Transcription
In the 300 and some years since the scientific revolution, the two deepest theories of nature that human beings have written down are the Standard Model of particle physics and Albert Einstein's theory of gravity, General Relativity. Both are examples of field theories, quantum in the first case and classical in the second, which describe the laws of nature in terms of fluctuating fields that permeate space and time.
In this video, I want to give you a quick introduction to the fundamentals of classical field theory, so that in the future, I'll be able to tell you more about those more advanced topics like General Relativity and Quantum Field Theory. First of all, what is field theory and why do we need it?
We all start out in physics by looking at a bunch of particles, writing down the total force on each one, and then setting it equal to ma. Think about the Sun and Earth, for example. According to Newton's law of gravity, the Sun exerts a force on the Earth that pulls it toward it. The force is proportional to the mass of each and goes as one over the distance squared between them. Then we can write down f equals ma for the Earth and solve for its orbit around the Sun. This was one of the first and most important accomplishments of modern physics.
But suppose that by some tragic calamity, the Sun suddenly exploded or just mysteriously popped out of existence. What would happen? According to Newton's theory, the gravitational force of the Sun on the Earth would suddenly vanish. It's as if the Earth had been swinging around on a rope, like a tetherball, when the cord suddenly snaps and the planet veers off on a straight line. Apparently, the disappearance of the Sun would instantaneously cause a dramatic change in the motion of the Earth, 92 million miles away. But that's total nonsense. For example, the last rays of light emitted from the Sun before its disappearance would take about eight whole minutes to travel that long distance to Earth and meet our eyes. As Einstein taught us about 200 years after Newton, when he wrote down his special theory of relativity, nothing can travel faster than the speed of light, including the information that the gravitational pull of the Sun has gone slack. We would have no idea of our precarious condition until those eight minutes had passed and we were plunged into darkness.
This is where field theory comes in. Field theories let us build locality in from the beginning, so that the only points in space where you would even know that the Sun had just vanished are the ones in its immediate vicinity. The data is stored in the gravitational field, which is a function of where you're located in space (x) and the time (t). As time goes on, the information about the Sun's unfortunate demise would propagate outward like a shock wave through the gravitational field to farther and farther corners of the solar system and the universe. This was Einstein's greatest achievement: constructing a relativistic theory of gravity that reproduces Newton's law in the circumstances where it's accurate and completes it in a consistent way where it fails.
And the applications of field theory go way beyond gravity. It's the framework we use to describe a huge range of phenomena in physics, including, of course, the electromagnetic fields E and B that you've probably already encountered. Those rays of light traveling toward us from the Sun are ripples in the electromagnetic field and likewise carry information along in a way consistent with relativity. And the Standard Model of particle physics, which is the most spectacularly predictive theory that physicists have ever come up with, is founded on field theory. The elementary particles like electrons, quarks, neutrinos, and so on, are all associated to fields, and the particles are interpreted as excitations of the fields. But we're getting ahead of ourselves. The Standard Model, General Relativity, and even classical electromagnetism are quite advanced examples of field theories. If you want to learn the subject, it's better to start with a simpler example that illustrates the key ideas.
The simplest example of a field theory is called the Klein-Gordon theory, and it's about the physics of a single real field, usually denoted phi. In the rest of this video, I want to explain the fundamentals of field theory using this simple example. It's still going to be challenging, but this is the language that we use to understand the deepest laws of nature that we've so far understood. So it's definitely worth your while. And as usual, you can get the notes that I've written going into further detail at the link in the description, so that you can work through things more slowly after you've watched.
In previous videos, I've told you about how instead of using f equals ma, we can formulate particle mechanics using a Lagrangian and applying the principle of least action. I'll link a video about that up in the corner. When it comes to solving classical mechanics problems with pendulums and springs and so on, whether you use f equals ma or a Lagrangian is more or less a matter of convenience. But in field theory, the Lagrangian approach is fundamental. We define a field theory by writing down its Lagrangian. So it'll be useful to quickly review how Lagrangians work for particles, and then we'll see how to generalize that for fields.
Say we have a particle of mass m with coordinate x. We define the Lagrangian by taking the difference between the particle's kinetic energy (one-half m times dx by dt squared) and its potential energy U of x. As the particle moves around with time, its kinetic energy and potential energy are both changing, and so the Lagrangian is a function of time. We define the action by integrating the Lagrangian over time. Suppose the particle is traveling from starting point xi at time ti to another point xf at time tf. Then the question is, what trajectory x of t will it follow to get there? I'm plotting the time on the horizontal axis here and the position on the vertical axis. For any path connecting the two points, the action is a number that you get by integrating the Lagrangian over it. The principle of least action says that the particle is going to follow the path for which S is smallest, or at least stationary.
