[Note: This is a six section storyline. Starting anywhere other than section I, may cause confusion]
So far we have considered particles moving around in real space. Space translations. Time translations. Space rotations.
Now we will investigate internal space transformations: U(1), SU(2), and SU(3). Call them gauge transformations.
Introductory chat about U(1), SU(2), and SU(3)
Electric charge, and electrons and positrons transforming as representations of U(1)
Weak charge, spin, and isospin, and electrons, positrons, quarks, and anti-quarks transforming as representations of SU(2)
Color charge, and quarks and anti-quarks transforming as representations of SU(3)
Gauge interactions (fermions undergo gauge transformations by emitting gauge bosons) and fermions transforming as fundamental representations of the group, antiparticle fermions transforming as complex conjugate representations of the group, and bosons transforming as adjoint representations of the group
SU(5), supersymmetry, and the gauge-gravity duality
There are three real dimensions of space (+ one of time) and six tiny, wrapped up dimensions. Gauge transformations have to do with rotating around in these little dimensions attached to space.
You can’t see the curled up dimensions, they are too small. They are called complex vector spaces. You might imagine them as like little knots or shapes attached to every single point of space and time. Sometimes they are called fibers. A 1-dimensional complex space is just a circle. U(1) is called the circle group and is closely related to the coulomb potential. Higher dimensional complex spaces are like fancier circles. Particles have orientations in, or states in, complex vector spaces, attached to space. You might say that particles have two lives, or pieces to their existence, one related to real space and one related to internal space.
Unitary matrices are defined over the complex numbers. That means that the individual entries of a matrix, and the individual entries of a vector upon which a matrix acts, are complex numbers. A U(1) matrix is just a 1x1 complex matrix, meaning one column and one row and thus one entry. That single entry is a unit magnitude complex number. As a matrix, that unit magnitude complex number, or phase, multiplies another unit complex number, which is a state in one of the aforementioned circles attached to space. The action of a U(1) matrix on a 1-dimensional complex number, rotates it around the circle. A complex number times a complex number is a complex number.
The special unitary groups SU(2) and SU(3) are just generalized rotations. SU(2) takes two complex number to two complex numbers, and SU(3) takes three complex numbers to three complex numbers. When particles transform in real space, they remain the same particle. When they undergo gauge transformations, they might change their particle type. For instance, an up spin electron might SU(2) rotate into a down spin electron, or an up quark might SU(2) rotate into a down quark.
U(1) | Multiplication by 1x1 U(1) matrix (a unit magnitude complex number). A U(1) matrix maps 1-dimensional complex numbers or probability amplitudes to 1-dimensional complex numbers or probability amplitudes.
SU(2) | Multiplication by 2x2 SU(2) matrix. An SU(2) matrix maps 2-dimensional complex numbers or probability amplitudes to 2-dimensional complex numbers or probability amplitudes.
SU(3) | Multiplication by 3x3 SU(3) matrix. An SU(3) matrix maps 3-dimensional complex number or probability amplitudes to 3-dimensional complex number or probability amplitudes
In real space, a particle can have a position and a time, or a momentum and an energy. The conserved quantities for real space are frequency or energy (time translations), wave number or momentum (space translations), and angular momentum (space rotations). Gauge transformations have a different kind of conserved quantity: Charge. Complex fields have charge. That's their nature. Our field theory is chalk full of complex fields. If we want to represent unitary operators, we'll need some of those. Unitary matrices are complex matrices.
“Charges of various kinds are always carried by fields which are complex.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 7, 1h 9m 55s | Stanford, YouTube
The conserved quantities, more specifically, are things like electric charge, weak charge/spin/isospin, and color charge. We are clumping weak charge, spin, and isospin together. There’s nuance there that will only take us away from the overall storyline. This is a complicated mathematical structure. These are the simplified punchlines. Roughly, spin and isospin are isomorphic, and weak charge, electric charge, and the z-component of isospin are related by an equation. If you know electric charge and the z-component of isospin, you know weak charge, and if you know electric charge and weak charge, you know the z-component of isospin. Let’s not worry too much about distinguishing: If you see one of these three, spin, isospin, or weak charge, you know we are talking about an object’s undergoing SU(2) rotation or its transforming as a representation of SU(2). We basically have three kinds of charge, electric charge, weak charge, and color charge. One is related to U(1), one is related to SU(2), and one is related to SU(3).
“No, no, no... not the dimensions of space. The dimensions of spin space. Not the dimensions of space. What I said to do was to forget about space. Forget for the moment the position of the electron. Forget its momentum. We’re thinking of an electron that somebody has nailed down to the wall so that it can’t move around, and all you can do is measure its spin.”
Leonard Susskind | New Revolutions in Particle Physics: The Basics Lecture 8, ~28m | Stanford, YouTube
In math speak, real space + these attached shapes or spaces = fiber bundle.
“It’s a fact of nature that there are gauge symmetries. It’s also an interesting fact of mathematics. That’s which symmetries are possible. To make it possible, you replace the ordinary derivative by what’s called the covariant derivative. This is a mathematical concept that comes from the theory of fiber bundles. Well, I don’t know whether the theory of fiber bundles was invented before or after the notion of gauge invariance in physics. I have a feeling it was invented afterwards, but somewhat independently.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 8, ~51m | Stanford, YouTube
Again, all of the group operations happening around you link and mingle. U(1), SU(2), and SU(3) again, are just generalized rotations. The unitary group is just the orthogonal group, over the complex numbers. The orthogonal group (and affine representation) pops out space translations, time translations, and space rotations. U(1) rotations in 1-dimensional complex space map quite nicely to SO(2) rotations in 2-dimensional real space, and SU(2) rotations in 2-dimensional complex space map quite nicely to SO(3) rotations in 3-dimensional real space, but in a 2-to-1 fashion.
