 How does the nucleus of an atom stay together? That's it! That's the question we will answer in this video, and in doing so we will uncover a whole new force of nature and the beautiful mathematical machinery behind it. This is our submission to the Summer of Math Exposition. At each step we will start with a tiny experimental fact, and then we will build the mathematics to understand it. This video is intended for people in their first year of college pursuing some degree in engineering, so we will assume you have some knowledge of physics, chemistry and linear algebra, but if you don't, if you are studying something like photography or graphic design, you might be more familiar with this topic than you realize, so this should be really fun! Let's start with a quick recap on atoms. Electrons are made of a nucleus of positive protons and neutral neutrons, with a cloud of negative electrons hovering around them. The reason electrons stay close to the nucleus is because of the electromagnetic force. The negative charge of electrons is attractive to the positive charge of protons, but electromagnetism cannot explain why protons and neutrons stay together. For starters, neutrons are neutral, so they don't feel electrical attraction in the first place, and if anything, protons should be repelling each other because they have equal electric charges, and gravity is not strong enough to overcome that repulsion, so what's going on? How do protons and neutrons stay together? From this simple observation, we can conclude that there must be another force, a force stronger than electromagnetism, and gravity, and this force is what's holding the nucleus together. At this point, we know virtually nothing about this force, except that it must be really strong, so let's call it the strong force, which is exactly what physicists did in the 1950s. To solve the mystery of the strong force, physicists started building the first particle accelerators in the 1960s, and the experiments they designed using those accelerators were extremely clever and beautiful, but using the length of this video, I have to cut corners somewhere, so let's skip to the discoveries, because at first, they made no sense. As it turns out, protons and neutrons are made of three other smaller particles known as quarks. In total, we have found six kinds of quarks, and we have given them six silly names, up, down, strange, charm, top, and bottom. Beautiful, no notes. Protons and neutrons are made of only two of these kinds of quarks. Protons are made of two upquarks and one downquark, while neutrons are made of two downquarks and one upquark. To understand why this was so confusing at first, we have to talk about Pauli's exclusion principle. And to talk about Pauli's exclusion principle, we have to talk about spin. Part one, Pauli's exclusion principle. Spin is called spin, because it's like if particles were spinning in one direction or the other, except it's also not like that at all, because particles are point-like, and points do not spin. And yet, they do have angular momentum. Look, the nature of spin is a really fun discussion to have, but it's also extremely complex and difficult, but thankfully, the mathematics of spin are actually quite easy to understand, so let's just focus on that. And if you want to learn more about the nature of spin, I'm gonna leave you some links in the description, and I wish you the best of luck. Spin is a quantity whose value, which we call the spin quantum number, changes in steps of one unit, never going above nor below a given absolute value, called the magnitude. For example, electrons, protons and neutrons all have spin with a magnitude of one half, so their allowed values are only one half and minus one half. We call these kinds of particles fermions. For example, let's say that we have an electron with a spin of minus one half, and then comes along a photon with a spin of one. If this electron absorbs that photon, the action of the photon will be to change that spin by one unit, from minus one half up to one half. But now this electron cannot absorb another photon with spin one, because it cannot go above one half. However, it can absorb a photon with spin of minus one, going back to minus one half, at which point it cannot absorb another photon with a spin of minus one, because it cannot go below minus one half. As you can see from this example, photons can carry more spin than electrons, protons or neutrons, and this is because their spin magnitude is actually one. We call these kinds of particles with integer spin bosons. Next, because quantum mechanics is inherently probabilistic, when we measure spin, we are never quite sure which one we are going to get. We can only know the probability of getting each result. We write such probabilities using something called the wave function. For example, let's say that this was the wave function of an electron. This strange arrow looking thing is called a ket. Physicists use it to specify when they are working with quantum probabilities. In this case, it means that if we measure the spin of this electron, the chances of measuring a spin of one half are of three