And that's actually not too hard to work out in practice. When you want to find the minimum or maximum of an ordinary function f(x), you just need to identify the points where its derivative vanishes. In other words, where the slope goes to zero. Said differently, when you take a little step away from a minimum by shifting x to x plus epsilon for some tiny epsilon, then the change in f is the derivative times the change in x, and so the change in the function is zero. When you take a tiny step away from a stationary point, that's just because the tangent to f at a minimum is a horizontal line, and when you take a step along a horizontal line, the height of the function doesn't change.
It's the same with the action, except that now we want to find the curve x of t for which S is minimized. So maybe we have a guess that the minimum curve looks like this. Then what we want to do is see how the action changes when we make a small variation of the curve by adding some little wiggles to it. Mathematically, we're taking our curve and shifting it by x of t goes to x of t plus epsilon of t, where epsilon is an infinitesimal function that adds little wiggles. If we indeed started at a minimum, then the action shouldn't change under this shift.
First, let's figure out how the Lagrangian changes. Again, it's just like taking the differential of a function, except that now the Lagrangian is a function of both x and dx by dt. From the kinetic energy term, we take the derivative and get m times dx by dt, and then we multiply that by the change in dx by dt, which is d epsilon by dt. And for the potential term, we take its derivative and get minus du by dx times the change in x, which is epsilon. So that's how the Lagrangian changes when we make a tiny variation of the path.
Next, we need to integrate the first term by parts. All that means is that we're going to use the product rule for derivatives. The derivative of this whole thing, m times dx by dt times epsilon, is m times the second derivative of x times epsilon plus m times the derivative of x times the derivative of epsilon. And we can flip that around to write m dx by dt d epsilon by dt like this. Now we put that back in for the first term in dL. What integrating by parts bought us is that we can now pull out the common factor of epsilon from the first two terms here. So that's how much the Lagrangian changes when we make a little change in the trajectory.
The change in the action is the integral of this over time. This is what we want to vanish in order to find a minimum of the action. The second term is easy, so let's deal with that first. It's the integral of a derivative, so all we're going to get is the difference of the thing in parentheses between the two endpoints, tf and ti. But this is zero. The reason is that we're not allowing just any variation epsilon of t here. We're looking at variations of the path connecting this initial point to this final point, and so we don't want the deformation to change the endpoints of the curve. That means that we require epsilon to vanish at ti and tf, and so this contribution to the change in the action is just zero.
So now our formula for dS is looking a lot simpler, and the extremal trajectory is the one that makes this vanish for any little variation epsilon of t that satisfies the boundary conditions. But the only way this integral can vanish for any epsilon of t is if the thing that's multiplying it in the integrand is zero. Therefore, the principle of least action tells us that the particle's actual trajectory is the path that satisfies m times the second derivative of x with respect to t equals minus du by dx. But that's just f equals ma. Well, remember the force is related to the potential energy by f equals minus du by dx. So that's how the principle of least action for a particle reproduces f equals ma, what we call the equation of motion for the particle.
Now I want to show you how to extend these ideas to field theory. This is challenging stuff, so remember that you can get the notes at the link in the description to really take your time processing all these ideas. Also, if you're finding my videos helpful and the kind of thing you'd like to see more of, it'd be fantastic if you could support me on Patreon. Even just a small contribution a month can go a long way. Thank you so much to all of you who have already contributed through Patreon and on my website. I appreciate it so much.
In field theory, the central object is no longer the coordinate x of t of a particle, but a field phi that assigns a number to each point in space at each time. A simple example could be a temperature field that tells you the temperature at position x at any time t. It's a function that takes a point in space and time and gives you back a number. Instead of the single degree of freedom we had corresponding to the position x of the particle, we now have an infinite number of degrees of freedom for the field, one for each point in space. That means field theories are complicated, but they're commensurately powerful.
The Lagrangian is the starting point for a classical field theory. We write down an L and take it as the definition of the theory. The action is again defined as the integral of the Lagrangian over time. Remember that we want the theory to be local; that was our main motivation for studying field theory in the first place, and we can ensure that by requiring that L is itself the integral of a Lagrangian density over space. Just like the Lagrangian for a particle was a function of x and its first derivative, curly L is a function of the field and its first derivatives at each point in space and time. Those are partial derivatives because phi is a function of two variables, x and t, but don't let them scare you off; they behave just like regular old derivatives.
For a particle, we define the Lagrangian as the kinetic energy minus the potential energy. We'll define curly L in essentially the same way, where curly K and curly U are the kinetic and potential energy densities of the field. In other words, energies per unit volume of space. Since we integrate over space to get the full Lagrangian, analogous to K equals one-half m dx by dt squared, curly K should go like the square of the derivative of phi with respect to time. c is the speed of light, and it's there to get the units right, as well as to ensure our theory comes out consistent with special relativity. Also, notice that there's no m here, unlike one-half mv squared. The field doesn't have a mass like a particle, per se, though in quantum field theory, we'll see how the field does give rise to particles with a certain mass.
