 Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection. This quote, attributed to German mathematician and theoretical physicist Hermann Weil, rings true in many aspects of society. Human minds have shown deep fascination with seeking and finding patterns and symmetries in nature, as well as forming our own symmetric creations that we equate with attractiveness or refinement. One interest of theoretical physics is the idea of symmetries in the universe. Group theory is a cornerstone in fundamental quantum theory. Using group theory, particle physicists can define and explain the different types of symmetry that they observe through the interactions of elementary particles. In this video, I will be presenting a brief overview of some of the connections between group theory and science, and explain how these applications deepen our understanding of quantum states in the universe. Let's begin by defining what a symmetry is. Simply put, a symmetry is a mapping of an object onto itself that preserves the structure of the object. It's a property where an object undergoes a process of operations and transformations that leaves it essentially unchanged. We can think of symmetries of the universe as transformations under which the laws of physics remain the same. Let me give an example. Suppose that you stand at the origin in a completely empty region of the universe. You have a ball and you give it a push along the x-axis so that it begins to move at a constant velocity. I stand at the position 4-4 and apply the same amount of force to an identical ball two seconds later along the line y equals x. So my question is, do the laws of physics remain the same in both scenarios? The answer would be, yes they do. There is nothing fundamentally different about this situation and this one. In both cases, we can predict what the behavior of the ball is going to be and we can think of these two cases as equivalent. This is because we say that the laws of physics are invariant under translation and rotation and invariant through time. That is, the universe follows a translational symmetry, it follows a rotational symmetry, and it follows a time symmetry. Emmy Noether, a brilliant German mathematician, proved this theorem named after her. Every differentiable symmetry of a physical system imposes the conservation of a physical quantity. In our example, conservation of momentum corresponds to translation, conservation of angular momentum corresponds to rotation, and conservation of energy corresponds to travel through time. We refer to these types of symmetries as global symmetries, but there are also local gauge symmetries in the world that correspond to fundamental forces. The four fundamental forces are gravitational force, electromagnetic force, strong force, and weak nuclear force. Gravity does not have a symmetry, but the other three forces do. The symmetry group that describes electromagnetism is U1. The symmetry group that describes the weak nuclear force is Su2, and the symmetry group that describes the strong interaction is Su3. In a moment here, we'll briefly go over each of these and what they mean, but first, these three groups are all what we call Lie groups, so let's talk about what those are. We can think of a Lie group as a type of continuous group that is also a smooth manifold. Topologically, a manifold is a space in which each point has some neighborhood around it that resembles Euclidean space, at least on a local level. So we can say that the neighborhood is, in fact, homeomorphic to an open subset of Euclidean space, where homeomorphic essentially means isomorphic in a topological context. In the field of quantum mechanics, we actually work in what we call Hilbert space, but this is simply an analog of Euclidean space that can be extended to infinite dimensions. We don't want to get too much into the weeds here, so I'll just say this. If we have a continuous group where element multiplication and the taking inverses are both differentiable or smooth, we have a Lie group. In an abstract algebra course, when students are first introduced to groups, they will typically begin by investigating groups with discrete structures like Zn, Dn, and Sn. However, Lie groups provide an ideal representation of the continuous symmetries that appear in quantum physics. Lie groups represent an intersection between algebra and geometry, allowing us to use algebra to describe physical phenomena. Furthermore, they are differentiable, which opens up the possibility of making several useful calculations on them using calculus. Another thing to note is that because Lie groups are continuous, they will have infinite order. So first, let's take a look at u1, the unitary group of degree 1. Un is the group of n by n unitary matrices. A unitary matrix is one whose conjugate transpose is its inverse. For u1, we only have a 1 by 1 matrix, which is really just a number. Geometrically, we can think of this group as simply the unit circle graphed onto the complex plane. Algebraically, we write this group as all complex numbers e to the i phi, where phi is a real number. We note that the conjugate of e to the i phi is e to the negative i phi, and these two multiplied together give us 1, which is the identity. One interesting thing is that u1 contains complex elements, but it's actually isomorphic to a real valued group, SO2. An orthogonal matrix is given by this definition, a matrix whose transpose is its inverse. We can kind of think of unitary matrices as the complex parallel to orthogonal matrices in that sense. The special orthogonal group only contains orthogonal matrices with a determinant of 1. We can define SO2 like this in this kind of matrix form. So here's our proof that matrices of this form are indeed elements of SO2. Geometrically, we know that SO2 is the same as u1, and we can clearly see that u1 maps to SO2 by this relation right here. We're not going to go through the proof of this, but this function is actually an isomorphism. Therefore, the two groups are isomorphic. Notice that only one parameter, the angle of rotation, is used to parameterize u1. This is because the angle of rotation is the only variable. It's the only quantity that changes in our equation for an element to u1, and this means that we can obtain any element in u1 by changing phi alone. Remember how before I said that u1 characterizes the symmetries of electromagnetism? Well, this single parameter of u1 actually corresponds to the electromagnetic force carrier, or messenger particle, namely the photon, and the symmetries of electromagnetic force correspond to the law of conservation of charge. Now by force carrier or messenger, I mean that in particle physics, we can think of a force as simply the transfer or exchange of an elementary particle. As an example, say that two electrons are traveling towards each other. How does the yellow electron communicate its presence to the green electron and vice versa? It's through the exchange of photons to each other, an exchange that causes the particles to be repelled through the electromagnetic force. So again, just like how u1 only has one parameter, electromagnetism only has one force carrier, also known as a mediating particle, and that is the photon. What do our