“Let’s come back to spin a little bit. Spin is more fun than we’ve dealt with so far. The reason I’m spending time with spin, is because the mathematics of spin, is what you really have to know such things as the mathematics of isospin, and mathematics of color, and the mathematics of all of the symmetries of particle physics. The mathematics is the same. The analogies between these different kinds of conserved quantities are formal mathematical analogies, but nevertheless, let's study the mathematics of spin a little more, and a little more about the Dirac equation, just a bit. And then I want to introduce the topic of isotopic spin, or isospin, which is a concept that dates back to the 30s.”
Leonard Susskind | New Revolutions in Particle Physics: The Basics Lecture 8, 21m 54s | Stanford, YouTube
We are not talking about the particle’s infinite dimensional wave function over real space. If those probability amplitudes get shifted around, the particle’s position gets shifted around. For U(1), it’s a 1-dimensional complex space. A particle’s taking on a state defined by one complex number or probability amplitude, is very different from a particle’s taking on a state defined by an infinite number of complex numbers or probability amplitudes. A 1-dimensional complex space is very different from an infinite dimensional one. With the infinite dimensional position vector, we had an infinite number of definite state options with probability amplitudes attached to them. Now we have one state option with a probability (amplitude) attached to it. Oddly, it's a trivial rotation. We have one state option. We always get the same thing. The electron has electric charge of -1, and rotates into an electron with electric charge of -1 under U(1).
U(1) or quantum electrodynamics or QED, is the simplest gauge theory. Gauge theories come equipped, roughly, with fermions, fermion antiparticles, and gauge bosons. U(1) being the simplest, comes with one of each: The electron (fermion), the positron (electron's antiparticle sibling), and the photon (gauge boson). The fermion transforms as the fundamental representation of the group, the antiparticle transforms as the complex conjugate representation of the group, and the gauge boson transforms as the adjoint representation of the group.
QED is important. That's how we get electrons to circle nuclei, i.e. atoms. It has something to do with an electron and a proton tossing a photon back and forth. Why might something like that pop out as a consequence of gauge symmetry? The physical act of an electron emitting a boson corresponds to the physical act of an electron undergoing a U(1) transformation. Two charged fermions can get into a nice harmony of emitting and absorbing bosons with one another, or tossing bosons back and forth between one another. This act can bind them and make them rotate around each other. As the electron circles the nucleus, it rotates in this internal U(1) space and trivially so. It just comes back to itself. This corresponds to multiplying the electron field by a phase, something like e^iθ. The antiparticle is up to the same tricks, it just goes the other way around the circle. It gets multiplied by e^-iθ.
“The electron wave function gets multiplied by e^iθ and the positron wave function gets multiplied by e^-iθ. They transform with complex conjugate group elements. So you would say that the electron and the positron are complex conjugate representations of the U(1) group. That’s the language.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 4, 35m 43s | Stanford, YouTube
SU(2) and SU(3) are generalizations. SU(2) boasts 2x2 complex matrices which act on 2-dimensional complex numbers. Now we call our particle, a spinor. It is now a two-component wave function which takes on a probability amplitude value for two definite state options, up and down.
With U(1), we trivially rotate to the same thing over and over again. With SU(2), there are two definite state options, up and down. Now we could, for example, go from having all of our probability tucked away in the up state, to having all of our probability tucked away in the down state. That would be an electron undergoing an SU(2) rotation from an up spin state to a down spin state. The act of shifting probabilities around between these two definite states, is the act of undergoing an SU(2) rotation. Physicists sometimes like to call it mixing. You might say that an SU(2) matrix, mixes up and down. It changes how we allocate our 100%. We will continue to butcher the distinction between probability amplitudes and probabilities. We mean probability amplitudes, but probabilities are more intuitive to discuss.
Electrons also transform as representations of SU(2). Particles can transform as the representation of more than one of these groups. The groups are very interrelated. All charged particles transform under the trivial U(1) rotation, but can often do more. As it relates to SU(2), we have two definite state options with probability amplitudes attached to them, or a 2-dimensional complex number (two complex numbers). We don’t have an infinite number of definite state options like the position space wave function and we don’t have one like with U(1): A two-component wave function or spinor contains two entries, one for up spin and one for down spin. Rotate from up to down, you undergo an SU(2) transformation: Probability is sent from being allocated to the up state, to being allocated to the down state. That is how SU(2) transformations work, they shift probability amplitudes around in this 2-dimensional complex space. The SU(2) generators are called the Pauli matrices and there are three of them.
Remember, this is quantum mechanics. When it came to position, we didn’t need our particle to take on a definite position, , or transform from one definite position to another. And now that we are dealing with the variable spin, we don’t have to go from say, a state of being definitely up, to say, a state of being definitely down. We might go from a state of being definitely up, to a state of being 50% up and 50% down. Our particle would be in a linear superposition of up and down. It is a mix of the two eigenstates, and SU(2) transformations can do the mixing.
Let us take a step back and talk through the geometry of an electron. First of all, it has a position (or momentum) wave function. That’s it’s life in real space. We call that, ψ(x). It is a function of position. Now it has this other life, in internal space, where we have an amplitude for being up and for being down (ψup,ψdown). That’s called a spinor. Now let's combine the two lives: (ψup(x),ψdown(x)). That gives us a spinor as a function of position. It’s like our old position space wave function, but with two components. The Dirac equation makes the necessary adjustments to the Hamiltonian to accommodate this nuance. In fact, we could toss in the antiparticle (the positron) and just discuss a four-component wave function which has four definite state options: Up electron, down electron, up positron, and down positron. All fermions in the Standard Model, electrons, quarks, and neutrinos, satisfy the Dirac equation and have antiparticles. They are all multi-component objects like this.