in five or sixty percent, while the chances of measuring minus one half are of two in five or forty percent. Notice that the probabilities are the absolute values of the squares of each coefficient, so the coefficients themselves could be negative or even complex, and it wouldn't be an issue. We can visualize this combination of probabilities using a plane. The x-axis will be how likely you are to measure one half, and the y-axis how likely you are to measure minus one half. Now we can visualize all possible combinations of probabilities in this plane. This also means that we can think of measuring one half as the vector one zero, and measuring minus one half as the vector zero one. Then we can rewrite the wave function as a sum of those vectors, which means the wave function is just a vector of probabilities. This is the reason why quantum mechanics is just like spicy linear algebra. It may look like we are just finding new ways to say the same thing, and we are, that's like eighty percent of mathematics, but the reason we do this is because it will be very useful when we look at systems made of more than one particle, like the electrons in this helium atom. Let's call them A and B. Each electron can have two values of spin, so when we measure the spin of both of them, there should be four possible outcomes as seen in this table. What we want is to assign a vector to each outcome, just like we did before, and there are a couple of ways we could do that, but the best one is to use an operation known as the tensor product. It is a little bit hard to explain what it does with words, but it is quite easy to explain visually. Using the tensor product, we can write each outcome as a base vector, and then we can add them up to write the wave function of both electrons as a vector of four different probabilities. We could calculate the coefficients in this vector using Schrodinger's equation, but that would be really difficult. Thankfully, we don't really need to do that. All that we care about is that the coefficients are the results of a function. Let's call it C of A and B, and this function takes as inputs the spin states of each electron, and then it does whatever calculations we don't want to do to get the right answers. Let me show you what that means with an example. If you give this function the inputs one-half and minus one-half, and it outputs the square root of one over four, that means there's a 25% chance you will find electron A to have a spin of one-half and electron B to have a spin of minus one-half. Ok, so all of that was just the background. Here is where it gets interesting. Let's go back to the tensor product. If electron A is in state one-half and electron B is in state minus one-half and we have A cross product B, then the result is the vector 0 1 0 0. But what if we have B cross product A? Then the result would be the vector 0 0 1 0. In summary, if we use A then B, we end up with one wave function, but if we use B then A, we end up with a slightly different wave function. And yet, this difference shouldn't really matter, it should be like measuring distance in meters or yards, just different ways to describe the same physical reality. Pauli wanted to know the relationship between these two different versions of the wave function and what he found is that it depended on the spin. If you're working with bosons, then it doesn't matter in which order you write the wave function, it will be the same either way, it is symmetric. But if you're working with fermions, like we are doing right now, then the order does matter. If you switch the order of the fermions in the wave function, it will gain a negative sign, it will be anti-symmetric. This is Pauli's exclusion principle, and it seems completely bonkers, doesn't it? Why can't it be symmetric every time? And why does it depend on the spin, what's going on? The answer lies deep in a connection between general relativity and quantum mechanics, known as the spin statistics theorem, which is really easy to explain. Here it goes. Ok, actually it's extremely difficult to understand and to explain, and you've seen the length of this video, so let's just take it as an axiom and move on with our lives. When we look at C of A and B with this restriction in mind, we find something very interesting. C of 1 half 1 half doesn't change if we switch the order of the fermions, and yet Pauli demands it has to be anti-symmetric, so the only option it has is to be zero, because only zero is equal to negative zero. Same with C of minus 1 half minus 1 half. This means there is exactly 0% chance of ever measuring the same spin in both of these electrons. Then we are left only with the states in which the electrons have opposite spins. They must have the same coefficients, but with opposite signs, and the probability must add up to 100%, which means there's always a 50% chance to measure either one. Think about what just happened. A mathematical restriction about the tensor product led us to discover real physics. Isn't that just lovely? But what happens if we have three electrons instead of only two? Because now, no matter what, at least two electrons will