other groups look like? Recall that SUn will be the n by n special unitary matrices. SU2 corresponds to the weak interaction and can be used to describe properties like the isospin of fermions such as leptons and quirks. It is isomorphic to the group of uniquiturnians, which is given by this formula right here, and its symmetries correspond to the law of conservation of lepton number. I'm showing three matrices here called the poly matrices. These matrices are very useful and very important to the SU2 group because when we multiply each one by i, we obtain three matrices that together with the identity matrix form a complete basis for the le algebra that corresponds to the SU2 group. As there are three matrices, there are also three associated elementary particles. z to the zero, which means that z has a zero chart, the w minus boson, and the w plus boson. These are all bosons and these are carriers of the weak force. Finally, let's take a look at SU3, the group that characterizes the strong force and its symmetries. We can utilize properties of this group and calculations in this group to investigate properties like the color of a quark, which we can think of as the strong force equivalent of electric charge. Color is conserved in strong interactions. In general, the SUN group will have n squared minus one generating matrices. We unknowingly observed this property in SU2 when we had three generators. Keeping with this pattern, SU3 has eight generators called the Gell-Mann matrices that look like this. Therefore, in keeping with our pattern here, we're going to anticipate eight mediators for the strong force and surely enough there are eight of them and they're called gluons. We name the gluons by different color combinations as shown here. These three groups provide the foundation for the standard model as it is today. The standard model can be thought of as SU3 cross SU2 cross U1, but what do these symmetries actually look like? Let's go through a few simple examples. Consider an electron that's traveling along and then releases a W minus boson. What happens to the electron? What does this look like in our system? And what needs to take place right here in the center? Let's think about a couple of the properties that need to be conserved in this situation. One quantum property that will be conserved is called lepton number, sometimes informally referred to as lepton-ness. As we see from our table of the standard model, an electron is a lepton, but a boson is not, so an electron will have a lepton number of plus one, whereas the W minus boson will have a lepton number of zero. As a side note, antileptons also exist with a lepton number of negative one, but for this scenario, we don't have to worry about that. Since we have an initial net plus one lepton-ness, we need a final net plus one as well, which means that we have another particle in our system when the W minus boson is admitted. However, electric charge also needs to be conserved. In elementary charge units, our electron has a charge of negative one, and so does our boson, which means that we need a neutrally charged lepton. That means that these top three leptons cannot be our mystery particle. We have other properties that also need to be conserved, and it turns out that the particle that fits all of these criteria is the electron neutrino. We can view the same situation with a slightly different framework. This diagram gives us a sort of unique visualization of the particles in the standard model and shows us how there is a natural symmetry embedded in these particle interactions. Using our example from before, let's start at the bottom of our picture with an electron. When it emits a W minus boson, we can consider this action as a kind of operation that we perform on the electron, which we can think of as a vector, by applying the generating matrices that we looked at previously. We also can consider it like a physical transformation of the electron, where when the boson is emitted, the electron undergoes a rotation in this space and becomes an electron neutrino. Another example we can use is the property of color in quarks. Color is not actually color in the typical sense that we think of it, but it is a property that is conserved under the strong interaction. The quark colors are red, blue, and green, and the quark anti-colors are anti-red, anti-blue, and anti-green. So let's consider a neutron that contains three quarks, a red, a blue, and a green. The net color of our neutron is white, or sometimes also called colorless, because red plus blue plus green gives us white. Now suppose that our blue quark emits a blue anti-green gluon, which the green quark then absorbs. Our system now looks like this, and though the colors between individual quarks have been adjusted, the net color of our system remains the same. Now mathematically, what is happening here? Let's take a second look at our Gelman matrices. You may have noticed earlier that the Gelman matrices are traceless and singular. That is, they don't have inverses. The fact is that these matrices themselves are not actually elements of SU3. However, they can generate elements of SU3 through what we call matrix exponentiation, which can be performed using this power series. So if we take a Gelman matrix and apply this equation right here, we actually get a matrix that is in the SU3 group. So in our quark example, we can think of the blue quark as a vector that is being multiplied by one of these matrices. And we can also think of the green quark as a vector that has the inverse matrix applied to it. And ultimately, color is conserved overall in our system. This is another way that we see symmetries in nature's most fundamental atoms. To wrap up here, I just want to mention that the way that physicists talk and think about groups differs from the way that mathematicians think and talk about groups. This is because their purposes are not the same. Physicists care a lot about the way in which groups are represented. They discuss concepts like adjoint, defining, and irreducible representations of groups, because representations are how we characterize groups as linear transformations in vector spaces that can then be manipulated, or in geometric environments that translate to the physical world. In fact, abstract algebra in theoretical physics often manifests itself as an overlap between group theory and our presentation theory. We don't see this kind of perspective or approach in pure mathematics. For a physicist, groups are like a means to an end. For a mathematician, the group itself is the end. This dissimilarity is a good thing and an interesting thing to be aware of, especially if this interaction between math and physics is something that peaks your curiosity and that you want to continue studying. That is all that I have for this video. My goal in creating this has been to provide a helpful and insightful resource that gives some simple to understand, not too technical examples of the applications of abstract algebra to particle physics, and I hope that this will prove to be useful for someone who is interested in these connections between math and science. Thank you so much for listening, stay curious, stay positive, and have a marvelous day!