When the world seems particularly crazy, I like looking into niche-controversies. A case where the nerds argue passionately over something that no one knew was controversial in the first place. In this video, I want to pick up one of these super-niche nerd fights: Are complex numbers necessary to describe the world as we observe it? Do they exist? Or are they just a mathematical convenience? That’s what we’ll talk about today.
So the recent controversy broke out when a paper appeared on the preprint server with the title “Quantum physics needs complex numbers”. The paper contains a proof for the claim in the title, in response to an earlier claim that one can do without the complex numbers.
What happened next is that the computer scientist Scott Aaronson wrote a blogpost in which he called the paper “striking”. But the responses were, well, not very enthusiastic. They ranged from “why fuss about it” to “bullshit” to “it’s missing the point.”
We’ll look at the paper in a moment, but first I will briefly summarize what we’re even talking about, so that no one’s left behind.
The Math of Complex Numbers
You probably remember from school that complex numbers are what you need to solve equations like x squared equals minus 1. You can’t solve that equation with the real numbers that we are used to. Real numbers are numbers that can have infinitely many digits after the decimal point, like square root of 2 and π, but they also include integers and fractions and so on. You can’t solve this equation with real numbers because they’ll always square to a positive number. If you want to solve equations like this, you therefore introduce a new number, usually denoted “i” with the property that it squares to -1.
Interestingly enough, just giving a name to the solution of this one equation and adding it to the set of real numbers turns out to be sufficient to make all algebraic equations solvable. Doesn’t matter how long or how complicated the equation, you can always write all their solutions as a+ib, where a and b are real numbers.
Fun fact: This doesn’t work for numbers that have infinitely many digits before the point. Yes, that’s a thing, they’re called p-adic numbers. Maybe we’ll talk about this some other time.
Complex numbers are now all numbers of the type a plus I time b, where a and b are real numbers. “a” is called the “real” part, and “b” the “imaginary” part of the complex number. Complex numbers are frequently drawn in a plane, called the complex plane, where the horizontal axis is the real part and the vertical axis is the imaginary part. i itself is by convention in the upper half of the complex plane. But this looks the same as if you draw a map on a grid and name each point with two real numbers. Doesn’t this mean that the complex numbers are just a two-dimensional real vector space?
No, they’re not. And that’s because complex numbers multiply by a particular rule that you can work out by taking into account that the square of i is minus 1. Two complex numbers can be added like they were vectors, but the multiplication law makes them different. Complex number are, to use the mathematical term, a “field”, like the real numbers. They have a rule both for addition AND for multiplication. They are not just like that two-dimensional grid.
The Physics of Complex Numbers
We use complex numbers in physics all the time because they’re extremely useful. There useful for many reasons, but the major reason is this. If you take any real number, let’s call it α, multiply it with I, and put it into an exponential function, you get exp(Iα). In the complex plane, this number, exp(Iα), always lies on a circle of radius one around zero. And if you increase α, you’ll go around that circle. Now, if you look only at the real or only at the imaginary part of that circular motion, you’ll get an oscillation. And indeed, this exponential function is a sum of a cosine and I times a sine function.
Here’s 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 a sine with a cosine… that’s a mess. You don’t want to do that. That’s why, in physics, we do the calculation with the complex numbers, and then, at the very end, we take either the real or the imaginary part. Especially when we describe electromagnetic radiation, we have to deal with a lot of oscillations, and complex numbers come in very handy.
But we don’t have to use them. In most cases we could do the calculation with only real numbers. It’s just cumbersome. With the exception of quantum mechanics, to which we’ll get in a moment, the complex numbers are not necessary.
And, as I have explained in an earlier video, it’s only if a mathematical structure is actually necessary to describe observations that we can say they “exist” in a scientifically meaningful way. For the complex numbers in non-quantum physics that’s not the case. They’re not necessary.
So, as long as you ignore quantum mechanics, you can think of complex numbers as a mathematical tool, and you have no reason to think they physically exist. Let’s then talk about quantum mechanics.
