Alright, it’s time to start wrapping things up a bit. I’ve been going on for some time now about how symmetries play a central role in our understanding of physics. Here’s a lightning review:

- In part 1, we thought about how the symmetries of space(time) restrict the form of our theories.
- In part 2, we saw how antimatter comes from a
**discrete**symmetry of spacetime (Charge-Parity) - In part 3, we introduced
**internal**symmetries that have nothing to do with spacetime, but that lead to a replication in the number of particles. This “explains,” for example, why there are three copies of the electron.

Here’s a summary in graphical form:

If you wanted a nice summary in the format of a nice TED talk, (I know Mike A. is a fan), then I recommend Marcus du Sautoy’s talk earlier this year:

[youtube 415VX3QX4cU]

Now I’d like to go over some more formal results with far-reaching effects in physics, i.e. some “advanced topics.” These are usually things which are derived rigorously in successively more advanced physics courses, but here we’ll just give heuristic explanations that highlight the physical relevance. Though the topics are somewhat high brow in their nature, they address very simple questions that I think should be very accessible.

**Where do conservation laws come from?
**

Emmy Noether was a prominent physicist and mathematician in the early 1900s when those fields were dramatically dominated by men. Today every undergraduate physics student learns **Noether’s Theorem** as part of analytic mechanics. The theorem can be summarized as this:

For every continuous symmetry, there is a conserved charge.

What does this mean? The first part refers to a continuous symmetry. These are like the spacetime symmetries that we discussed in part 1: rotations, translations and their relativistic generalizations (Lorentz transformations). The word **continuous** means that you can perform the symmetry by any arbitrary amount, as opposed to discrete symmetries (such as those in part 2).

The second part says that if you have a continuous symmetry, then you have a conserved quantity which we call **charge**. This is something you’re already familiar with: we know that electrons carry electric charge and that this charge is conserved: it is neither created nor destroyed, and every interaction between particles must have the same charge going out as it did going in. For example, if ten physicists entered a bar and only nine left by closing time, then the number of physicists is not conserved. (Maybe one of them had a change of heart and became a mathematician.)

This is really neat, because now we can **explain** the existence of conserved charges in terms of the existence of a symmetry in nature. Here are a few well known examples from non-relativistic classical physics:

- The laws of physics are the same over time (time translation symmetry). This implies the existence of a conserved quantity that doesn’t change with time. We call this
**energy**. i.e. the energy of a system of constant in time. - The laws of physics are the same at every point in space (space translation symmetry). This implies the existence of a conserved quantity that doesn’t change with space. We call this
**momentum**. - The laws of physic are the same no matter how we change the direction of or coordinates, this leads to the conservation of
**angular momentum**.

(I once convinced myself that if you think about this for a while, it makes sense ‘intuitively’ without any mathematics. However, this depends on what you mean by ‘intuitive.’) This is now really useful because physicists building theories can generate conserved charges just by imposing that the theory obeys some symmetry. (more…)