String Theory Emerges from "Almost Nothing"

If you could take an apple and break it into smaller and smaller parts, you would find molecules, then atoms, followed by subatomic particles like protons and the quarks and gluons that make them up. You might think you hit the bottom, but, according to string theorists, if you keep going to even smaller scales—about a billion billion times smaller than a proton—you will find more: tiny vibrating strings.

Developed in the 1960s, string theory proposes that everything in the universe is made from invisible strings. The theory arose as a possible solution to the problem of "quantum gravity," the quest to align quantum mechanics, which describes our world at the smallest scales, with the general theory of relativity, which explains how our universe works on the largest scales (and includes gravity). Researchers have tried to reconcile the two theories—asking, for example, how gravity behaves in the quantum realm—but their equations go berserk, or in mathematical terms, go to infinity.

String theory is a mathematical solution that tames the unruly infinities. It purports that all particles, including the graviton—the hypothetical particle believed to convey the force of gravity—are generated by very small vibrating strings. The math behind string theory requires the strings to vibrate in at least 10 dimensions, rather than the four we live in (three for space and one for time), which is one of the reasons some scientists are not convinced that string theory is correct. But perhaps the biggest challenge for the theory is the ultrahigh energies required for testing it: Such an experiment would require a particle collider the size of a galaxy.

What is a physicist to do? One way they can probe the theory is to turn to a "bootstrap" approach, in which researchers start with certain assumptions they believe to be true about the universe, and then see what laws emerge out of those assumptions. In a new paper titled "Strings from Almost Nothing," accepted for publication in Physical Review Letters, Caltech researchers, and their colleagues at New York University and Institut de Fisica d'Altes Energies in Barcelona, have done just that. From a couple of basic assumptions about how particles should scatter off one another at very high energies, they derived the elements of string theory.

"The strings just fell out," says Clifford Cheung, professor of theoretical physics and director of the Leinweber Forum for Theoretical Physics at Caltech. "We didn't start with any assumptions about strings at all, but then the solution contained the cornerstone signatures of strings."

Though the work does not amount to experimental evidence for string theory, it is "very suggestive from the theoretical viewpoint, since the general assumptions could have yielded infinite solutions, but they yielded only one," Cheung says.

This bootstrap approach helps physicists home in on the defining traits of string theory, explains Hirosi Ooguri, the Fred Kavli Professor of Theoretical Physics and Mathematics at Caltech and the Kent and Joyce Kresa Leadership Chair of the Division of Physics, Mathematics and Astronomy, who is a string theorist though not an author on the paper. "It also helps researchers come up with alternative theories. If string theory is not true, and we want to find another model, then what basic assumptions do we need to remove?" Ooguri says.

Particles in Harmony

One of the key signatures of strings that "fell out" of the team's analysis is known as the string spectrum. Discovered by Italian theoretical physicist Gabriele Veneziano of the European Organization for Nuclear Research (CERN) in the late 1960s, the spectrum is an infinite tower, or ladder, of particles, in which the masses and spins increase in discrete steps.

"At Veneziano's time, particle colliders were seeing this spray of junk come out of the collisions, particles of different masses. It was fascinating and nobody had any idea what was going on. Veneziano wrote down a function to describe all the masses, revealing an infinite tower of particles," Cheung says.

Other researchers later came to realize that Veneziano's tower of particles corresponds to a harmonic series of a vibrating string. If you pluck a violin string, you'll get a series of notes representing the fundamental note and overtones that follow a similar pattern.

String theory was born, but it was not until 1974 that Caltech's John Schwarz, the Harold Brown Professor of Theoretical Physics, Emeritus, and his colleague Joël Scherk, a French physicist, realized that the theory included gravity, thereby forming the first connection between string theory and general relativity. "Like all particle physicists in that era, we had no prior interest in gravity. String theories are well-behaved at very high energies, unlike Einstein's general theory of relativity, which survives as a low-energy approximation. Therefore, even though much was not yet understood, we were very excited that some version of string theory could provide a unified quantum theory of everything," Schwarz says.