“How do we represent a wave function that has position and spin? Okay, another question. Supposing now we return to the real electron, which can move around in space. It can move around in space, so it has a position, or a momentum, and a spin. Spin and position, spin and momentum can be simultaneously measured. What can’t? Position and momentum, and different components of the spin. How do we represent the wave function, or the state vector, of an electron, given the fact that it also has a position? It has a position and a spin. Answer is, we turn these two-component objects... they’re called spinors, this is what a spinor is... which just represents up spin and down spin... we just turn the spinor components, to functions of position. So let’s uhh... the electron is free to move around, the full state of the electron is not described just by a simple function of x, but by two functions of x: (ψup(x),ψdown(x)). What’s the meaning of this? The meaning of this is the following, that you don’t just ask, what’s the probability that the electron is up or is down, you ask a more refined question: If the electron is found at position x, what is the probability that it is up? That may depend on x. The probability that the electron is up, may depend on where you look. Another way of saying it, classically... it corresponds to the classical statement, that the spin can vary from place to place. Quantum mechanically what varies is the probability to find it up and the probability to find it down, so at every point in space, you have a spinor, and the spinor tells you, at that point in space, what is the probability that the electron is up.”
Leonard Susskind | New Revolutions in Particle Physics: The Basics Lecture 8, ~51m | Stanford, YouTube
“Student question: There were two parts on your expose on groups... seemed to be some abstract mathematical object that has some kind of multiplication table. Then part on how those groups act on various objects. Can we disassociate the two?
Answer: Well they are completely related. There’s the abstract notion, let’s take the case of the rotation of a spin. There’s the abstract notion of rotation in space. I don’t tell you what it acts on. I just say, that is an operation you can do. Physics has to respect the symmetry under rotations of space, so one must then say, how does the rotation of space act on the quantities which are physically relevant? If talking classically, we might just say a rotation rotates the direction of a particle. In quantum mechanics, the states of the system are always represented by a linear vector space. There’s the abstract notion of the state of a system, then there’s the concrete representation of it as column vectors. How big are those column vectors? What’s number of entries? Depends on system, but it’s the number of mutually orthogonal possibilities for that system, If system in question only consists of an electron spin, then there are only 2 mutually orthogonal states. If two electrons, and we are only interested in spin, then there are four states. If we are talking a more complex object, for example the entire motion of an electron, its position as well as its momentum, then there are an infinite number of states. Then the number of states that it takes to describe the orbital location... the number of mutually orthogonal states of the electron... talking about its position... it could be anywhere in space. Or you could choose to describe it in terms of momentum. So column vectors describing an electron’s position, are infinite dimensional. An amplitude for every possible location or for every possible momentum. So there must be representations of the rotation group, of this group of rotations, which are infinite dimensional matrices. And there are. Just what are the matrices which act on the column vectors, which describe the system you are interested in? The size of those matrices, will depend on the number of mutually orthogonal states. So let’s come back to the spin of an electron (and) 2x2 matrices. The group elements are the abstract rotations represented by 2x2 matrices.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 4, ~2m | Stanford, YouTube
The definite states are what we measure classically. Remember, when it came to position, we couldn’t see a particle’s full fledged quantum state. All of those probabilities sit over space. Even though the particle was whatever, 25% over here, 50% over there, and 25% somewhere else, we could only see it in specific places at specific moments of time. We keep looking, accumulating statistics, and compute the classical average. Here, we have two definite state options. If you tried to measure a spin that was 50% up and 50% down, you would only ever see it up or down. Half the time you would see it down, and half the time you would see it up. On average, you would actually see an object with zero spin. The average values or the expectation values, behave classically. In real space, we can only measure eigenvalues and eigenvectors. Things measure each other over and over again, exposing eigenvalues, and average values are calculated. Statistics are gathered from tons of measurements. Quantum mechanics is a statistical theory. We are real stuff, trying to measure complex stuff, and we can’t see it all. We approximate it.
“When we (measure the spin), we always get +1 or -1 (+1 = spin up, -1 = spin down). Randomly, but in such a way, that if you do it a great number of times, the average is zero. As many plus ones as minus ones. Every one is +1 or -1, but on the average, it adds up to zero.”
Dr. Leonard Susskind | The Theoretical Minimum Lecture 4, 37m 45s | Stanford, YouTube
Spin is interesting in that we use it to differentiate fermions and bosons, and it is isomorphic to SO(3), i.e. rotations in 3-dimensional space. We have a direct connection between internal space and real space. Half spin particles are fermions. Integer spin particles are bosons. Half spin particles obey the Pauli exclusion principle and don’t like to be in the same state. Integer spin particles like to be in the same state, sometimes leading to classical force fields.
“The representations of SU(2) are the spin states, spin 0, 1/2, 1, 3/2, 2, 5/2 and so forth. Those are the representations of rotations.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 4, ~32m | Stanford, YouTube
SU(2) maps to SO(3), but in a 2-to-1 fashion. SU(2) is the double cover of SO(3). They share the same topology thus defining a projection map. When an electron orbits the nucleus of an atom transforming as a representation of SU(2), it goes from up to down after one full rotation, and back down to up after another full rotation. It takes two real space rotations to achieve invariance. This takes us back to the fiber bundle stuff. Check out more about them in this great read: “Fiber Bundles and Quantum Theory”.
“Fiber bundle consists of a base space, a total space, and a map that projects each point in the total space onto a point in the base space. The set of all the points in the total space that are mapped onto the same point in the base is called a fiber. The total space can resemble a sheal of bundle of fibers. Every fiber in a fiber bundle must have the same topological structure. And so all the fibers can be represented by a single ideal fiber.”
Herbert J. Bernstein and Anthony V. Phillips | “Fiber Bundles and Quantum Theory”
You might summarize much of this like so: Stuff can’t just wildly or arbitrarily transform. Physical quantities show up in what are called, multiplets. There are sets which transform among themselves, using symmetry operators. Only specific kinds of transformations are taking place, which belong to a handful of interrelated groups. Quarks transform under SU(2) from say, up to some mix of up and down. Quarks also transform under SU(3) from say, red to some mix of red, green, and blue. Quarks do not, however, transform from red to down. No such thing. Only certain things transform to certain things in certain ways. SU(3) concerns a 3-dimensional complex vector space, which is not the 2-dimensional complex vector space of SU(2). These groups do independent work.