have the same spin. I mean, there are only two possibilities and three particles, at least two of them have to share. This is not anti-symmetric, because we could switch the two electrons with the same spin and the wave function wouldn't change. Well, this is not a big problem, because regardless of how much I like to talk about it, nothing is not the only quantum number. In this case, at least one of the electrons will have a different energy level, written as the quantum number N. This will be a new variable in the function for the coefficients, and it will take the responsibility of making the wave function anti-symmetric. If you remember your chemistry classes, this is the reason why you can only have two electrons in each energy level. Now, you are finally ready. Here is why quarks were so confusing at first. Quarks also have spin with a magnitude of one-half, and yet you need three of them to make protons and neutrons, which means spin cannot make them anti-symmetric, and they are too close together to have different energy levels, so how can they possibly be anti-symmetric? Regardless of how much I like to talk about it, spin is not the only quantum number. Not the only quantum number. Not the only quantum number. Hmm, that's it. There must exist yet another quantum number making the wave function anti-symmetric. Part 2, color charge. At this point, we know nothing about this new quantum number, but we can deduce how it should work. We know it needs to have at least three different values, one for each quark, and we can represent them as vectors from the get-go, one-zero-zero, zero-one-zero, and zero-zero-one. But it will be useful to give the names, red, green, and blue. These names may seem very random now, but they come from a metaphor that will make a lot of sense when we get to it. Let's say we have a proton made of three quarks, Q1, Q2, and Q3, and let's say that each of them is in a different state of this property we are calling color. You know that if you switch the color of any two quarks, the wave function should get a minus sign. By testing more and more combinations, we can start to see a pattern. It is like a cycle. If the order of the colors flows with the cycle, then the whole thing is positive. But if it flows against it, then it is negative. This is the color cycle, and it will be very important. Turns out that mathematicians have used an object with this property for a long time. They call it the Levy-Civita symbol, also known as the antisymmetric symbol. We can write the wave function of a proton using these color states and the Levy-Civita symbol like this. In each case, the Levy-Civita symbol will be one if the colors flow with the cycle, minus one if they flow against it, and zero in all other cases, like if one of them is repeated or something. After removing all the outcomes that end up with zero probability, we end up with only six. Notice how if you switch the order of two colors in one outcome, you will find another one with that order of colors but the opposite sign. Part three, white is color neutral. We got carried away with antisymmetry, but remember that the whole point was to figure out how the nucleus of an atom stays together, and maybe we just figured it out. In the same way that electric charge interacts with electromagnetic fields, this new color charge interacts with strong fields. Each quark must be attracting the other two, and this is how they form protons and neutrons. But wait, no, something just plainly makes no sense here. If this force is so strong, and if it interacts with all protons and neutrons, why is this the first you are hearing about it? Shouldn't you see it everywhere in your everyday life? I mean, electromagnetism forms molecules which form cells and living creatures like us. Gravity forms planets, stars and galaxies, but the strong force seems to stop forming things at the scale of atomic nuclei. At larger distances, it seemingly disappears. To understand what's going on, we could try to find the formula that describes the attraction between color charges, the equivalent of Newton or column loss, and other scientists certainly did that, but let's take another approach. Let's look at neutrons. Neutrons are made of two down quarks with a charge of minus one-third and one up quark with a charge of two-thirds. At short distances, each of those quarks can feel electrical attraction, but at long distances, the effect of each quark can sell out, because their charges add up to zero, and this makes the neutron, well, neutral and immune to electrical attraction, the ace of the particle world. Gravity, having three different color charges, acts in a similar way. Maybe at short distances, each quark feels the strong force, and this is how they form protons, neutrons and atomic nuclei, but at long distances, they cancel each other out, making protons and neutrons colorless, or I guess following this metaphor, white. We can write this idea with a simple equation, red plus blue plus green is equal to zero. But this means that red should be equal to minus green minus blue, and the same goes for all the other colors. Each one is equal to having the