In physics, the conundrum known as the “few-body problem,” how three or more interacting particles behave, has bedeviled scientists for centuries. Equations that describe the physics of few-body systems are usually unsolvable and the methods used to find solutions are unstable. There aren’t many equations that can probe the wide spectrum of possible few-particle dynamics. A new family of mathematical models for mixtures of quantum particles could help light the way.

“These mathematical models of interacting quantum particles are like lanterns, or islands of simplicity in a sea of complexity and possible dynamics”, said Nathan Harshman, American University associate professor of physics and an expert in symmetry and quantum mechanics, who along with his peers created the new models. “They give us something to grip onto to explore the surrounding chaos”. The work was published in Physical Letters X.

The researchers’ key insight is using a simple case and start working in abstract, higher dimensions. For example, the equation describing four quantum particles in one dimension is mathematically equivalent to the equation describing one particle in four dimensions. Each position of this fictional single particle corresponds to a specific arrangement of the four real particles. The breakthrough is to use these mathematical results about symmetry to find new, solvable few-body systems, Harshman explained. By moving particles to a higher dimensional space and choosing the right coordinates, some symmetries become more obvious and more useful.

Coxeter models, as Harshman calls these symmetric, few-body systems, named for the mathematician H.S.M. Coxeter, can be defined for any number of particles. So far, only rarely do solvable few-body systems have experimental applications. What comes next is to implement the Coxeter models in a lab to help unravel some of the most complex concepts in physics, like quantum entanglement. (1)

We cannot solve even simple equations.

And yet we believe we can describe how the planets move.