In string theory, different vibrational modes of the tiny strings give rise to the different particles. For instance, a photon arises from an open-ended string vibrating in its fundamental mode while the graviton is thought to result from the fundamental vibrational mode of a closed string.

Credit: AI-generated art by Clifford Cheung

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Credit: AI-generated art by Clifford Cheung

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Artwork illustrating how string theory emerges from a few simple mathematical assumptions about particle collisions. Image credit: AI-generated art by Clifford Cheung.

From the Ground Up

In the new study, the researchers looked at something called scattering amplitudes, which describe the probabilities of possible outcomes of particle collisions. When researchers formulate scattering amplitudes at higher and higher energies using the tools of general relativity, the unruly infinities crop up. In mathematical terms, this means the results do not make sense and cannot be right.

"If you take general relativity and scatter at very high energies at the so-called Planck scale—that is roughly 19 orders of magnitude greater than a proton's mass—you get a result that makes no sense. Everything completely breaks down," Cheung says.

This is where string theory shines. It prevents the math from going to infinity in a few ways, one of which is called ultrasoftness, whereby the strings soften, or smear out, interactions at extreme high energies, making them more manageable mathematically.

"In a string theory framework, as you increase the energy transfer between particles, you will see a swift fall off in the probability that the particles will scatter. It's like the particles don't even want to scatter off one another, but rather pass freely," Cheung says. "The scattering amplitudes don't go to infinity. It's better behaved."

The researchers took this ultrasoftness trait for particle behavior as one of their starting assumptions. They didn't assume anything about strings but took it to be true that particles have a lower probability of scattering at high energies—something that is needed to tame the unwanted infinities in quantum gravity theories.

In addition, they made another assumption about particle behavior called "minimal zeros," which is more complex. "Remarkably, consistency requires scattering amplitudes not only to interact but also to not interact at special kinematic points called 'zeros.' The assumption of 'minimal zeros' demands the sparsest number of such vanishing points mathematically allowed by the equations," Cheung says.

Starting with the math describing these two assumptions, the researchers rigorously proved that the only mathematical functions that satisfied these assumptions amounted to the defining hallmarks of string theory. These hallmarks include the full spectrum of particle masses and spins as defined by string theory, along with their detailed interaction strengths.

"The precise details of string theory emerged automatically, including the infinite tower of massive spinning particles that form the 'harmonics' of the string that the theory is famous for," says co-author Grant N. Remmen (PhD '17), the James Arthur Postdoctoral Fellow at New York University.

The researchers' bootstrap approach is somewhat like a sudoku puzzle: You start with just a few rules to follow about how to place numbers in a grid and, from these basic rules, search for the one unique solution to the puzzle.

"The deep irony is that this bootstrap idea that we're pursuing now with modern tools and modern ideas is super retro. It's an old idea," Cheung explains. "The original discovery of the Veneziano spectrum, and John Schwarz's work, took a similar approach. They didn't start with string theory models but rather the solutions came out of basic principles."

Cheung also points to Caltech's Steven Frautschi as a pioneer of the bootstrap approach. Frautschi, a professor of theoretical physics, emeritus, at Caltech, and his colleague, the late Geoffrey Chew, formerly of UC Berkeley, were the first to develop the bootstrap theory in particle physics in the 1960s (Chew came up with the name after the phrase "pulling oneself up by one's bootstraps"). Frautschi and Chew found early evidence for the infinite tower of particles later discovered by Veneziano.

"The bootstrap idea had become obsolete but now people like Cliff are reviving and modernizing it," Ooguri says. "We now have a better understanding of the basic assumptions we can make, as well as stronger techniques for translating these assumptions into properties of scattering amplitudes and other observables."

The study "Strings from Almost Nothing" was funded by the US Department of Energy, the Walter Burke Institute for Theoretical Physics, the Leinweber Forum for Theoretical Physics, the James Arthur Postdoctoral Fellowship at New York University, and the Next Generation EU. Other authors include Francesco Sciotti of Institut de Fisica d'Altes Energies in Barcelona and Michele Tarquini, a graduate student at Caltech.