Field theory 
I've been working my way through Leonard Susskind's wonderful lectures on theoretical physics. I just want to publicly thank Prof. Susskind. It takes a tremendous amount of work to put those lectures together. He'll never know the joy that all that work is bringing to people like me.

I want to try and share some of the insights that Susskind's lectures are giving me. If I do this right then hopefully you'll be as amazed as I am right now about the nature of the universe we live in. This post is a summary of over a dozen lectures, which are themselves a summary of over 50 years of theoretical physics. The information content is necessarily quite dense but if you make it through to the end I hope the reward will be as breathtaking for you as it has been for me. I'll include some technical asides in square brackets [like this]. You can skip these either on a first pass, or if you're not too interested in the details. I'm not going to bother with links to web pages, you're as capable of googling any technical terms that come up as I am.

Let's start with Newton's second law: force equals mass times acceleration.

F = ma

If you know the form of F then you can calculate the motion of a particle from one instant to the next and figure out its path through space. Once you've calculated the path though, something else becomes clear.

As well as things like acceleration and velocity, there's another mathematical quantity we can attach to a particle. It's called the Lagrangian. As the particle moves through a field of force, this number varies. When you add all the values of the Lagrangian up, you find that the total value for the actual path (the Action) is always less than it would be for any other possible path. This is called the Principle of Least Action.

[For a particle moving through space under the action of a field of force, the Lagrangian is just the kinetic energy minus the potential energy due to that force: L = T - U. The Action is the integral of this quantity over the path of the particle. Mathematically, minimising this integral leads to something called the Euler-Lagrange equations. When you apply the Euler-Lagrange equations to T - U, you get Newton's second law back.]

The Lagrangian is like the grand daddy of everything in physics. For a particle, or for its generalisation to a collection of particles, it is equivalent to Newton's laws - it's just a different way of writing the same thing. However, the Lagrangian tends to be much easier to deal with mathematically. Problems that would be quite tricky in the Newtonian view, often turn out to be easy using the Lagrangian formulation. Crucially however, the Lagrangian can be generalised to the fields of force themselves and it's this that lies at the core of all modern physics.

It's from the Lagrangian that we can derive Noether's theorem: for every symmetry of the Lagrangian there is a corresponding conserved quantity. What does this mean? Let's take our example of a particle moving through space. We can formulate the Lagrangian as a function of the particle's position. For every co-ordinate (x,y,z) that the particle passes through, the Lagrangian has a unique value. But our choice of (x,y,z) is arbitrary - we can place our co-cordinate axis anywhere we like. We could move it a bit in the x direction or a bit in the y direction. We wouldn't expect the particle to be affected by this choice. In other words, the Lagrangian is invariant under a translation in space. We have a symmetry under space translation. What Noether's theorem does is give us a formula. When you've found one of these symmetries in the Lagrangian, it tells you what the corresponding conserved quantity is. In the case of symmetry under translation in space, the conserved quantity is linear momentum.

If you've followed all this then your jaw should be dropping right about now. Think about it. From considerations of symmetry we can derive a fundamental principle of nature. It gets better. When you take account of symmetry with respect to rotation, conservation of angular momentum pops out. Symmetry in time (your choice of time co-ordinate) gives conservation of energy.

Now we get a bit more imaginative. Suppose, instead of the Lagrangian of our particle just having one number associated with it, it has two. It's not too hard to imagine such a thing. At every point in our atmosphere we can define two numbers: temperature and pressure. They're related, but can still take on a wide range of independent values. The temperature and pressure, to a good approximation vary continuously from one point in the atmosphere to the next. This "double Lagrangian" is a bit like that. It's a pair of numbers attached to a particle that are related and vary continuously as the particle moves (but they're nothing to do with temperature and pressure).

[The pair of numbers are actually a complex scalar field and apply throughout spacetime whether there is a particle present or not.]

