Graduate Student Extends Quantum Uncertainty Principle to Higher-Dimensional Fractals

Background on the Fractal Uncertainty Principle

The uncertainty principle in quantum mechanics states that precisely knowing a particle's position reduces knowledge of its momentum, and vice versa. Mathematician Semyon Dyatlov at MIT began studying whether quantum particles could follow fractal-like paths in chaotic situations, similar to classical objects. To address this, a new uncertainty principle was needed that could account for fractals, which are shapes with consistent complexity at any zoom level.

Initial Breakthrough and Subsequent Challenge

In 2016, Dyatlov, with contributions from Jean Bourgain, proved the fractal uncertainty principle for one-dimensional fractals. These one-dimensional fractals can represent paths of objects moving in two dimensions. A workshop was held to extend this proof to higher dimensions, which would allow for the study of three-dimensional systems and create a universal mathematical tool. However, attendees, including Frédéric Naud from Sorbonne University, found the task too difficult at the time.

Doctoral Student's Achievement

Years later, Alex Cohen, then a doctoral student at MIT, successfully extended the fractal uncertainty principle to all higher dimensions. This achievement formed the basis of his thesis and led to an assistant professorship at New York University. His paper is scheduled for publication in 2025 in the Annals of Mathematics.

Significance of the Discovery

Peter Sarnak of the Institute for Advanced Study described Cohen's result as a "foundational result" and a "remarkable achievement for a guy in his thesis." This extended principle offers a new mathematical tool for understanding how quantum particles behave when trapped in chaotic, fractal-like trajectories, addressing a long-standing problem in quantum mechanics.