Next, what should we choose for the potential energy density? Again, similar to a particle, where we can pick any potential U of x and it determines the kind of system that we're studying, our choice for curly U will depend on the theory that we want to construct. But there is one contribution that we always want to appear: in relativity, time and space are supposed to enter on equal footing, and so since we have a term that goes like d phi by dt squared, there should also be a term that goes like d phi by dx squared. This is sometimes called the gradient energy. It's a very reasonable potential to include; it means that a field configuration with lots of wiggles in space will have a larger energy. In general, there'll also be terms for the y and z directions, but I'll mainly just write x here to keep things from getting too complicated.
The rest of the potential energy depends on the theory that we want to study. For a particle, the most fundamental potential energy function is probably the harmonic oscillator potential, U equals one-half k x squared. My last video was about that. Then, likewise, a simple and important example of a field theory is obtained by choosing the rest of the potential to be quadratic. I'll write it as one-half kappa squared times phi squared, where kappa is a parameter that controls the strength of the potential, with dimensions of one over length. These kinetic and potential energy densities define the Klein-Gordon field theory, so that all together, the Lagrangian density is given by their difference, plus the additional terms with the y and z derivatives that I haven't written. The minus signs are there for the same reason that L equals K minus U in regular old particle mechanics. If we flip them around to plus signs, we'll get the total energy density again.
This Lagrangian defines the Klein-Gordon field theory, and when we quantize it to make a quantum field theory, it describes the relativistic quantum physics of free scalar particles of mass m equals h-bar kappa over c. We'll see where that mass comes from in a minute, but let's continue to work out the classical theory. All we've really done so far is define it. Next, we want to figure out its equation of motion by applying the principle of least action. Very similarly to what we did for a particle, we start with some field configuration phi of x and t, and then we make a little variation of it by adding on an infinitesimal field epsilon of x and t that deforms phi by a little bit. If phi is going to be a stationary configuration of the action, then S had better not change to first order in epsilon when we make any such variation.
Let's first work out the change in curly L. In the first term, we take the derivative and get one over c squared times d phi by dt, and then we multiply that by the change in d phi by dt, which is d epsilon by dt. Likewise, in the second term, we take the derivative and get d phi by dx, and then we multiply that by d epsilon by dx. In the last term, the derivative is kappa squared phi, which we multiply by the change in phi, which is epsilon. Just like before, we'll integrate by parts, only now we have to do it twice for the first two terms. We get minus one over c squared times the second derivative of phi with respect to time times epsilon, and then plus the second derivative with respect to x times epsilon. There'll also be total derivative terms that vanish again after we integrate over space-time to get the change in the action, so I won't bother writing those here.
Therefore, the leading change in the action when we make a small variation of the field configuration looks like this. As usual, integrating by parts has enabled us to pull out the common factor of epsilon. And now, if we want this to vanish for any suitable variation epsilon, we conclude that in order for a field configuration phi to extremize the action, it has to satisfy minus one over c squared times the second derivative of phi with respect to time plus the second derivative of phi with respect to space equals kappa squared times phi.
This is the Klein-Gordon equation. It's quite famous. It's a generalization of the wave equation, since if you set kappa equal to zero, it becomes the equation of a wave traveling at the speed of light. Since the Klein-Gordon equation is so similar to the wave equation, let's try solving it by plugging in a simple wave: phi equals A times cosine of kx minus omega t, where A, k, and omega are constants: the amplitude, wave number, and angular frequency of the wave. We want to see if we can choose them so that this guess really does solve the equation. Actually, it's a huge pain in the neck to deal with cosines and sines; it's much more convenient to work with complex exponentials, e to the i times kx minus omega t. At the end of the day, you can just pick out the real part to get a real solution, remembering the identity that e to the i theta equals cosine theta plus i times sine theta. These solutions are called plane waves because the profile is constant in the y-z plane, that's perpendicular to the direction of motion along x. But again, this is just a guess. Let's see what happens when we actually plug it into the Klein-Gordon equation.
When we take the derivative with respect to x, it brings down a factor of ik from the exponent, and when we take the derivative a second time, it brings down another power, giving us ik squared, or minus k squared, times the thing that we started with. Similarly, the second derivative with respect to time gives us back minus omega squared times phi. So when we plug this guess into the Klein-Gordon equation and cross out the common factors of phi, we get omega squared over c squared minus k squared equals kappa squared. So the plane wave will indeed solve the Klein-Gordon equation as long as omega and k are related by this formula.
