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### Symmetry in Physics, Part 1: Spacetime Symmetry

One of the reasons why physicists often wax poetic about the beauty of physics is that so much of the field has based on symmetry, and humans find symmetry beautiful. But it is partly because people have an intuitive sense of what symmetry is that there aren’t many attempts to really explain the central role that it has played in theoretical physics.

Over the next few posts I’d like to go over some of the ways symmetry shapes how we think about physics. I’ll be necessarily heuristic, but I hope to give an honest sense of what physicists mean by symmetry and why it’s been the primary guiding principle in theoretical particle physics for the past 50 years.

Already at an intuitive level it’s probably not hard to believe that symmetries constrain and simplify things. If I tell you to draw a blob, you might draw any unexpected shape. On the other hand, if I told you to draw a blob with rotational symmetry, you are constrained to draw circles:

Similarly, if I tell you that a theory of gravity is spherically symmetric, then you know that the gravitational potential only depends only on the radial distance from the source and is independent of angular information. If you calculate the field at some distance r, then you know that the potential is the same for all other points of the same distance r. [A more advanced discussion: because the force is proportional to the change in, or gradient of, the potential, you know that the force can only be in the radial direction.]

The mathematical language of symmetry is group theory. A discussion of group theory would take us too far astray, but I heartily recommend that young physicists make a point to learn group theory when they have a chance. (The earlier the better, especially since it doesn’t require much fancy math background.)

Actually, this is a lesson in having a broad background. Murray Gell-Mann, one of the heroes of American physics and co-developer of the quark model, actually re-invented group theory for himself when trying to describe the spectrum of hadrons discovered in the 1960s. He called it the eightfold way (he was very good at naming things) and only later realized that there already existed a mathematical language for his method.

Symmetries of Spacetime

Let’s talk about the symmetries of space and time.

• Translational symmetry: the laws of physics are the same everywhere and at all times
• Rotational symmetry: the laws of physics don’t depend on how we orient our coordinates
• Boost symmetry: the laws of physics don’t depend on what inertial frame we’re in. In other words, the laws of physics for someone standing still are the same as for those who are moving at a constant velocity.

Note that I didn’t say that physical observables obey these symmetries. I can survive on the surface of the Earth but not at the sun’s core. Gravity tells me that “up” and “down” are two very different directions. And if I stand still I’m fine but if I’m moving at a constant velocity I’ll eventually run into something like a building or a tree.  Our universe clearly doesn’t obey the symmetries listed above. That’s fine.

The point is that the laws of physics do obey these symmetries. You can think of “the laws of physics” as a set of fundamental laws governing the dynamics of nature. Even though nature itself isn’t symmetric (that’s a whole different story), its rules are. Classically, the law “F=ma” holds no matter where you are, when you measured it, and how you oriented yourself. Sure, the numbers might change depending on how you’re oriented, but the relation between the numbers remains true.

What does this all mean for particle physics? A particle is a particle is a particle! Suppose I had a particle. Let me draw it as a penguin (don’t ask me why). The symmetries of spacetime tell me that if I put the penguin somewhere else, or if I rotate it, or if I give it a nudge so that it’s moving at a constant velocity… then the penguin is still the same penguin that I started with. I don’t need a new framework to describe “penguin in motion rotated by 30 degrees” than I would to describe the original penguin.

This statement is so ridiculously trivial that it probably ends up sounding more complicated than it is. Obviously we know that these are all versions of the same penguin, I’ve just moved them around in ways allowed by the symmetries of space. But ‘technically’ they are all different penguins: one is tilted one way, another one is moving… if we were very, very naive we’d think “these are different things — why are you calling them the same?” The answer is that spacetime is symmetric.

[Let me say this in a different way: the symmetries of spacetime are so deeply ingrained in us as infants that we don’t even think about a penguin as being different from a penguin tilted on its side. Technically, though, they are different: one is tilted! So without any prior input, why should physics treat the tilted penguin the same way as it treats the original penguin?]

Okay… still with me? Good. Now go back and re-read the last three paragraphs with the word “penguin” replaced by “particle.” The take-home message is this: symmetries allow us to treat [naively] different objects as the same object.

Now we’re starting to get somewhere.

Before we go any further, let me stop and drop some fancy words for more advanced readers. For others, just think of these as words that you can use to impress your friends. The “full” meaning of these words is based on group theory, but we’ll try to give a sense of what they mean. The symmetries of spacetime are collectively called the Poincare symmetry. Particles are irreducible representations of the Poincare group, meaning they change in well-defined ways under Poincare symmetry. The subset of Poincare symmetry dealing only with rotations and change of inertial frames is called Lorentz symmetry. It is formulated to automatically obey the symmetries of Einstein’s special relativity (e.g. effects like time dilation and length contraction are already “built-in”). Quantum fields, which “give rise” to particles, are irreducible representations of the Lorentz group. [If you feel a little lost, don’t worry; this paragraph is just an aside.]

Epilogue, Part 1

Unfortunately, this was probably enough of a bite-sized (blog-sized) discussion for today. In terms of learning ‘sexy new truths about the universe,’ we haven’t gotten very far. What we have done, though, is lay down some foundational principles about the role of symmetry in physics. In summary,

• Symmetries simplify things by constraining their form (e.g. a rotationally symmetric blob must be a circle)
• Symmetries allow us to identify “naively different” objects as really being part of the same object

In our next post we’ll follow these symmetry principles and consider the discrete symmetries of spacetime. Without spoiling too much ahead of time, we’ll see that we are naturally led to the idea of antiparticles.

Flip