10-30-08, 02:21 PM #1
Fun Problem: Field Theory on S2!
Ok, I worked this problem out the other night. It's kind of cute, and straightforward for a scalar field.
Consider field theory on . Given the six dimensional Klein-Gordon equation for a massless scalar field:
derive the Kaluza-Klein mode expansion.
This problem is pretty easy, so let's complicate it! Define an operation on the sphere such that , where is the azimuthal angle. What does this do to the sphere? What happens to the points at the north and south pole of the sphere?
Next, define a parity operation on the six dimensional scalar field, such that
Now what do the mode expansions look like, given that can be either even or odd under the operation?
For extra credit, work out the normalizations of the scalar field so that it is canonically normalized.
Last edited by BenTheMan; 10-30-08 at 04:18 PM.
10-30-08, 03:22 PM #2
I assume the Kaluza-Klein mode expansion of the solution is
Let's call your operation the Ben Twist. It seems to rotate the the sphere on it's polar axis by one half revolution. Solutions (even at the north and south pole) pick up a phase factor of .
So the parity is even for even and odd otherwise.
Or I could be completely off base since I never took a graduate physics course.
Last edited by rpenner; 10-30-08 at 04:55 PM. Reason: Swapped ranges on m and l
10-30-08, 04:17 PM #3
Anyway, the "Ben Twist" is actually an orbifold---something about which AN knows quite a bit
It's not a proper manifold, because you have these singular points (conical singularities) called fixed points. These are points which are unaffected (i.e. map to themselves) by the orbifold action.
The poles are fixed points in this construction, I think, because they are poorly defined by the spherical coordinate system. You might ask whether the fixed points are still fixed if you put a different coordinate system on the sphere---I think the answer is yes, because any coordinates will have places where they don't work.
10-30-08, 05:22 PM #4
I'm lost on terminology. I thought the orbifold was the quotient space not the operation -- since BenTwist is an action faithful to the non-trivial element of Z_2 (such that ). So, I think that's written as -- Seeing as I don't know how to make the mass spectrum out of this six- dimensional Kaluza-Klein theory or even the basics of orbifolds, I'll shut up now.
Last edited by rpenner; 10-30-08 at 10:08 PM. Reason: Grammar
10-30-08, 08:13 PM #5I thought the orbifold was the quotient space not the operation
Seeing as I don't know how to make the mass spectrum out of this six- dimensional Kaluza-Klein theory or even the basics of orbifolds, I'll shut up now.
10-30-08, 10:11 PM #6
10-31-08, 01:18 AM #7
Sure it's easy.
Presumably you know how to do separation of variables?
Let's start with an easier example. Suppose you have a scalar field on a something like . The five dimensional Klein-Gordon Equation for a massless scalar looks like
Explicitly, we can write this as
Now take an ansatz for .
What kinds of functions should f(y) be?
10-31-08, 01:20 AM #8
Also, this is a bit tricky---where does the minus sign in come from?
(This one ALWAYS screws me up)
10-31-08, 02:23 AM #9
Since it follows that for fixed n, a solution to looks a lot like .
Last edited by rpenner; 10-31-08 at 02:32 AM. Reason: Oops, made an odd number of sign errors. And a missing i. Fixed.
10-31-08, 03:12 AM #10
10-31-08, 04:15 AM #11
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10-31-08, 11:37 AM #13
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