opposite of the other two, which is weird. I mean, for starters, what even is the physical meaning of a negative color charge? Well, just like there are positive and negative electric charges, there are positive and negative color charges. Quarks only have positive color charges, but anti-quarks, the anti-matter versions of quarks, have only negative color charges. And of course, minus red minus blue minus green is equal to zero. Again, each negative color charge is equal to having the opposite of the other two. For example, anti-red is equal to green plus blue. It's perfectly natural if this sounds absolutely bonkers! That was my first impression when I learned about this, but if you find this reasonable and even familiar, then you are probably a photographer or a graphic designer. Part 3.5 The RGB Color Space RGB stands for Red, Green, Blue, and it is a system used to show colors in screens. Each pixel is made of three tiny LEDs. One is red, the other is green, and the last one is blue. The screen can make each LED shine with different intensities, ranging from being completely off, which is called zero, to shining as bright as the device allows, which is called 255 for engineering reasons. The combination of these different intensities can give us the illusion of many different colors. For example, if all three LEDs are at their maximum intensities, we will see it as white. But then, if we turn off the blue LED, we will see it as yellow. This happens because we have three kinds of color receptor cells in our eyes. They are tuned to trigger at specific frequencies of light, but there is some overlap. For example, the cells that detect red and green light are both triggered by yellow light. So, when we see red and green at the same time, our brains assume we are just seeing yellow light, even if there is no yellow light reaching our eyes. This system was first invented for color televisions in the 1960s, and what do you know? These discoveries about quarks were being made in the 1960s. Just like we have done a couple of times now, we will visualize colors in the RGB system as vectors. For example, white will be 255, 255, 255, and yellow will be 255, 255, 0. But wait a second, the vector 00255 is blue, right? And if we add this yellow vector, then we get white, huh? In a sense, it's like they cancelled each other out. In this system, yellow is the opposite of blue, but yellow is made of red and green. We can do the same for the other colors. Green and blue make cyan, which is the opposite of red, while red and blue make magenta, which is the opposite of green. Can you see the similarities? The mathematics used in the RGB system are the exact same mathematics followed by the color charges of the strong force. This incredible coincidence is the reason the color charges are called color charges in the first place, but I know something is nagging you at the back of your head. How come two completely different things ended up being described by such similar mathematics? You'll figure it out soon in the next section, I promise. Part 4, SU3. This is all insanely cool, isn't it? Using Pauli's exclusion principle, you deduce the existence of a new property of matter, and you use the fact that it cancels out at large distances to deduce that it follows the same mathematics as the RGB system, but there is still one problem, magnetism. Even if protons and neutrons have zero color, even if they are white, they could still interact with the strong force. The easiest way to think about this is to look at neutrons again. They might be electrically neutral, but they still have magnetic fields. You can see this with the magnets in your fridge. They don't shock you when you touch them because they have neutral electric charge, but they can still interact with the magnetic part of the electromagnetic force, and quite strongly. The reason this happens lies deep in quantum electrodynamics, but in summary, it happens because the electromagnetic force is transmitted using photons, and photons are neutral. They do not have electric charge. If photons have electric charge, then there would be no magnetism. The strong force is transmitted using particles called gluons. They have this name because they keep the nucleus glued together. Get it? No? Whatever, I mean it's not like it's my joke, I didn't come up with it. Anyway, if gluons were neutral, like photons, then there would be a sort of strong magnetism. But we see nothing like that in nature, which means gluons must not be neutral, they must be carrying color charge. This is one of the cool differences between math and physics. In physics you can just say, hmm, I looked really hard for something and I couldn't find it, so it doesn't exist. But in math, people can fail to find something for 4000 years and they will still not accept that it doesn't exist. Just like my soulmate, I know she's out there. Going back to gluons, consider this Feynman diagram. Here we have two op quarks, one is green and the other one is blue, and they interact through the strong force by exchanging a gluon. Maybe this green quark emitted its green charge and it was absorbed by this other