“Up quarks and down quarks are symmetric with respect to each other in the same mathematical way, that a spin up and spin down are symmetric. In case of spin, it actually had to do with the symmetry of space, rotating the axes, and in the case of up quarks and down quarks, it's just a mathematical manufactured space that you can imagine, where you take an up quark to a down quark by flipping some imaginary direction in your head. But all you are doing is interchanging up quarks and down quarks. (In) thinking of up quarks and down quarks as mathematically the same, or isomorphic to up spins and down spins, you come to the topic of isotopic spin. Isotopic spin is the analog of spin, but not up and down in sense of z-axis and flipping spin, but up and down in the sense of up quarks and down quarks. Really nothing up and down about it. Just interchange of two labels. So one invents the concept of isotopic spin, and for all mathematical purposes, the replacement of up quarks and down quarks if very analogous to the replacement of an up spin by a down spin.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 2, ~27m | Stanford, YouTube
Quarks, thought of as particles or fields, transform as representations of SU(3). The generators of SU(3) are the Gell-Mann matrices. There are eight of them.
“A quark was an object which we called 'qj'. 'qj' could stand for the quantum field of a quark. And it has an index. It’s not the 'up/down' index or the 'charmed/strange' index… we’ll come to that sooner or later. It is the color index: Red, green or blue. 'i’ takes on three values. And, the symmetry operation is not multiplying by e^iθ, but multiplying by a special unitary matrix, ‘Uij’. That was the symmetry operation (he writes ‘Uij’ acts on ‘qj’ to give ‘qi’), which is a kind of rotation in a kind of 3-dimensional complex space. Don’t confuse it with ordinary 3-dimensional space. And, one would say that the ‘qs’... think of them as particles if you like… that they form a representation of SU(3)... which is called the fundamental representation. It has three entries: Red, green, and blue. Sometimes it’s called the defining representation. It’s the smallest non-trivial representation. It can just be thought of as three component vector and the unitary matrices can be thought of as 3x3 matrices.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 5, ~16m | Stanford, YouTube
Here’s a table of all of the fermions in the Standard Model (excluding antiparticles), and how they get mixed up under gauge transformations. You could even say that there is one fermion field, and that the separate particles are its different components. Watch Susskind walk through this table in the passage below.
“Let’s go back to the quarks for a moment and let me just make a table of the different kinds of quarks. Or make a sequence here which will be red, blue, and green. These are the colors of quarks. Let me list the up quark, the down quark, the charmed quark, the strange quark, the top quark and the bottom quark. Each one of these boxes is filled. There are red, blue, and green up quarks. Let's put an X there, there's an X here, there's an X here and so forth... these X's represent all the possible quarks that exist. Now quantum chromodynamics has to do with symmetries which connect red to green to blue... has to do with these unitary transformations which mix up the colors of quarks. They do not... those symmetries do not mix horizontally. An SU(3) symmetry... a color symmetry... may rotate an up red quark into an up blue quark, or an up green quark... and the nature of the symmetry, is to act vertically. To mix up things this way (vertically). At the same time, we’ll mix the different down quarks... the different colors of down quarks. The same symmetry will mix the different colors of charmed quarks and strange quarks, and so forth. So the color symmetry acts vertically... it mixes up the different rows here. The unitary matrices mix up red, green and blue. The weak interactions are associated with symmetries which act horizontally in this picture. They mix up with down. They mix... at the same time that they mix up with down, they mix charmed with strange, and they mix top with bottom. Let me just remind you that an up quark has the same properties, apart from its mass... same properties as a charmed quark or a top quark. The down quarks are similar to strange quarks and bottom quarks. Ups have charge ⅔, downs have charged -⅔... same here and here. So these are symmetries which mix, if you like, upness with downness. At the same time, charmness with strangeness, and topness with bottomness. Would you care to speculate on what group might be involved? SU(2). It acts on things... on doublets. We simply have three doublets here. They don't take up to charmed, or up to top... they simply act on up and down, horizontally, among pairs of things. A group SU(2). And it is a gauge symmetry... in other words, it also comes together with forces, with gauge bosons…. with interactions... and with gauge fields. We're going to get to those short enough, but we've left out something from this table here. It's kind of as if there was a fourth color. Now the reason the fourth row here is not usually identified as a color, and the reason is because the particles do not interact with the gluons, but nevertheless it is another row here... another row in which there are particles which are filled in here... and they are also doublets and also get mixed up under this SU(2) symmetry, which moves things horizontally. What are they? what are those particles? They're the leptons. So in a sense, the fourth color could be thought of as lepton number. But what are the leptons? Analogous to the down, there is the electron. The electron is a lepton. But its partner... its partner is the neutrino. But there are different neutrinos. There's not only one neutrino... this is called the electron neutrino. What comes in the next column? The muon. The muon, in every respect except for mass, is the same as an electron. Just as in every respect except for mass, the charmed quark is like an up quark, or the strange quark is like a down quark. And together with it, there's its own neutrino... the muon neutrino. And finally the last one is called the tau. All of the charged leptons... incidentally neutrinos of course, as you might guess from their name... are electrically neutral. The electrons, muons and taus have charged -1. They are put under the down column here, not because their charge is the same as the down charge, but because the difference between the charges of the two is the same as the difference up here. What's the difference between the charge of an up and a down? ⅔ - (-⅓)... so the difference between this column and this column, is +1 unit of charge. The difference between this column and this column is also 1 unit of charge… 0 - (-1). So as you go from here to here, you decrease charge by one unit... as you go horizontally.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 5, ~56m | Stanford, YouTube
We just discussed one gauge theory or three interrelated gauge theories. The lie group SU(3)xSU(2)xU(1) is a product of 3 lie groups, SU(3), SU(2), and U(1).
