When the world seems particularly crazy, I like to delve into niche controversies. A case where the nerds argue passionately about something that no one knew was even controversial. In this video, I’d like to take up one of those super-niche nerd battles: are complex numbers necessary to describe the world as we watch it? Do they exist? Or are they just a math convenience? That’s what we’ll talk about today.
The most recent controversy arose when an article entitled “Quantum Physics Needs Complex Numbers” appeared on the preprint server. The paper contains proof of the claim in the title in response to an earlier claim that the complex numbers can be dispensed with.
Next, the computer scientist Scott Aaronson wrote a blog post in which he described the newspaper as “suspicious”. But the answers weren’t very enthusiastic. They ranged from “why a fuss about it” to “bullshit” to “the point is missing”.
We’ll check out the paper in a moment, but first I’ll quickly summarize what we’re talking about so that no one is left behind.
The math of complex numbers
You probably remember from school that complex numbers are what you need to solve equations like x square equal to minus 1. You cannot solve this equation using the real numbers you are used to. Real numbers are numbers that can have infinite digits after the decimal point, like the square root of 2 and π, but they also contain whole numbers and fractions and so on. You can’t solve this equation with real numbers because they always square to a positive number. So when you want to solve such equations, introduce a new number, usually called “i,” which squares its property to -1.
Interestingly, it is enough to just give the solution to this one equation a name and add it to the set of real numbers to make all algebraic equations solvable. No matter how long or how complicated the equation is, you can always write all of its solutions as a + ib, where a and b are real numbers.
Complex numbers are now all numbers of type a plus I time b, where a and b are real numbers. “A” means “real part” and “b” means “imaginary part” of the complex number. Complex numbers are often drawn in a plane called the complex plane, with the horizontal axis being the real part and the vertical axis being the imaginary part. I myself am conventionally in the upper half of the complex plane. However, this looks just like drawing a map on a grid and naming each point with two real numbers. Doesn’t that mean that the complex numbers are just a two-dimensional real vector space?
No they are not. And that’s because complex numbers are multiplied by a certain rule that you can calculate by taking into account that the square of i is minus 1. Two complex numbers can be added like vectors, but the law of multiplication distinguishes them. Complex numbers, to use the mathematical term, are a “field” like real numbers. You have a rule for both addition and multiplication. You’re not just like this two-dimensional grid.
The physics of complex numbers
We are constantly using complex numbers in physics because they are extremely useful. It’s useful for many reasons, but the main one is. If you take a real number, let’s call it α, multiply it by I and put it in an exponential function. You get exp (Iα). In the complex plane, this number exp (Iα) is always on a circle with a radius of one around zero. And as you increase α you go around that circle. If you now only look at the real or only the imaginary part of this circular motion, you will get an oscillation. Indeed, this exponential function is a sum of a cosine and I times a sine function.
Here is the thing. If you multiply two of these complex exponentials, say one with α and one with β, you can just add the exponents. But if you multiply two cosines, or multiply a sine by a cosine, that’s a mess. You do not want that. That is why in physics we calculate with the complex numbers and then take either the real or the imaginary part at the very end. Especially when describing electromagnetic radiation, we have to deal with a lot of vibrations, and complex numbers are very useful.
But we don’t have to use it. In most cases we could only do the calculation with real numbers. It’s just awkward. With the exception of quantum mechanics, which we will discuss in a moment, the complex numbers are not necessary.
And as I explained in a previous video, only when a mathematical structure is actually necessary to describe observations can we say that they “exist” in a scientifically meaningful way. This is not the case for the complex numbers in non-quantum physics. You are not necessary.
As long as you ignore quantum mechanics, you can think of complex numbers as a mathematical tool and you have no reason to believe that they physically exist. Then let’s talk about quantum mechanics.
Complex numbers in quantum mechanics
In quantum mechanics, we work with wave functions, usually denoted by Ψ, which have complex values, and the equation that tells us what the wave function does is the Schrödinger equation. It looks like that. You will immediately see that there is an “i” in this equation, so the wave function must have a complex value.
However, you can of course decompose the wave function and this equation into a real and an imaginary part. In fact, you often do that by solving the equation numerically. And let me remind you that both the real and imaginary parts of a complex number are real numbers. If we now calculate a prediction for a measurement result in quantum mechanics, this measurement result is always a real number. So it looks like you can get rid of the complex numbers in quantum mechanics by dividing the equation into a real and an imaginary part, and that will never make a difference to the result of the calculation.
This finally brings us to the paper I mentioned at the beginning. What I just said about the decomposition of the Schrödinger equation is of course correct, but you didn’t see it in the newspaper, that would be pretty lame.
Instead, they ask what happens to the wave function when you have a system that consists of several parts, in the simplest case of several particles. In normal quantum mechanics, each of these particles has a wave function with a complex value, and from these we construct a wave function for all particles together, which is also complex. What this wave function looks like depends on which particle is involved with which one. When two particles are involved, it means their properties are correlated, and we know experimentally that this entanglement correlation is stronger than what you can do without quantum theory.
The question that you are dealing with in the new work is whether there are ways of involving particles in normal, complex quantum mechanics that you cannot build from particles that are completely described by real-valued functions. Previous calculations have shown that this is always possible when the particles come from a single source. However, the new paper looks at particles from two independent sources and claims that there are cases that you cannot reproduce with just real numbers. They also suggest a way to measure this specific entanglement experimentally.
I must warn you that this paper has not yet been peer-reviewed. Perhaps someone will find a flaw in their proof. Assuming their result holds, it means that if the experiment you propose finds the specific entanglement predicted by complex quantum mechanics, you cannot describe observations using real numbers. It would then be fair to say that complex numbers exist. That’s why it’s cool. You have found a way to experimentally test whether complex numbers exist!
Something like that. Here’s the fine print: This conclusion only applies if the purely real-world theory is supposed to work in the same way as normal quantum mechanics. If you are ready to change quantum mechanics so that it does not become more local than it already is, you can still create the necessary entanglement with real numbers.
Why is it controversial? Well, if you are at a standstill and calculate the bearing, that statement is completely irrelevant. Because complex numbers have absolutely nothing to complain about. That’s why half of the people say, “What’s the point?” Or “Why all the fuss about it?”. On the other hand, if you find yourself in the camp of people who believe that something is wrong with quantum mechanics because it uses complex numbers that we can never measure, then you are now caught between a rock and a hard place. Either accept complex numbers or accept that nature is not yet more local than quantum mechanics.
Or, of course, the experiment could be inconsistent with quantum mechanics predictions, which would be the most exciting of all possible outcomes. I am sure this is a topic we will hear about again.