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.

**Where do forces come from?**

There’s a particular kind of symmetry that particle physicists are particularly interested in: **gauge symmetries**. These symmetries are special because they give us **gauge bosons**, otherwise known as force particles. Thus we can also say that symmetry is the reason why we have forces between particles.

There are two complementary ways in which one can view gauge symmetry. The first way is to think of it as an additional feature of a regular global symmetry. For example, we can say that there is an internal symmetry that causes us to have three ‘colors’ of each quark. We can then “promote” this symmetry to a gauge symmetry and this forces us to introduce a new force particle (the gluon) which connects particles which have color charge. (This color charge, of course, comes from Noether’s theorem above.) This works by saying that we are promoting a global symmetry into a local symmetry, meaning that we allow ourselves to do different symmetry transformations at different points in space(time).

For example, electric charge comes from a simple symmetry called **U(1)**. This is the symmetry of a circle. So we can imagine a little circle living at every point in spacetime, and at every point there is some point on the circle which tells a property of the electron field. What is this property? It’s a complex phase, basically a complex number of length 1 that multiplies the electron variable in our theories. This isn’t a “real” circle since it’s an internal symmetry, we’re just imagining this to help us understand what’s going on. A **global** symmetry would tell us that if we move the point on the circle by the *same *amount *everywhere* in space, then the laws of physics should be the same:

A **gauge** symmetry, however, is one where we can do an *independent* rotation at *each *point in space (i.e. a **local** transformation):

If a symmetry is a gauge symmetry, then this transformation should still leave the laws of physics the same. Note that the **local** symmetry is much more restrictive than the **global** symmetry. Why does this give us force particles? If all of the arrows are mutually misaligned, it turns out that we need to define a way to go from the orientation of an arrow at point *x *to the orientation of the arrow at point *y*. In differential geometry this is called a **connection**. In physics it’s called a **gauge field**. This “gauge field” object which tells us how the arrows point is precisely what we call force particles! So the photon is just the “gauge field” which connects the U(1) orientations of electrically charged particles. Particles that aren’t charged don’t have a U(1) orientation, and so the photon doesn’t “see” them.

As a side note, the second way to think of this is to say that if the arrows can point in any direction then this piece of information is actually a redundancy in our theory. In other words, gauge symmetries can be thought of as not arising from some natural principle, but as a human-made invention that introduces redundancy to makes nature more mathematically tractable. If you think about this for a while this becomes something of an ontological question, which is *not *what we want to get into; but the point is that quantum effects tend to break lots of symmetries. If we create a gauge symmetry that is redundant, we ought to make sure that quantum effects don’t mess it up or else our theory in completely inconsistent. The quantum effects are called **anomalies**, and this leads to a powerful tool in theoretical physics called **anomaly cancellation**. (It was a cancellation of anomalies in 1984 by Green and Schwarz that led to the first superstring revolution.)

**Where do massless particles come from?
**

These are a weird questions Why should I ask where massless particles come from? Why is this different from asking where *any* particle comes from?

Massless particles are somewhat special in quantum field theory. It turns out that they can lead to apparent theoretical inconsistencies (so-called “infrared divergences”) and quantum mechanics usually conspires to give particles mass unless (ta-da) such masses are protected against by some symmetry. (An example of this is chiral symmetry, i.e. the left-handed and right-handed structure of the Standard Model fermions, that prevents fermions from getting mass other than by the Higgs mechanism. If that didn’t make sense, then don’t worry about it.)

This is related to **Goldstone’s theorem**, which tells us that:

If a continuous global symmetry is broken, then there are massless particles in the theory.

By “continuous global symmetry” we mean something like the U(1) symmetry of the circle described above or its higher-dimensional generalizations. I’ll have to be necessarily sketchy here, but the picture that everyone always draws is the so-called “Mexican hat potential”:

The idea is that we can imagine a quantum field as a marble rolling around in that Mexican-hat-shaped bowl. The position of the marble represents the “vacuum expectation value” of the field. Here ‘**vacuum expectation value**‘** **(vev) is code for whether or not the particle exists everywhere in space without any additional quantum effects. Most potentials don’t have that dimple in the middle, so the ‘marble’ wants to sit right in the center where the field takes zero vacuum expectation value. This is what most particles do. For the **Higgs**, however the vacuum expectation value is nonzero — the Mexican-hat shaped bowl pushes the Higgs to live on the ring where the hat is flat. This means that the Higgs “exists everywhere” in space even without having to be created through quantum interactions. Indeed, it is the interaction of particles with this “omnipresent” Higgs field that gives those particles mass.

Now, Goldstone’s theorem is the observation that the shape of the bowl actually corresponds to the mass of the particle rolling around in it. This is part of the structure of all quantum field theories, and you can take it as a stated fact. The key observation is that the Higgs particle still has a direction where it is free to roll around, namely the angular direction. Goldstone tells us that these directions (in “field space,” i.e. in the space of possible Higgs vacuum values, not physical space) correspond to massless excitations of the Higgs field.

So you can think about it this way: the fact that the Higgs field doesn’t live at the center of the potential (it’s pushed out by the dimple) tells us that the Higgs lives everywhere in space. Particles interactions with this “everywhere in space” field gives them mass. In addition, however, there is a certain way that we can jiggle this “everywhere in space” field that corresponds to massless particles. This is just like the ripples one gets from throwing a rock into a pond.

**Where do massive force particles come from?**

This massless Higgs field coming form Goldstone’s theorem relates to a different problem in field theory related to gauge bosons, i.e. force particles. It turns out that if we write a theory with *massive* force particles (as we observe with the *W* and *Z* bosons), this usually leads to apparent theoretical inconsistencies (a “breakdown of unitarity”), e.g. probabilities that are greater than 100%. However, it turns out that if the force particles get their masses in a very specific way, i.e. via a process called **spontaneous symmetry breaking**, then everything is okay. Further, it turns out that everything is okay because—you guess it—there is symmetry in the theory.

The point is this: the *W* and *Z* bosons get their masses from “eating” the massless Higgs particles. What this means is that the physical *W* and *Z* bosons actually contain part of the Higgs field! More technically, the *W *and *Z* fields mix quantum mechanically with the massless Higgs particles. This becomes a rather technical point, so I’ll leave it at that. Don’t worry if you’re unhappy with the word “eat” in this context. It is commonly used even in quantum field theory textbooks and bugged me for a long time until I felt like I properly understood what was going on.

**Summary and Outlook
**

I’ve been going on for quite some time now about symmetries and how they’re important in particle physics. Let me close with the following summary:

- Symmetries play a key role in how we define our theories and, in turn, restrict their structure.
- Antiparticles are related by charge-parity conjugation, a discrete symmetry of spacetime.
- The Standard Model is composed of several symmetries: spacetime, family, color, electroweak.
- Symmetries can be very interesting when they break. The Higgs boson is intimately connected to
**electroweak symmetry breaking**, which ends up giving the*W*and*Z*bosons mass through its Goldstone particles and gives mass to the matter particles through interactions between left and right handed fermions. - Local, or “gauge,” symmetries are responsible for the existence of force particles.
- Symmetries are responsible for conservation laws, such as the conservation of momentum or the conservation of electric charge.

In future posts I’ll spend more time talking about my research, in which the theoretical physics community has been using symmetry in novel ways to go beyond the Standard Model. To whet your appetite, this includes extending spacetime symmetry (extra dimensions), imposing a matter particle/force particle symmetry (supersymmetry), and incorporating all of the Standard Model gauge symmetries into a single symmetry (grand unification).