Do Complex Numbers Exist?

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 nerdfights are complex numbers necessary to describe the world as we observe it do they exist or are they just 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 a 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 aronson wrote a blog post in which he called the paper striking but the responses were well not very enthusiastic they ranged from why fuss about it to 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 you probably remember from school that complex numbers are what you need to solve

equations like x squared equals minus one 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 pi but they also include integers and fractions and so on you can't solve this equation with real numbers because they're 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 minus 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 plus 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 the thing they're called periodic numbers maybe we'll talk about this some other time complex numbers are now all numbers of the type a plus i times 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

are 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 one two complex numbers can be added like they were vectors but the multiplication law makes them different complex numbers 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 we use complex numbers in physics all the time because they are extremely useful they're useful for many reasons but the major

reason is this if you take any real number let's call it alpha multiply it with i and put it into an exponential function you get e to the i alpha in the complex plane this number e to the i alpha always lies on a circle of radius one around zero as you increase alpha you go around that circle now if you look only at the real or imaginary part of that circular motion you'll get an oscillation and indeed this exponential function is the 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 alpha and one with beta 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 in quantum mechanics we work with wave functions usually denoted psi which are complex valued and the equation that tells us what the wave function does is the schrodinger 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 of 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 an imaginary part and that will 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 schrodinger

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's 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 aren't

tangled 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 calculations 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 the 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 and it will then be fair to say that complex numbers exist so this is why it's cool they figure out a way to experimentally

test if complex numbers exist well kind of here's the fine print this conclusion only applies if you want the purely real value 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 are 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 already is or of course it might be 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'm sure that this is a topic we'll hear about again this video was sponsored by brilliant which is the website and app that offers interactive courses on a large variety of topics in science and mathematics if you want to really understand complex numbers and their role in quantum mechanics brilliant is a great starting place for that to get more background on this video's topic have a look for example at their courses on complex numbers like all of their courses you can check your understanding along the way by answering questions in

the exercises they also have courses on the properties and behavior of quantum objects to support this channel and learn more about brilliant go to brilliant.org sabine and sign up for free the first 200 subscribers using this link will get 20 off the annual premium subscription thanks for watching see you next week