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This is a very brief "part 4 and a half". I feel like the one big thing that was missing from my description of conformal symmetry was an actual visual image to go with my text description, because it's otherwise pretty hard to visualize or explain how to visualize.

Fortunately, Wikipedia has a great page on this:

http://en.wikipedia.org/wiki/Conformal_pictures

So I'm just linking there and embedding one of their best images of a conformal transformation (also called a conformal mapping) here. This is an example of a 2-dimensional conformal transformation.

Before the conformal transformation, start with any old image (you have to imagine that this pattern repeats forever and fills the entire 2-dimensional infinite plane):



After the conformal transformation 1/z:



Believe it or not, even though it looks pretty weird and warped, all of the angles are locally the same in the final image as they were in the original image. For example, if you draw a right angle somewhere on the first image, it will still be a right angle at the point it gets mapped to in the second image. In other words, the angle won't have gotten squeezed or expanded.

Keep in mind that the fact that this image looks different after the transformation means that it does *not* have conformal symmetry. If you wanted to find an image that has conformal symmetry, you'd have to find one that remains *exactly* the same before and after the same remapping of points. Obviously, this puts a huge restriction on the image and it has to be very symmetric.

Hopefully this does a better job of getting it across than my text description... I know long text descriptions about abstract things like this can get confusing.

Comments

( 17 comments — Leave a comment )
nasu_dengaku
Jul. 20th, 2010 02:23 am (UTC)
A while back I was thinking about what the world would look like if we put the origin (0,0,0) inside my body and then mapped every point (x,y,z) to (1/x, 1/y, 1/z). The rest of the universe would be confined to a limited ball, surrounded by my flesh. Appendages would reach in and manipulate it. I'd love to do some 3D animated videos of what this would look like.
spoonless
Jul. 20th, 2010 03:14 am (UTC)
Yeah. Looking at these pictures actually gives me some ideas of what to experiment with next in Mathematica. We have so many image mapping and manipulation functions, I could probably make an animation of these mappings very easily. (We also have tools that let you fairly trivially make slider buttons to control animations like that, which is even more fun!)

This kind of experimenting is generally encouraged at Wolfram, if nothing else we can use it for demos.

Incidentally, this picture does exactly what you describe... maps every point of distance r from the origin to 1/r.

Did you think about how your internal organs would be mapped to fill up the entire rest of the universe? =)
nasu_dengaku
Jul. 20th, 2010 03:20 am (UTC)
Hey, I'd love to see such a video... even if it's something simple like a guy walking down a street with a few buildings.

> Did you think about how your internal organs would be mapped to fill up the entire rest of the universe? =)

Of course. I would never consider myself the center of the universe... I just want to occupy the rest of it. :-)
spoonless
Jul. 20th, 2010 03:30 am (UTC)

Hey, I'd love to see such a video... even if it's something simple like a guy walking down a street with a few buildings.

Well that, I'm not sure how to do with Mathematica. What I was thinking was simpler... taking a single image, and making a video of it morphing from the original to the transformed version and back.

The problem with working on an entire video is you have to have something that reads a standard video format and can work on it pixel by pixel, frame by frame. The may be some way of doing this, although I'm not sure how. Maybe if the original format is just an animated .gif?
nasu_dengaku
Jul. 20th, 2010 03:40 am (UTC)
Oh... what I meant was taking a 3D rendered scene of something ordinary (eg a man walking through various rooms in a house), remapping every vertex, line, and face in the 3D space to its inverse, and then re-rendering the scene in 3D.
vaelynphi
Jul. 20th, 2010 04:48 am (UTC)
You can just use a video as a texture on a 3D surface, then manipulate that surface as needed; OpenGL and DX both support this kind of thing, but it's a little tricky to program.
onhava
Jul. 20th, 2010 02:58 am (UTC)
Surely the only conformally invariant image is a solid color everywhere?
spoonless
Jul. 20th, 2010 03:23 am (UTC)
It would seem that way, although I don't know. In my earlier post I mentioned that the only two examples I could think (for 3d Euclidean space) were a single point at the origin or the entire space.