quark. However, a quark cannot remain colorless, again that would produce a strong magnetism. To fix this, maybe this blue quark also emitted its blue charge, and in this way they switched colors. At first, it looks like they emitted two gluons, because two kinds of color charge were exchanged. But, since this was a single interaction, they must have exchanged a single gluon, and that gluon must have been carrying two color charges. In fact, gluons always carry two color charges. Let me show you. Maybe this green quark became blue, and to ensure that the total amount of color charge remained the same, it emitted a gluon carrying two color charges, green and anti-blue. Because when we subtract the color charge of the gluon from the color charge of the quark, it adds up to blue. Then the blue quark absorbed the emitted gluon, blue and anti-blue cancelled out, and this quark was left only with green charge, and in this way they switched colors. Or maybe it happened the other way around. Maybe the blue quark emitted a gluon carrying blue and anti-green. Hmm, how can we know? That's the fun part, you don't! We must always consider gluons as the two possibilities they could have been. They too are vectors of probabilities. And now a question starts forming in your mind. It may be a little hard to express, so let me help you out. How many different combinations do we need to create all possible gluons? Or in terms of linear algebra, how many linearly independent combinations of gluons are there? This is a very important question. It will tie everything together once we solve it. But to solve it, we must go back to spin. I know, it always goes back to spin. Part 4.5 Singlets and Triplets Okay, I know this video has been long enough, but this is the home stretch, we can do it. Consider a hydrogen atom, a proton and an electron. Once again, there are four possible spin combinations, but since there are many differences between them, they are allowed to have the same spin. Let's imagine both of them have a spin of minus one-half. Then the total spin of the atom is minus one. Next we shoot a photon with spin one at this atom, and it is absorbed by either the electron or the proton. We don't know which one absorbs it, but we know the action of this photon is to change a spin from minus one-half to one-half. Remember from the beginning of the video? Once again, we can represent this as a combination of two possibilities, one in which the electron absorbed the photon and one in which the proton did. Regardless, in either case, the total spin of the atom is zero. Next, we shoot another photon with spin one. Once again, we don't know which one absorbed it, but we know it had to be whichever particle had a spin of minus one-half, because their spin cannot go above one-half. We can apply the action of the photon to both states, and whether you know, they end up being identical, so we can add them up and the total spin is one. In the end, this atom has a spin with a magnitude of one, it is a boson, and for that reason, it is free to move up and down between these three states. Together, we call them a triplet state. But what I want you to notice here is that the photon is secondary. Its action is what really matters. These three states are related to each other by the action of going up or down in spin. Remember that. Let's go back to when the spin was zero. What if this plus sign was a minus sign instead? Well, if we apply the action of going up, we end up with two equal states, just like before, but now they don't add up, they cancel out. And yet, the wave function cannot be zero because the atom cannot just stop existing. Instead, if this atom absorbs a photon, it will simply move the electron farther or closer to the proton, but the spin state will remain the same. In other words, this atom is behaving like a boson, but a boson with a spin magnitude of zero, and for that reason, it only has one allowed state, zero. This is called a singlet state. Finally, finally we are ready. Everything in the video has been leading up to what comes next. Let's look at one possible gluon. This one is carrying either red anti-blue or blue anti-red. Now, if we look at the color cycle, we can see that before red is blue, and before blue is green. For that reason, if we went down the color cycle, we would find the state blue anti-green plus green anti-blue. Next if we continue going down, we will find the state green anti-red plus red anti-green. Here we have three states with a sort of relationship between them regarding going up or down. This is analogous to the triplet state from before. As you can imagine, this means we should have a state analogous to the singlet, blue anti-green minus green anti-blue. However, there's a small caveat here. In the case of spin, one of the states has to be flipped to be negative, right? Well, in this case we do that, but we also must flip the whole thing around the complex plane by multiplying everything by negative i. I would love to go into detail about why we have to do this, but this is one of those things that should be its own video. Hopefully, the notion that we have to flip the vector in more