Graduate Student Proves a Quantum Uncertainty Principle for Fractals
A new mathematical proof extends the fractal uncertainty principle to all higher dimensions, revealing a fundamental difference between quantum and classical chaos. Alex Cohen, a graduate student at MIT, solved a problem that stumped experts for nearly a decade, using insights from the late mathematician Jean Bourgain. The result, published in the Annals of Mathematics, has been hailed as a foundational achievement.
So there’s something different about quantum and classical [chaos], and one ingredient that you can try to use for that would be uncertainty principles.
- openasocket
For those that aren't aware, the uncertainty principle goes pretty deep. You can actually define the uncertainty principle as an inequality involving the integral of a function vs the function's Fourier transform: https://en.wikipedia.org/wiki/Uncertainty_principle#Harmonic... . And in a quantum system you can construct the momentum of a particle as the Fourier transform of the position (up to a constant) and the uncertainty principle falls out because of this. This article doesn't state that super explicitly. So the real interesting thing being done here is getting a Fourier transform that works on fractal spaces.
- akkartik
"In 2017, Dyatlov and Long Jin from Tsinghua University in Beijing used the one-dimensional fractal uncertainty principle to prove that you can never trap a wave on a hyperbolic surface; it will always spread out until it touches every corner. To do so, they imagined a region on the surface that a wave never enters, even after having infinite time to spread out. When they removed all the trajectories that entered that region, what remained was the same sort of fractal dust that appeared in the pinball example. Since the fractal uncertainty principle forbids a wave from being trapped on a fractal, no such region can exist — the wave must spread everywhere."
As a non-mathematician this is a wonderfully evocative summary. I'm curious if mathematicians find it a reasonable characterization of the proof.
- linuxhansl
As a physics layman I find it fascinating how quantum mechanics are tied to information theory.
For example, take quantum decoherence (which, IMHO, is the most logical explanation for the collapse of the wave-function - by saying it does not actually collapse). Quantum decoherence is almost like a giant constraint resolution system - once a particle randomly interacts with another they become entangled and both now have fewer degrees of freedom. When it interacts with many particles, like any macro-effect it has essentially no degrees of freedom anymore. It's all about who knew about what and when. The experiments around this fascinating. (Note that there are other theories, like the many-worlds interpretation, that also explain the collapse of the wave function)
This seems to be another example of this. Anyway, as I said, just a layman.