Gauge theories allow for representations of unitary groups. Gauge theories have, roughly, three kinds of particles or fields: Fermion fields, antiparticle fermion fields, and gauge boson fields. U(1) is the simplest and has one of each particle type. The fermion field (electron) transforms as the fundamental representation of U(1), the antiparticle fermion field (positron) transforms as the complex conjugate representation of the group, and the gauge boson field transforms as the adjoint representation. All gauge theories have a similar build. SU(2) has quark fields (e.g. up quark), anti-quark fields (e.g. anti-up quark), and three gauge fields, the W+, W-, and Z bosons. SU(3) has quark fields (e.g. red quark), anti-quark fields (e.g. anti-red quark), and eight gauge fields, the eight gluons. SU(2) has three generators, the three Pauli matrices, and SU(3) has eight generators, the eight Gell-Mann matrices.
“Incidentally...there are many other gauge theories of interest. They are all very similar to each other. The group might not be SU(3)...it might be SU(2), SU(4), SU(10), could be anything, and the objects of the group might not be called quarks...the fundamental objects might be called something else, might have different name, but would be objects which have a single index. The anti-objects would also have a single index, but of the complex conjugate kind, and the analog of the gluons, which are called gauge bosons, the Maxwell like fields, always have 1 index of the particle type and another index of the antiparticle type. So that means, talking SU(n). That means how many generators? n^2. Then -1 for the trace. n^2-1. So n^2-1 gluons.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 5, ~23m | Stanford, YouTube
When it comes down to the actual mechanics of gauge theories and how the deed is done, it comes down to particle processes. A fermion undergoes a gauge transformation, by emitting a gauge boson. So do anti-fermions. The emission of the gauge boson is in 1-to-1 correspondence with the actual action of the gauge transformation. Gauge theories are rooted in interaction processes or scattering processes. Particles coming together, annihilating each other, and creating other kinds of particles. The intersection points, where particles go in, and particles come out, are called Feynman vertices. That’s how particles transform into new kinds of particles, through these scattering processes.
“That’s the mathematical structure of SU(3) cross SU(2) or SU(3)xSU(2). If you add the photon. The photon emission, is associated with the symmetry which changes the phase of every charge carrying field. Every charge carrying field transforms with a phase under U(1). And if you add those 3 together, you have the basic ingredients of SU(3)xSU(2)xU(1). It’s a product structure. And that’s where the terminology… I guess we can call it terminology… that the Standard Model is a gauge theory based on the gauge group SU(3)xSU(2)xU(1). It’s more complicated than this, but it’s roughly… SU(3) for color (gluon emission), SU(2) for W emission, and U(1) for electrodynamics. It’ll get slightly more complicated.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 6, ~7m 51s | Stanford, YouTube
One thing we’ve ignored is the fact that these quantum fields, are operators. We won’t harp on that too much, as it will only generate confusion. Nothing changes regarding these beautiful gauge symmetries stuff, we just have our symmetry operators act on quantum field operators. Sometimes we call this move to field operators, second quantization. Particles can be created and annihilated. The number of particles is actually a variable. The operator twist allows for all the perks of what we’ve been doing, plus the ability to toggle particle counts. That introduces something called Fock space. It also makes the number of particles, an observable. For simplicity’s sake, let’s just say that these special unitary matrices, say SU(3), can act on plain ol’ quarks, or objects which can create plain ol’ quarks.
“Should we go over again basic simple construction of quantum field? Let’s do again but slightly different. As I said, trick of introducing fields, from particles. Can go two ways. (1) Start with particles and reconstruct why described by fields, or (2) start with fields and describe why described by particles or quanta. We’ve started with particles. Fine to talk about 1, 2, or 3, etc. number of particles, some fixed number of particles. Why not talk about all possible numbers of particles? I’m talking about a single species of particle. One kind of particle for the moment, and only one kind. Let’s make a theory where we can simultaneously talk about many particles and all things multiple particles can do, even when the number of particles change. Why? In real life, the number of particles changes. The number of photons in this room is constantly changing, being emitted by the electric light bulbs. Other times, particles decay, turning into other kinds of particles. The neutron can decay into a proton, an electron, and neutrino. In that process, the number of neutrons, protons, electrons, neutrinos, and the total number of particles changes. We have to think about that number as changeable. (We need a) flexible tool that embraces any number of particles and even a changing number of particles. That’s the point of inventing fields. The math isn’t hard but is abstract.”
Leonard Susskind | Advanced Quantum Mechanics Lecture 7, ~0m | Stanford, YouTube
“The entire configuration of this oscillating string, this quantum mechanically oscillating string, would be characterized by a collection of integers, which tells us how much quantized energy there is in each mode of oscillation. So basically what we have to do mathematically, is repeat the mathematics of the harmonic oscillator, infinitely many times, one for each mode of oscillation. And the result is a quantum field. That’s what a quantum field is, it is a collection of harmonic oscillators, a collection of creation and annihilation operators, which add and subtract energy in each mode of oscillation. It is the mathematics of that collection of oscillators which is called quantum field theory.”
