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a comparison of two subjects in physics

I don't know if they do this at any other colleges, but when I was an undergrad at Georgia Tech they used to call the Physics I that a lot of non-physics majors have to take "Particle Dynamics" (affectionately nicknamed Party-Die by most of the students). This made me laugh, even at the time, since I was informed enough to know that the subject being taught (Newtonian mechanics) dealt with the motion of large objects like baseballs and levers and pulleys which has nothing whatsoever to do with particle physics. But now that I'm finally taking my second quarter of Particle Physics, 9 years later, I find it even more humorous looking back upon.

Here is a typical problem that might have been given on a homework in Georgia Tech's "Particle Dynamics" class:

A baseball is thrown with an initial speed of 10m/s at an angle of inclination of 30 degrees. Neglecting air resistance, how long does it take to hit the ground? How far from the feet of the thrower does the ball fall?

And here's a question off of my current homework assignment, which is actual particle physics:

Consider a theory of three scalar fields with an SO(3) internal symmetry, under which they transform as a 3-vector. Write down the most general Lagrangian with at most two derivatives invariant under this symmetry. Keeping only the terms of two lowest order polynomials in the fields, find the condition on the parameters such that the classical minimum breaks the symmetry. Is there an unbroken subgroup? What is it? How many Nambu-Goldstone bosons are there? Answer the same questions for SO(n). Answer the same questions for SU(2) with scalars transforming in the complex doublet representation (isospin 1/2). Prove that in this last case the Lagrangian actually has an SO(4) symmetry (introduce real and imaginary parts of the two component complex doublet to get 4 real fields).

They might as well have named their Newtonian mechanics class "Knitting and Crochet", it would have been no less relevant to the subject matter. Although they did rename it just to "Physics I" in recent years, it still gives me a chuckle.


( 5 comments — Leave a comment )
Jan. 28th, 2005 09:23 pm (UTC)
Nice. :)

I recognize some of the words in your current homework assignment, but not well enough to do anything with them. ;)
Jan. 29th, 2005 05:15 am (UTC)
Now to make things more interesting, SU(2) is "doubly isomorphic" to SO(3) so what happened?
Jan. 29th, 2005 05:54 am (UTC)
Yeah, that's the thing I keep wondering about the last part. I've only done the problem up to finding the condition that breaks the SO(3) symmetry. The rest I was planning on working on tomorrow, but I had to re-read the end part several times before I was sure I read it write. How can SU(2) be both SO(3) and SO(4)? I don't know! But don't tell me yet, cause I haven't really worked on it yet. Once I've thought about it more maybe I'll be ready for a hint.
Jan. 30th, 2005 05:38 am (UTC)
ok, I've thought about it a bit
Somehow this SU(2) has got to really be SU(2) x SU(2), otherwise it's just not big enough to work. The only thing I can think of is... maybe the only way to make a Lagrangian which is invariant to SU(2) is to make it also invariant to a second SU(2) (perhaps this is why spinors have to have two kinds of indicies... dotted and undotted?) Am I on the right track?

Either it's something like that, or the question is screwy. Or maybe it's something about being locally isomorphic but not isomorphic? But I don't believe that, because it doesn't even have the right number of degrees of freedom.
Feb. 1st, 2005 02:47 am (UTC)
Re: ok, I've thought about it a bit
Sorry got bogged down in other things. So yeah you have to think about what the SU(2) symmetry is of. Notice that it says, "...with scalars transforming in the complex doublet representation..." In other words the internal symmetry is isospin 1/2. If something is a complex doublet then it has 4 real variables. The thing to focus on is that the geometric invariant for all spaces is the inner product. All SO(n) means is that it is rotationally invariant in that number of Real dimensions. But what is "it"? The "it" is the spatial symmetry of the fields. But an internal symmetry is a totally different space altogether. It can be whatever it wants with the exception that if it is a spin like internal symmetry its invariant inner product will hence be an inner product of a Complex space. But invariant Complex n dimensions are invariant Real 2n dimensions. So they want the derivative to be invariant under the SU(2) symmetry or at least thats what I gather from the question. This is different because derivatives act on the physical coordinate space typically. In physics we often protect our internal symmetries by modifying the spatial derivatives with connection coefficients of the requested internal symmetry space.

Hope this makes sense as well as helps.
( 5 comments — Leave a comment )


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