String theory first emerged in the 1960s as a possible way to solve one of physics' biggest problems: combining quantum mechanics, which governs the smallest particles, with general relativity, Einstein's theory describing gravity and the large-scale structure of the universe. Scientists have long struggled to unite the two because the equations often spiral into mathematical infinities when gravity is included at quantum scales.

String theory offers a potential way around that problem. In the theory, every particle, including the hypothetical graviton that would carry the force of gravity, comes from different vibrations of tiny strings. The mathematics also requires the strings to exist in at least 10 dimensions rather than the four dimensions humans experience.

One major obstacle remains. Testing string theory directly would require energies so extreme that researchers would need a particle collider as large as a galaxy.

Bootstrap Physics and String Theory

Since direct experiments are impossible with current technology, physicists are exploring other methods. One promising strategy is known as the "bootstrap" approach. Instead of assuming a detailed theory from the start, scientists begin with a few broad principles they believe nature must obey and then determine what laws naturally emerge.

In a new study titled "Strings from Almost Nothing," accepted for publication in Physical Review Letters, researchers from Caltech, New York University, and Institut de Fisica d'Altes Energies in Barcelona used this strategy to investigate particle behavior at extremely high energies. Starting from just a couple of assumptions about how particles scatter during collisions, they unexpectedly arrived at the core features 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."

Although the findings do not prove string theory experimentally, Cheung says the results are striking because many different mathematical outcomes could have been possible. Instead, the calculations pointed toward only one solution.

The Infinite Tower of Particles

One of the most important features to emerge from the calculations is known as the string spectrum. In the late 1960s, Italian theoretical physicist Gabriele Veneziano at CERN developed a mathematical function describing a mysterious "tower" of particles seen in collider experiments. The particles appeared in a sequence where mass and spin increased in orderly 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.

Researchers later realized this pattern resembles the harmonics of a vibrating string. When a violin string is plucked, it produces a main tone along with a series of overtones. String theory proposes that particles arise from similar vibrational patterns.

In 1974, Caltech physicist John Schwarz and French physicist Joël Scherk recognized that string theory could also include gravity. That discovery created one of the first meaningful links 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.

According to string theory, different vibrational modes generate different particles. A photon, for example, comes from an open string vibrating in its simplest mode, while the graviton is thought to arise from a closed vibrating string.

Why Quantum Gravity Breaks Down

The new study focused on scattering amplitudes, mathematical expressions describing the outcomes of particle collisions. When scientists use general relativity to calculate collisions at extremely high energies near the Planck scale, the math stops working properly and produces infinities.

"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.

String theory avoids these infinities through a property called ultrasoftness. At extremely high energies, the strings effectively spread interactions out, preventing the violent behavior that normally causes the equations to fail.

"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 used this ultrasoft behavior as one of their starting assumptions. They also included another condition called "minimal zeros," which limits the number of points where scattering probabilities vanish.

"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.

Using only these assumptions, the team showed that the resulting mathematics naturally reproduced the defining characteristics of string theory, including its famous spectrum of particle masses and spins.

"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.

Reviving an Old Idea With Modern Tools

Cheung compares the bootstrap approach to solving a sudoku puzzle. A few simple rules are provided at the start, and those rules eventually guide you to one unique solution.

"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."

The study also builds on earlier work by Caltech physicist Steven Frautschi and UC Berkeley physicist Geoffrey Chew, who pioneered the bootstrap approach in particle physics during the 1960s. Their work provided some of the earliest hints of the infinite particle spectrum later connected to string theory.

"The bootstrap idea had become obsolete but now people like Cliff are reviving and modernizing it," says 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. "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" received funding from 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. Additional authors include Francesco Sciotti of Institut de Fisica d'Altes Energies in Barcelona and Michele Tarquini, a graduate student at Caltech.