ways than one is enough for now. Anyway, we have a triplet and a singlet, just like before, but who said we were done? We can keep going down the color cycle or up for that matter. If we do, we are not gonna find any new states, just the same three over and over, but we do find that we can group them in three sets. Each of these sets is like a triplet, and thus it should have the corresponding singlet. Of course, these six states are not triplets and singlets exactly, but they are related to those concepts, and we will see why in a bit. We are so close. Here I mentioned that colorless gluons must not exist to avoid strong magnetism. Now I can say more precisely that the problem would be colorless gluons that are not affected by the actions of moving through the color cycle or switching colors. Only those gluons could interact with colorless protons and neutrons. For example, one of these colorless gluons would be the symmetric state red, anti-red plus blue, anti-blue. Because if we switch red and blue, it remains the same. But the worst of them all would be this one. This is the real color singlet, red anti-red plus green anti-green plus blue anti-blue. No matter what actions you do to this, which direction you move through the color cycle nor how many times, it just never changes. If gluons could take this state, the universe would be a very different place. However, we can totally have colorless gluons as long as they are affected by these actions. Take for example, red anti-red minus blue anti-blue. If we switch red and blue, we end up with an anti-symmetric state. Therefore, this gluon can exist because it could not interact with colorless particles. As it happens, we can create all colorless gluons using only two of them. So we'll use this one and that one. In the end, we are left with only eight gluons. No matter what experiment we study or what process we describe, whatever gluons are involved, they can always be described as combinations of these eight gluons. But the thing is that this is not the only group of eight gluons we could use. To find the other alternatives, we can once again move through the color cycle. These new sets look slightly different, but the relationship between the gluons in each of them are the same every time. In other words, this set is symmetric under the action of moving through the color cycle. When we find a set that is symmetric under a certain kind of actions, we call it a group. The group we found here has a name, the special unitary group of order 3, but among friends they call it SU3. The fact that we found SU3 in here is not a coincidence. What we've been trying to understand about the strong force is a consequence of the fact that nature follows this symmetry. SU3 must be represented using eight different elements, so there have to be eight gluons. There must be a cyclic relationship between six of them, so there have to be six color charges and they have to be antisymmetric. The other two elements have to be outside of that cycle, so the six color charges have to cancel each other out somehow. The analogs for triplets and singlets exist because SU3 contains three copies of a smaller group known as SU2 and it just so happens that spin is described by SU2. The reason the RGB color space has a similar math to the strong force is because we have three kinds of color receptors in our eyes and the combinations of those three things is also described by SU3. It's beautiful. It is so beautiful. It was SU3 all along and it would have gotten away with it had it not been for those meddling kids. Okay, I've calmed down now, but my point still stands. How mind blowing is it to realize that all of the properties of a force of nature are the consequence of a mathematical symmetry, a symmetry that can exist in other things like in our eyes by sheer coincidence and the strong force is not the only one. Electromagnetism and the weak force are the result of symmetries known as U1 and SU2. But what about gravity? Don't mention gravity, we were having a good time. In fact, when you combine all these symmetries into a single equation, you get the standard model of particle physics, that's what it is. It's describing the properties of these three symmetries and it's also the best description of nature humanity has devised so far. I want us to just admire the beauty of you all, the beauty of the mathematics and the fact that we can know them. The universe sticks around us, flowing, changing. And now, do you understand those sticks just a little bit better? Joy, euphoria, knowledge. I would love to keep talking about the strong force and symmetry, but the video is long enough already. Thank you for staying with this explanation until the end. I hope to have really broadened your horizons. The strong force is a topic that is rarely mentioned in physics and chemistry courses because, since its effects are limited to the nucleus, it's not seen as relevant, depriving people of learning about something so beautiful. But now, you are ready. Now you have the tools you need to continue learning about the strong force, spin, symmetry and so much more. So go out there and keep on learning.