Leonard Susskind | New Revolutions in Particle Physics: The Basics Lecture 2, 1h 47m 28s | Stanford, YouTube
“There are rather three important representations for our purposes, one is a 3-d representation. The three components that go into the vector, can be thought of as the three field operators… let’s call them… quark red, quark blue, (and quark green). We can either think of these matrices as mixing up states of a single quark, or we can think of it as mixing up the field operators that create single quarks, red, blue, and green.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 4, ~32m | Stanford, YouTube
A particle can be moving along, and spontaneously break into particles by way of these vertices. This is the nature of gauge theories. Again, there’s a 1-to-1 correspondence between a fermion emitting a gauge boson, and its undergoing a gauge transformation. For instance, a down quark can emit a W- boson to SU(2) rotate into an up quark. A gauge boson is kind of like two fermions fused into one, one fermion and one antiparticle. It’s kind of like it comes equipped with whatever you are, and anti-whatever you are about to become. The boson is a tensor. It has two vector indices. The fermion is a vector, it has one index. FYI, antiparticles can strangely be thought of as particles which move backwards in time. That's how the anti-up quark below comes in from the future into our interaction setup (as part of the W-boson, if we think about the gauge boson in this way).
U(1) is the same thing, but trivial. An electron can emit a photon and trivially rotate into an electron. There are other legs as well: For instance, an electron and positron can come together, annihilate, and spit out a photon. It conserves charge because the electron and positron have opposite charge and cancel each other out.
“The other example is something that is real experimental physics, real observational physics. You start with electron and a positron… we haven't talked about anti particles yet. Let's just ignore the fact that we haven't talked about them for a moment… two kinds of particles, electrons which have negative charge and positrons which have positive charge… they have exactly the same mass, precisely the same mass. They are in every respect similar, except for the fact that they have opposite electric charge… because they have opposite electric charge, they can combine and disappear they have plus charge and a minus charge the net charge is zero. Nothing prevents them from disappearing… they have energy, each one has an energy… let's suppose these two particles are at rest. We bring them together, each one has an energy equal to mc^2, so the total energy is twice mc^2. Well mc^2 for each particle. We bring them together… that energy can't disappear. Energy is conserved. What happens to that energy? That energy becomes photons. Photons go out. You bring an electron next to a positron, you let them annihilate…. poof out go photons. The photons, in some sense are energy. They can be absorbed by a material and heat the material. They can be converted to all forms of energy. And how much energy do you get by annihilating an electron and a positron? The answer is, twice the mass of an electron times the speed of light squared. It’s a tiny amount of energy… I mean, if you just had one electron and positron you couldn't do much with it. You need, you know, huge numbers of them to heat a cup of coffee but but still in principle, that's what happens and of course this is an observational fact. That's the meaning of E=mc^2. And another way to really say it, which i think is the right way to say it, is to say, energy and mass, at least for an object at rest, are the same thing. Why do we have to have an equation? Really it's just a conversion of units from what we call mass units to what we call energy units and this tells you the conversion. Roughly, like the conversion from meters to centimeters or whatever. This is the conversion between energy units and mass units.”
Leonard Susskind | New Revolutions of Particle Physics: The Basics Lecture 1, ~1h 1m | Stanford, YouTube
SU(3) works similarly.
“The color symmetry, the SU(3) symmetry, color is the symmetry group which acts on this triplet here and mixes them up into each other. That symmetry group, has generators. Those generators correspond to the infinitely small transformations, but it also corresponds to the gluons. The symmetry group which takes red to green to blue, is associated with the existence of gluons, and the gluons, when they’re emitted, from one quark to another, for example from red to green, you create a red, green bar gluon. This transformation from red to green, is simply one of the generators of SU(3). So the generators of SU(3), the transformations, are in 1-to-1 correspondence with the emission of a gluon. You can think of the emission of a gluon as a physical act, physical phenomenon, which does the same thing as the symmetry group which mixes the particles into each other. That’s the character of gauge bosons. Gauge bosons are a sort of manifestation of the symmetry group in which the emission of a gauge boson, really does change a particle from one thing to another in exactly the same pattern, as the symmetry group would change one thing to another. So that’s red, green, and blue. And when you perform one of these transformations, you do it on all of the quarks simultaneously. You don’t just do it on the up down family. If you want the symmetry to be correct, you want it to be a real symmetry of the theory, then you must do it simultaneously on all of the quarks, and that corresponds to the statement, that exactly this same gluon is (always) emitted.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 6, ~2m | Stanford, YouTube
U(1) concerns the photon.
SU(2) concerns the W/Z bosons.
SU(3) concerns the gluons.
Quanta of charged fields exchange photons if they have electric charge.
Quanta of charged fields exchange W/Z bosons if they have weak charge.
Quanta of charged fields exchange gluons if they have color charge.
When a charged particle undergoes a U(1) rotation, it emits a photon.
When a charged particle undergoes an SU(2) rotation, it emits a W or Z boson.
When a charged particle undergoes an SU(3) rotation, it emits a gluon.
We diced up the Susskind passage below into some of the snippets used thus far. Watch the passage uninterrupted with its full nuance, to drive home your understanding of these symmetries.
“Let’s get back to SU(3) and SU(2). SU(3) acted vertically on our table, it took us from 1 color, red green and blue, and mixed them up. We made a table of all the fermions in the Standard Model. And, not so much that they’re fermions, but just happens that they’re kind of the constituent particles of the atom in the nucleus. Let’s write them down again. The 6 different families of quarks. The families refer to, up and down. That’s called a family. Then charm strange. Strange family. Top and bottom. That’s the horizontal structure. Then we have red, green, and blue. And we have a quark in each one of these slots.
The color symmetry, the SU(3) symmetry, color is the symmetry group which acts on this triplet here and mixes them up into each other. That symmetry group, has generators. Those generators correspond to the infinitely small transformations, but it also corresponds to the gluons. The symmetry group which takes red to green to blue, is associated with the existence of gluons, and the gluons, when they’re emitted, from one quark to another, for example from red to green, you create a red, green bar gluon.
