After the 1940s, physicists realized they didn’t need a revolution after all. They learned how to better use a formula, known as a Lagrangian, that specified all the particles that might come into being during a collision and how they might interact with each other. The formula could be used to infer what was really happening during a collision, even during the unobservable interim moments. With this deeper understanding, physicists could sidestep the problematic infinities and wade straight into the fray of a particle collision.
But in the 1960s, quantum theory failed physicists once again. Experiments had uncovered a dizzying array of new particles resembling protons and neutrons, but with a wide assortment of masses and charges. A complete Lagrangian would need to account for this multitude of “hadron” particles, but their sheer numbers were overwhelming. Paolo Di Vecchia, an emeritus professor at the Nordic Institute for Theoretical Physics, recalled the sentiment of the time. “Should we write a Lagrangian containing all the hadrons? It did not seem the thing to do.”
Once again, physicists dreamed of revolution. Soon enough, Geoffrey Chew, a physicist at the University of California, Berkeley, championed an audacious new approach. He and his followers hoped to ignore the complicated hadron interactions that might arise only momentarily during a particle collision. Instead, they would focus on the outcomes that could be observed in a detector.
“From Berkeley came the idea that we should forget Lagrangians,” Di Vecchia said, “and try to construct directly the S-matrix: that is, the quantity more directly related to experiments.”
Geoffrey Chew, a physicist at the University of California, Berkely, attempted to describe a bonanza of newly discovered particles by guessing their S-matrix from logical requirements.
Lawrence Berkeley National Laboratory, courtesy AIP Emilio Segrè Visual Archives, Physics Today Collection
To do so, they observed that any S-matrix must reflect a handful of unavoidable physical and mathematical principles. For example, the probabilities derived from the S-matrix, like all probabilities, must be between zero and 1. Chew’s group hoped to use general logical requirements such as these to infer specific results, as if lifting themselves up by their own bootstraps. They called their program the S-matrix bootstrap.
Chew’s group identified promising theories in their hunt, showing that their bootstrap generated genuine lift. Numerous physicists joined the S-matrix movement, which produced the S-matrix that would eventually lead to string theory.
But in the race for a theory of hadrons, the bootstrappers were overtaken. Physicists soon realized that the messy and varied hadrons were in fact composed of just a handful of fundamental particles: quarks and gluons. Researchers managed to pack these particles and their interactions into a Lagrangian in the 1970s.
With the success of the new theory of quarks and gluons, Lagrangians were once again in ascendance. Physicists would go on to identify a master Lagrangian that encapsulates the observed behavior of all known particles to date. We now call this Lagrangian the Standard Model of particle physics.
A Modern Revival
Now, physics again faces an uncertain future. Physicists know the Standard Model Lagrangian to be incomplete. But it might be decades or even centuries before experiments produce a result beyond what current theories can handle. But when they do, physicists expect their view of the world to shatter. Reconciling quantum mechanics and gravity seems to demand a revolution even more dramatic than those of previous eras, and no one knows what the building blocks of the next reality might be. In such uncertain times, physicists are once more returning to the one idea they know will remain: the S-matrix.
Some, like the CERN postdoc Lucía Córdova, are broadening the search launched by Chew’s group. Using numerical techniques, she and her colleagues are mapping out the space of viable S-matrices. Such exploration might be the only way to understand theories with no Lagrangian, which may be needed for quantum gravity.
Others, like Sebastian Mizera, a physicist at the Institute for Advanced Study in Princeton, New Jersey, see renewed hope in Chew’s original dream: bootstrapping a single S-matrix capturing the behavior of specific particles. He and other researchers have recently focused on massless particles like gluons.
The S-matrix bootstrap was born from the study of massive hadrons, and naïvely it seemed that massless particles would be much harder to understand in this way. But “decades later, precisely the opposite turned out to be true,” Mizera said. The unexpected simplicity of massless particles means that their interactions can be bootstrapped from just a few straightforward rules. In this way, physicists have fully defined theories of gluons and other particles purely with S-matrices, avoiding Lagrangians.
At a workshop in March, Mizera met with fellow S-matrix enthusiasts. Inspired by recent progress, they continue to seek more universal rules that govern all S-matrices. Like Heisenberg and Chew, they focus on observations, striving to discover truths that will endure even when the shape of future theories remains uncertain.
