## What is Heisenberg uncertainty, anyway?

#### January 2023

Like relativity, uncertainty is a deeply beautiful topic that is widely misunderstood. Many people are familiar with the idea that you cannot simultaneously know, for example, a particle’s position and momentum to perfect accuracy. This is often thought of as being due to the fact that you need to interact with a system in order to measure it, and in doing so you alter its state, forcing you to pick what you want to measure before you change it. This makes it sound like both position and momentum in reality coexist perfectly fine and if you just had some better position or momentum measuring technology, you could approximate them both increasingly well.

In fact the problem is much deeper than that.

Imagine you want to calculate what frequencies are present in a sound. As the width of the time window you are looking at goes to zero, you just have a single point. There’s no frequency content of a single point! It’s just a number. As you look at a longer time window, you can see what frequencies are present over the window, but you lose some sense of time “resolution” over which that is meaningful. This is fundamental and inherent in the Fourier transform.1

Put differently, you can’t have a signal that is simultaneously both “band-limited,” meaning it can be described as the sum of less than an infinite number of pure sine waves, and “time-limited,” meaning the signal has a finite duration. All conjugate variables — variables which are related via a Fourier transform — including position and momentum, have this limitation. The problem is not in our measurement tools, but rather somehow in the math itself!

This gives a feel for uncertainty but it is not the whole story. For example, it doesn’t explain why the uncertainty bound on position and momentum is $\hbar / 2$ as opposed to something else, or why position and momentum are Fourier duals in the first place. It is also not the only way to describe uncertainty phenomena mathematically; the quantum people usually talk about orthonormal eigenbases rather than Fourier duals, and there are other ways, too.2

That’s all outside the scope of this post, though: beyond the physical intuition, the other main thing I want to convey is the feeling of strangeness that a limitation of these mathematical tools is actually borne out by the universe. This is so weird! Just because the Fourier transform, which is a pure math concept not otherwise overtly related to physics, has this property doesn’t in principle mean that the universe needed to actually have that limitation! And yet, to the best of our understanding, it does.

1. I originally wrote this as a Hacker News comment, but figured it was interesting enough to deserve a permanent article.

2. This is where a professional physicist appears to say yeah well uncertainty breaks down in [insert extreme physical scenario]. To which I respond: sure but just how crazy is it that we get a hierarchy of consistent abstractions at different energies in the first place? Just appreciate how beautiful this whole thing is at low energies, man.