This transformation from red to green, is simply one of the generators of SU(3). So the generators of SU(3), the transformations, are in 1-to-1 correspondence with the emission of a gluon. You can think of the emission of a gluon as a physical act, physical phenomenon, which does the same thing as the symmetry group which mixes the particles into each other. That’s the character of gauge bosons. Gauge bosons are a sort of manifestation of the symmetry group in which the emission of a gauge boson, really does change a particle from one thing to another in exactly the same pattern, as the symmetry group would change one thing to another. So that’s red green and blue. And when you perform one of these transformations, you do it on all of the quarks simultaneously. You don’t just do it on the up down family. If you want the symmetry to be correct, you want it to be a real symmetry of the theory, then you must do it simultaneously on all of the quarks, and that corresponds to the statement, that exactly this same gluon, let’s say this is from up quark to up quark.
Incidentally the gluon emission does not change horizontally, only vertically. For example same gluon which would be emitted when a red up quark becomes a green up quark, that’s exactly the same gluon, as would be emitted, when let’s say when a top red quark, would be a top green quark.
So the [color] symmetry takes whole rows and mixes them into each other.
Then there is a 2nd symmetry… we’ve talked about electrodynamics and what it does to the...what the emission of a photon, what the symmetry associated with electrodynamics is.
But now we are encountering another symmetry which is usually called flavor symmetry, or family symmetry sometimes. Takes ups to downs, at the same time that it takes charm to strange and top to bottom, keeping the color unchanged. So for example, there will be processes in which, an up quark, let’s say of the blue variety, emits a W+ boson, and becomes a bound quark, also of the blue variety. So what has this W boson done? It is a physical manifestation of the transformation which takes you horizontally, back and forth. Exactly the same W+ would mediate, or would be emitted, same kind of W+, would be emitted when a green charm quark, becomes a green strange quark. Same W+... And so forth.
The gluons never move you horizontally. The W bosons never move you vertically. And this is also the character of the symmetry groups. The symmetry groups move you horizontally, that’s SU(2), it doesn’t move you across families obviously…
That’s the mathematical structure of SU(3) cross SU(2). If you add the photon. The photon emission, is associated with the symmetry which changes the phase of every charge carrying field, explained that...every charge carrying field transforms with a phase, under U(1), and if you add those 3 together, you have the basic ingredients of SU(3) cross SU(2) cross U(1). Cross simply means, a product structure, where SU(2)s don’t mix up the colors, and the SU(3)s don’t mix up the upness and downness.
More complicated than this. But roughly:
SU(3) for color.
SU(2) for W emission.
U(1) for electrodynamics.
Will get slightly more complicated. Emission of W boson...incidentally of course W bosons were only discovered late in game, postulated some time in 60s. Only achieved traction in 70s. But theory of weak interactions, goes back much earlier than that, suppose to 30s, the fermi theory of weak interactions.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 6, ~0m | Stanford, YouTube
Without gauge transformations, chemistry would not exist and things would be very boring. This is how atoms come together and form molecules and such. Two charged particles can get into a nice harmony of emitting and absorbing particles between one another. Quarks bind together into protons and neutrons by tossing gluons back and forth. Protons and neutrons make up nuclei. Nuclei bind to electrons by tossing photons back and forth. Atoms stick together by tossing electrons back and forth. In quantum mechanics speak, we say that an electron is in a quantum linear superposition of being around one atom's nucleus, and the other atom's nucleus. In chemistry speak, we say that the two atoms are sharing an electron. Or we call them covalent bonds.
Quantum field theory is all about these vertices. These vertices pop out of the equations of motion. Remember, it all comes back down to solving differential equations. Equations of motion are differential equations. They come in one of two forms, the Hamiltonian or the Lagrangian. Strictly speaking, an equation of motion is a Hamiltonian, which is generated from a Lagrangian, but both involve the field and derivatives of the field.
Complex fields, fields which take on values in little complex spaces attached to space and can undergo local gauge transformations like this, have equations of motion where one swaps the plain ol’ derivative, with the gauge covariant derivative.
We are just complicating the idea of a derivative a tad, to accommodate complex fields. Now the Lagrangian not only has terms related to plain ol’ derivatives, which we might call kinetic terms, but now it also has terms related to the gauge covariant derivative and gauge symmetries. We call the other terms, gauge interaction terms, or potential terms.
“It’s a fact of nature that there are gauge symmetries. It’s also an interesting fact of mathematics. That’s which symmetries are possible. To make it possible, you replace the ordinary derivative by what’s called the covariant derivative… this is a mathematical concept that comes from the theory of fiber bundles. Well, I don’t know whether the theory of fiber bundles was invented before or after the notion of gauge invariance in physics. I have a feeling it was invented afterwards but somewhat independently.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 8, ~51m | Stanford, YouTube
This adjustment incorporates our field’s ability to transform locally and adds on our gauge field(s).
“Ultimately it’s an experimental fact that all of the interactions of nature are associated with gauge symmetries like this. Not all of them, but all the gauge symmetries, all the gauge interactions. Photons, z bosons, w bosons, all are associated with symmetries not just of the kind where you can rigidly everywheres do the transformation, but symmetries where you gradually...well it doesn’t even have to be gradual...an arbitrary variation of the symmetry parameters from place to place. Okay so let’s see what’s the ingredient you have to add...you have to add a new collection of fields.”
Leonard Susskind | New Revolutions in Particle Physics: The Standard Model Lecture 7, ~1h 20m | Stanford, YouTube
The Lagrangian is built out of gauge covariant derivatives, and is symmetrical under local gauge transformations. The gauge covariant derivative is ∂ + iA. Derivative plus an interaction term with the gauge field. We already knew about ∂. Those are kinetic terms.
The other one is the interaction term. This introduces a gauge field A, or in mathematics what is called a connection. U(1)’s gauge field, is the photon field. The Lagrangian also includes the dynamics of the gauge field itself, i.e. its kinetic terms. The interaction term links them with the fermionic fields.
There are roughly two kinds of terms in Lagrangian. Kinetic terms, and potential terms. The potential terms, are tied to these nonlinear interaction terms related to gauge transformations.