Complex Numbers in Quantum Mechanics
In quantum mechanics, we work with wave-function, usually denoted Ψ, which are complex valued, and the equation that tells us what the wave-function does is the Schrödinger equation. It looks like this. You’ll see immediately, there’s an “i” in this equation, which is why the wave-function has to be complex valued.
However, you can of course take the wave-function and this equation apart into a real and an imaginary part. Indeed, one often does that, if one solves the equation numerically. And I remind you, that both the real and the imaginary part of a complex number are real numbers. Now, if we calculate a prediction for a measurement outcome in quantum mechanics, then that measurement outcome will also always be a real number. So, it looks like you can get rid of the complex numbers in quantum mechanics, by splitting the equation into a real and imaginary part, and that’ll never make a difference for the result of the calculation.
This finally brings us to the paper I mentioned in the beginning. What I just said about decomposing the Schrödinger equation is of course correct, but that’s not what they looked at in the paper, that would be rather lame.
Instead they ask what happens with the wave-function if you have a system that is composed of several parts, in the simplest case that would be several particles. In normal quantum mechanics, each of these particles has a wave-function that’s complex-valued, and from these we construct a wave-function for all the particles together, which is also complex-valued. Just what this wave-function looks like depends on which particle is entangled with which. If two particles are entangled, this 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 which they look at in the new paper is then whether there are ways to entangle particles in the normal, complex quantum mechanics that you cannot build up from particles that are described entirely by real valued functions. Previous calculation showed that this could always be done if the particles came from a single source. But in the new paper they look at particles from two independent sources, and claim that there are cases which you cannot reproduce with real numbers only. They also propose a way to experimentally measure this specific entanglement.
I have to warn you that this paper has not yet been peer reviewed, so maybe someone finds a flaw in their proof. But assuming their result holds up, this means if the experiment which they propose finds the specific entanglement predicted by complex quantum mechanics, then you know you can’t describe observations with real numbers. It would then be fair to say that complex numbers exist. So, this is why it’s cool. They’ve figured out a way to experimentally test if complex numbers exist!
Well, kind of. Here is the fineprint: This conclusion only applies if you want the purely real-valued theory to work the same way as normal quantum mechanics. If you are willing to alter quantum mechanics, so that it becomes even more non-local than it already is, then you can still create the necessary entanglement with real valued numbers.
Why is it controversial? Well, if you belong to the shut-up and calculate camp, then this finding is entirely irrelevant. Because there’s nothing wrong with complex numbers in the first place. So that’s why you have half of the people saying “what’s the point” or “why all the fuss about it”. If you, on the other hand, are in the camp of people who think there’s something 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 embrace complex numbers, or accept that nature is even more non-local than quantum mechanics.
Or, of course, it might be that that the experiment will not agree with the predictions of quantum mechanics, which would be the most exciting of all possible outcomes. Either way, I am sure that this is a topic we will hear about again.
Complex numbers are now all numbers of the type a plus I time b, where a and b are real numbers. “a” is called the “real” part, and “b” the “imaginary” part of the complex number. Complex numbers are frequently drawn in a plane, called the complex plane, where the horizontal axis is the real part and the vertical axis is the imaginary part. i itself is by convention in the upper half of the complex plane. But this looks the same as if you draw a map on a grid and name each point with two real numbers. Doesn’t this mean that the complex numbers are just a two-dimensional real vector space?
No, they’re not. And that’s because complex numbers multiply by a particular rule that you can work out by taking into account that the square of i is minus 1. Two complex numbers can be added like they were vectors, but the multiplication law makes them different. Complex number are, to use the mathematical term, a “field”, like the real numbers. They have a rule both for addition AND for multiplication. They are not just like that two-dimensional grid.
The Physics of Complex Numbers
We use complex numbers in physics all the time because they’re extremely useful. There useful for many reasons, but the major reason is this. If you take any real number, let’s call it α, multiply it with I, and put it into an exponential function, you get exp(Iα). In the complex plane, this number, exp(Iα), always lies on a circle of radius one around zero. And if you increase α, you’ll go around that circle. Now, if you look only at the real or only at the imaginary part of that circular motion, you’ll get an oscillation. And indeed, this exponential function is a sum of a cosine and I times a sine function.