This new type of Lagrangian has all the symmetries that the old one had: symmetry in space translation, time translation and space rotation, but now we can play with an extra symmetry. Just as we can imagine a set of (x,y,z) axis to locate the particle in space, so we can imagine a pair of axis to define the particle's two Lagrangian components: call them q1 and q2. As the particle moves through (x,y,z) it's Lagrangian varies over the (q1,q2) plane. But just as our choice of the (x,y,z) axis is arbitrary, so is our choice of (q1,q2). We could rotate these axis any way we like. We wouldn't expect the particle to pay any attention to our choice. This "double Lagrangian" has given us a new symmetry. Just like before, we can apply Noether's theorem and out pops a new conserved quantity: electric charge.

[Actually, at this stage it just forms a good candidate for electric charge: it obeys the continuity equation. The "rotation" is just multiplying the complex field by e^(i*theta), where theta is an angle in the complex plane. Because all the factors in the Lagrangian involve a complex number multiplied by its conjugate, the e^(i*theta)s cancel with e^(-i*theta)s and the Lagrangian remains unchanged. Mathematically, this candidate for electric charge looks similar to an angular momentum, except it's in (q1,q2) space rather than real space. If you have a three number Lagrangian, instead of two, you apparently get conservation of Isotopic Spin (Susskind only mentioned this in passing - I haven't seen the maths and may have misheard him). I think these extra dimensions are the ones that String theorists go on about.]

[Update - I've since read a bit more on Quantum Field Theory. The three number Lagrangian is indeed the one used for Isotopic Spin, but Isotopic spin is an inexact symmetry. The exact symmetry gives the weak nuclear force.]

There's a thought experiment we could do with the particle and it's (x,y,z) trajectory. Instead of rotating the (x,y,z) axis, we could keep the axis fixed and rotate the universe instead. The outcome ought to be entirely equivalent. Well, we can imagine something similar with the (q1,q2) axis. Instead of rotating the axis, we could rotate the universe. Unfortunately, the universe doesn't allow us to rotate the entire thing simultaneously. The theory of relativity tells us that no signal can propagate faster than the speed of light. To be consistent with this principle we can only rotate things locally and then watch the rotation propagate throughout the rest of spacetime. So if we want to perform our thought experiment on (q1,q2) for our particle we have to rotate the universe a little bit locally and watch that rotation spread out through spacetime. This means that, instead of our (q1, q2) rotation being the same everywhere at once, it now becomes dependent on spacetime - the (q1,q2) rotation is different at different places.

[This is called a Gauge Transformation. e^(i*theta) now becomes e^(i*theta(x,y,z,t)), i.e. the rotation is a function of spacetime. The rotations no longer cancel because all sorts of extra terms appear in the derivatives.]

This messes up the maths. When you make the (q1,q2) rotation spacetime dependent, our neat result, that electric charge is conserved, disappears. You have to muck about with the Lagrangian and add something else to it. You have to add electric (E) and magnetic (B) fields giving a Lagrangian that depends on (q1,q2,E,B). With the electromagnetic field added, the Lagrangian restores charge conservation.

[What we actually have to add is the four-vector potential, A, consisting of the electric potential (a scalar) and the magnetic potential (a three vector).]

To me, this is absolutely stunning. We've conjured up electric charge just by imagining an extra couple of dimensions attached to each point of spacetime and considering the extra symmetry that it introduces. Then, in order to let the resultant field vary in a way that is consistent with relativity, we find that the electromagnetic field must exist. Essentially, electric charge and electromagnetism boil down to geometry. The other forces of nature, gravity excepted, apparently arise in analogous fashion: an internal symmetry gives rise to a conserved quantity. Flows of the conserved quantity must propagate at a finite speed which in turn results in a characteristic force law.

[None of this is quantum mechanical - it's an entirely classical theory of electromagnetism. AFAIK, Leonard Susskind hasn't done a similar series of lectures on Quantum Field Theory yet. I hope he does. I'm told that when you quantise the (q1,q2) field, it's quantum is the electron. When you quantise the electromagnetic field it's quantum is the photon.]