These plane wave solutions are very simple and special, but the beauty of them is that the general solution to the Klein-Gordon equation can be written as a sum of plane waves over all possible values of k. In the quantum theory, each plane wave is identified with the wave function psi of a single particle with momentum p equals h-bar k and energy E equals h-bar omega. The condition relating k and omega then implies that p and E have to satisfy E squared over c squared minus p squared equals h-bar squared kappa squared, or solving for E, we get this. And that might look familiar. It's Einstein's equation for the energy of a particle of mass m and momentum p, where m equals h-bar kappa over c. In this case, when p equals zero, this is the famous E equals mc squared, the relativistic rest energy of a particle of mass m. This is why the Klein-Gordon theory describes the relativistic quantum mechanics of particles of mass m.
Speaking of relativity, the Klein-Gordon theory is totally compatible with special relativity, though it's not totally obvious at first glance. But we can make it obvious by introducing some better notation, like you might have seen before if you studied some special relativity. We combine the position coordinates x, y, and z, and the time coordinate t into a four-component space-time coordinate x mu, the position four-vector, where mu is an index that can be 0, 1, 2, or 3. x0 is defined as ct, and x1, x2, and x3 are x, y, and z. We include the factor of c in x0 equals ct so that all the components have the same dimensions of length. Then the derivatives with respect to x mu, d by dx0, d by dx1, and so on, are therefore given by one over c d by dt, d by dx, d by dy, and d by dz.
Next up, let's define a four-by-four matrix eta with components minus one and then ones all along the diagonal, with everything else equal to zero. It's very simple as far as matrices go. The time component is eta 00 equals minus one, the space components like eta 11 are all one, and all the other components are zero. Eta is called the Minkowski metric, and it's what tells us how to measure distances in space-time, like I explained in the video I shared with you about the action for a particle in special relativity. For present purposes, eta is useful because it lets us sum up the pairs of space-time derivatives in the Klein-Gordon equation like so. Remember that eta is a diagonal matrix, so most of the terms here are actually zero. All we get are the diagonal terms: eta 00 times d by dx0 d by dx0 plus eta 11 d by dx1 d by dx1 plus the y and z terms. Eta 00 is minus one, and x0 is ct, so we get minus one over c squared times d squared by dt squared in the first term, and likewise d squared by dx squared in the second term, and so on. But these are exactly the derivatives that show up on the left-hand side of the Klein-Gordon equation. So we can write it like this.
The basic reason this equation is compatible with relativity is because the mu and nu indices in the derivatives are paired up with the indices in the metric eta mu nu, and because phi itself is a scalar. You'll learn more about how all that works as you study more special relativity. This relativistic combination of second derivatives is called the d'Alembertian operator, and it's usually written simply as d squared or even just box, and so you'll often see the Klein-Gordon equation written even more compactly as box phi equals kappa squared phi. We can also use this relativistic notation to clean up the Klein-Gordon Lagrangian that we started with, again by summing up the derivatives with eta mu nu. We usually write this even more compactly as minus half d phi squared minus half kappa squared phi squared, where it's implied here that we're summing up the derivatives of phi with the Minkowski metric.
That was a very rapid tour of the simplest example of a relativistic field theory, and it's the best place to start learning field theory. Like I mentioned, in quantum field theory, the counterpart of this Klein-Gordon system describes the physics of spin-zero particles, where the spin refers to how the particles and fields transform under the symmetries of space-time. But that would take at least a whole other video to explain. Most of the elementary particles of nature are not spin-zero, however. The only spin-zero particle in the Standard Model of particle physics is the Higgs boson, which was finally discovered at the Large Hadron Collider in 2012 after its theoretical prediction almost 50 years earlier. The more familiar particles have different spins, like electrons with spin half and photons with spin 1. These are described by their own fields with different Lagrangians and equations of motion.
Spin-half fields are described by the Dirac Lagrangian, which looks like this, and whose equation of motion is the Dirac equation. I won't explain what any of these symbols mean right now; I just want to give you a quick lay of the land of field theory. But if you're interested in learning more, please leave me a comment letting me know. Spin-1 fields, like the photon, are described by the Maxwell Lagrangian, whose equations of motion are Maxwell's equations, which you'll learn about in a college class on electromagnetism, though probably using a notation that makes them look much uglier. Finally, coming back to where we started at the beginning, gravity is described by a spin-two field with the Einstein-Hilbert Lagrangian and whose equation of motion are Einstein's equations. Again, I haven't said what any of these symbols mean, but let me know in the comments if you want to learn more about this in future videos, and make sure you're subscribed to the channel.
You should definitely go and get the notes for this video so that you can work through all these challenging concepts for yourself. They'll stick much better that way. Those are at the link in the description. Thank you so much again if you're able to support me on Patreon or through my website, and most of all, thank you for watching. I'll see you next time with another physics lesson.