If this term wasn’t there, a gluon and a quark would just wiggle right through each other undeformed. They would have their own kinetic terms and that’s it. Particles which don’t interact with each other, go right through each other. If the two fields are linked by a nonlinear interaction term, then they have a nonzero probability of scattering each other or deforming each other. The scattering process is coded in the interaction term. The interaction term below for instance, allows an electron and positron to annihilate each other, or an electron to emit a photon and U(1) rotate into… well an electron again, and so on.
The quantum field operators create and annihilate particles, and electric charge is a conserved, dimensionless probability amplitude. The charge speaks to how likely an electron is to emit a photon. The stronger the charge, the more likely and frequent, the emission. The weaker, the less likely. This along with a few other things like the mass of propagators, makes the length scales different. The strong force is the strongest and keeps quarks rotating around extremely close. It operates on very short, tight distance scales, and fast time scales. The others are weaker. The strong force is the strongest, shortest range force. The weak force is weaker than the strong force, but stronger the electromagnetic force. And the electromagnetic force is the weakest, excluding gravity. Include gravity, and gravity is the weakest and longest range of them all.
There’s another charged field that we didn’t mention which also transforms under SU(2). It couples to fermion fields, and makes them oscillate between right and left handed spins, giving them mass. That field is called the Higgs field. Mass is rest energy. It concerns oscillations which do not have to do with motion in real space. This is called a fermion’s chiral oscillation. The role of the Higgs field in giving fermions mass, is called the Higgs mechanism.
That pretty much sums up the Standard Model and everything one would care to know about the physics of everyday life. In order to extend the Standard Model so as to accommodate its limitations, we sprinkle in a little more geometric seasoning and add in some more fields. This helps with stuff like dark matter, which plays virtually no role in the nuance of everyday physics. That is not to say that dark matter, doesn’t matter. If we were talking the accelerated expansion of the universe or something like that, it would be matter a ton.
“Okay, I think what you are asking me is, how complicated the Lagrangian of the world is. At least in physicists minds, there’s one Lagrangian that governs everything. Now that might not really be true, but let’s say it is. And it contains a large number of fields, it contains all of the fields that describes all of the particles and all of the interactions and it may have the sum of many, many terms like this, describing lots and lots of different interactions, but it is one Lagrangian… maybe composed out of sums of pieces that individually look like recognizable pieces, but the whole thing is one big Lagrangian that governs all of nature.”
Leonard Susskind | New Revolutions in Particle Physics: The Basics Lecture 9, ~1h 53m | Stanford, YouTube
Supersymmetry comes along with two more gauge bosons (X and Y gauge bosons), which allows fermions to transform to bosons and vice versa.
If we want to get fancy with SU(3)xSU(2)xU(1), we can say that it drops out of, or fits nicely inside of, the group SU(5).
Most recently, physicists have been trying to sync the quantum mechanical mathematics of quantum field theory, with the classical mathematics of gravity or space and time itself. Remember we separated that out? We’ve been ignoring gravity given its negligible impact on particle physics. Einstein’s relativity tells us about classical gravity. Quantum mechanics tells us about everything else being made of little quantum pieces or wave functions, transforming under the action of U(n) or SU(n) operators.
String theory gives us a quantum mechanical theory of gravity. It says that gravity breaks down into little pieces called gravitons, which are spin 2 bosons.
Most recently, Juan Maldacena’s discovery of something called the gauge-gravity duality seems to explain why gravity behaves classically, while everything else behaves quantum mechanically. Dualities come about when the same mathematics has two different, but equivalent descriptions. His groundbreaking 1997 paper on this duality has been the most cited paper in all of physics for the past couple of decades. Here’s Susskind on it:
“Let me begin with a concept that is sort of overtaken physics, and has for a long time. It’s called duality. It’s a preliminary thing for my talk today to know that there are such things as dualities and what they are. Duality is a very simple concept. There are simply two equivalent descriptions of the same thing. And typically one of those descriptions involves wildly fluctuating variables. It could be quantum fluctuations and it could be thermal fluctuations. And the other description involves very unfluctuating simple, almost classical descriptions. … ~3m … A number of years ago...not sure how long, we began to recognize that there was a special kind of duality. This kind of duality replaced...or it was a duality between very strongly coupled highly quantum mechanical systems of many many degrees of freedom on the one hand...on the other hand, a system described by gravity. That sounds crazy, and it did seem crazy...but it appears to be true, and when I say gravity, I mean quantum gravity. This connection has really been at the heart of an explosion in theoretical physics and it’s an explosion which is not just about quantum gravity, it seems to be infecting just about every single area of theoretical physics.”
Leonard Susskind | Entanglement and Complexity: Gravity and Quantum Mechanics | Stanford, YouTube
Juan’s paper is roughly about a geometry, which on the inside behaves like classical gravity, and on the outside boundary behaves like a U(n) or SU(n) quantum mechanical gauge theory. When the inside is classical, the outside is quantum mechanical. A duality connects the two pieces of the total geometry (the inside and the boundary). Here are some of the main points from Juan himself. Watch the entire talk for the full nuance.
“The gauge gravity duality is a relationship between (1) gravity in the interior of some space (left)…that is equal to a (1) field theory that lives only on the boundary. The duality is valid as a quantum mechanical statement. We have here on right a (2) quantum mechanical theory on the boundary and to the left we also have a (1) quantum mechanical theory of gravity.”
“The observables of the field theory will be correlation functions of local operators. The field theory to the right could be like a statistical mechanics problem.”
“Typically this theory to the right will be a U(n) gauge theory, or SU(n) gauge theory.”
“On the left side we will compute observables which correspond to deforming the metric or the behavior of the fields far away. The tiniest deformation of the geometry creates a gravity wave that moves through the interior."
Juan Maldacena | Emergent Geometry: The Duality Between Gravity and Quantum Field Theory | Institute for Advanced Study, YouTube