Here’s 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 a sine with a cosine… that’s a mess. You don’t want to do that. That’s why, in physics, we do the calculation with the complex numbers, and then, at the very end, we take either the real or the imaginary part. Especially when we describe electromagnetic radiation, we have to deal with a lot of oscillations, and complex numbers come in very handy.
But we don’t have to use them. In most cases we could do the calculation with only real numbers. It’s just cumbersome. With the exception of quantum mechanics, to which we’ll get in a moment, the complex numbers are not necessary.
And, as I have explained in an earlier video, it’s only if a mathematical structure is actually necessary to describe observations that we can say they “exist” in a scientifically meaningful way. For the complex numbers in non-quantum physics that’s not the case. They’re not necessary.
So, as long as you ignore quantum mechanics, you can think of complex numbers as a mathematical tool, and you have no reason to think they physically exist. Let’s then talk about quantum mechanics.
Complex Numbers in Quantum Mechanics
In quantum mechanics, we work with wave-function, usually denoted Ψ, which are complex valued, and the equation that tells us what the wave-function does is the Schrödinger equation. It looks like this. You’ll see immediately, there’s an “i” in this equation, which is why the wave-function has to be complex valued.
However, you can of course take the wave-function and this equation apart into a real and an imaginary part. Indeed, one often does that, if one solves the equation numerically. And I remind you, that both the real and the imaginary part of a complex number are real numbers. Now, if we calculate a prediction for a measurement outcome in quantum mechanics, then that measurement outcome will also always be a real number. So, it looks like you can get rid of the complex numbers in quantum mechanics, by splitting the equation into a real and imaginary part, and that’ll never make a difference for the result of the calculation.
This finally brings us to the paper I mentioned in the beginning. What I just said about decomposing the Schrödinger equation is of course correct, but that’s not what they looked at in the paper, that would be rather lame.
Instead they ask what happens with the wave-function if you have a system that is composed of several parts, in the simplest case that would be several particles. In normal quantum mechanics, each of these particles has a wave-function that’s complex-valued, and from these we construct a wave-function for all the particles together, which is also complex-valued. Just what this wave-function looks like depends on which particle is entangled with which. If two particles are entangled, this 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 which they look at in the new paper is then whether there are ways to entangle particles in the normal, complex quantum mechanics that you cannot build up from particles that are described entirely by real valued functions. Previous calculation showed that this could always be done if the particles came from a single source. But in the new paper they look at particles from two independent sources, and claim that there are cases which you cannot reproduce with real numbers only. They also propose a way to experimentally measure this specific entanglement.
I have to warn you that this paper has not yet been peer reviewed, so maybe someone finds a flaw in their proof. But assuming their result holds up, this means if the experiment which they propose finds the specific entanglement predicted by complex quantum mechanics, then you know you can’t describe observations with real numbers. It would then be fair to say that complex numbers exist. So, this is why it’s cool. They’ve figured out a way to experimentally test if complex numbers exist!
Well, kind of. Here is the fineprint: This conclusion only applies if you want the purely real-valued theory to work the same way as normal quantum mechanics. If you are willing to alter quantum mechanics, so that it becomes even more non-local than it already is, then you can still create the necessary entanglement with real valued numbers.
Why is it controversial? Well, if you belong to the shut-up and calculate camp, then this finding is entirely irrelevant. Because there’s nothing wrong with complex numbers in the first place. So that’s why you have half of the people saying “what’s the point” or “why all the fuss about it”. If you, on the other hand, are in the camp of people who think there’s something 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 embrace complex numbers, or accept that nature is even more non-local than quantum mechanics.
Or, of course, it might be that that the experiment will not agree with the predictions of quantum mechanics, which would be the most exciting of all possible outcomes. Either way, I am sure that this is a topic we will hear about again.