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Posts Tagged ‘Symmetry’

Symmetry in Physcs, Pt. 4

Friday, November 6th, 2009

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:

symmetrynotes

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…)

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Now time for another installment of “symmetry in physics.” For those of you tuning in late (or who have forgotten what we’ve been discussing), we started out in part 1 with a very general discussion of the symmetries of spacetime and how this constrains the form of our theories. Next, in part 2 we looked at discrete symmetries and how they relate the notion of antimatter to charge and parity conjugation. We’ll be using some of the jargon of part 2, so make sure you brush up and remember what “CP” means. Now we’d like to address another mystery of the Standard Model: why is there so much repetition?

Family Symmetry

Let’s review the matter content of the Standard Model:

standardmodelmatter

The top two rows are quarks, the bottom two are leptons (charged leptons and neutrinos). Each row has a different electric charge. The top row has charge +2/3, the 2nd row has charge -1/3, the third row has charge -1, and the last has charge 0. As discussed in part 2, there are also the corresponding anti-particles with opposite charges [note 1]. Just about all of the matter that you’re used to is made up of only the first column. All atoms and everything they’re made of are more-or-less completely composed of up and down quarks and electrons (the neutrinos haven’t done much since early in the universe).

The replication of the structure of the first column is known as family symmetry. For each particle in the first column, there are two other particles with nearly the same properties. In fact, they would have exactly the same properties, except that they are sensitive to the Higgs field in different ways so that the copies end up having heavier masses. Technically the Higgs discriminates between different generations and breaks this symmetry, but we are still left with the question: why are there two other families of matter?

(more…)

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Particle Physics Nobels!

Wednesday, October 8th, 2008

This week the Nobel Prize for Physics was given to three theorists in our field: Nambu, Kobayashi, and Maskawa. If you want to read more about them, the science behind their discoveries, or the coverage of the award, I recommend this post on the Knight Science Journalism Tracker. You can also check out the post on Cosmic Variance, which discusses some of the controversy in the physics community over this award and has a link to a good explanation of spontaneous symmetry breaking.

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Physics and Wikipedia

Wednesday, August 13th, 2008

As Peter once noted, a contribution to Symmetry Magazine means a free blog post. Now it’s my turn; I wrote the commentary for the new issue, and here it is:

If you’re interested in contributing to physics articles on Wikipedia, WikiProject Physics is a good place to start. If you want to see what I’m up to on Wikipedia, you can look at my user page; bear in mind that my edits there are entirely my personal responsibility.

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