However, now that I think of it, I must be wrong about the single point at the origin. I was thinking you couldn't map it anywhere else just by doing scaling or rotation, but at least in 2d the obvious counterexample would be e^z which maps it to the unit circle.

What I'm confused about though is, if it's true that there is nothing but "the entire space is homogenous" that is conformally symmetric... then why are conformal field theories interesting at all? Why are they not all just completely trivial?
onhava
Jul. 20th, 2010 03:47 am (UTC)
"Conformal field theory" means that the conformal group acts on the states, not that every state is conformally invariant. Observables are conformally covariant, not invariant. Think about ordinary QFTs: the theory is invariant under time translations, but the states aren't; if they were, they would all have zero energy.
spoonless
Jul. 20th, 2010 02:35 pm (UTC)
I wasn't implying all states in the theory had to be invariant. But I thought that things like the action, the S-matrix, the Green's functions, and the vacuum (assuming it isn't spontaneously broken) had to be.

I'm glad you pointed out about observables being covariant rather than invariant though. One of the analogies I made in an earlier post is not quite right because of this distinction. (Although I think it is probably good enough at the vague level I was trying to get things across.)

The main thing I wanted to talk about here was scale invariance... the rest of the conformal group was just a tangent. And for scale invariance, I think the term "invariant" would still be appropriate to use for observables in a CFT, right? (In other words, they should not change if you re-scale. Or can they?)
onhava
Jul. 20th, 2010 03:00 pm (UTC)
And for scale invariance, I think the term "invariant" would still be appropriate to use for observables in a CFT, right? (In other words, they should not change if you re-scale. Or can they?)

Think about some examples here. For instance: say I have an operator O1 of dimension d1 and an operator O2 of dimension d2. Then conformal symmetry constrains their two-point function to look like:
<O1(x) O2(0)> = c/x^(d1+d2).
This isn't invariant, it's covariant; if I do a rescaling of coordinates, x -> ax, and the two-point function between ax and 0 differs from the one between x and 0 by a power of a.
spoonless
Jul. 20th, 2010 03:17 pm (UTC)
Oh, right I was being dumb. Thank you.

So Greens functions and observables do not have to be invariant.

It's only the action, the S-Matrix, and the vacuum? Isn't the vacuum state at a non-trivial fixed point "non-trivial" though? Or is it still just a flat background of fields?

[I'm now wondering if this issue of S-Matrix being invariant while Green's functions are not has anything to do with the seemingly similar situation of gauge invariance. In regular field theory the Green's functions are gauge invariant, while in string theory they are not, right?]
spoonless
Jul. 20th, 2010 02:49 pm (UTC)
I guess the way to say what I'm finding so surprising/confusing here is...

There are often interesting non-trivial fixed points of the beta functions, where the theory does very interesting interacting stuff. And even the "trivial" fixed points seem more interesting than "an entire space of one color" which is the only fixed point of this 2d Euclidean space example.

Hmmm... maybe what you said is the right thing to say though... somehow in making that comparison I must be forgetting that the thing that has to be invariant in a CFT is much simpler than the thing that would have to be invariant in this image. It's really only one (or a few) complicated combinations of things that have to be invariant, rather than everything. So thanks... your comment did lead me to the answer I just didn't see it immediately.
vaelynphi
Jul. 20th, 2010 04:52 am (UTC)
Conformal symmetry under discrete transformations (think fractals) is easier to construct.

For continuously conformally symmetric spaces, I can't off the top of my head think of something that isn't trivial; perhaps I need to break out a textbook or two?
spoonless
Jul. 20th, 2010 02:40 pm (UTC)

For continuously conformally symmetric spaces, I can't off the top of my head think of something that isn't trivial; perhaps I need to break out a textbook or two?

I think onhava is right that the entire space being the same color is really the only conformally symmetric configuration here. I was sort of doing the reverse of what most physics analogies do... thinking in terms of how it works abstractly in field theory, and using that to explain how it works on a more simple concrete example =) That got me into trouble, because in the concrete example, it's a lot more trivial.
vaelynphi
Jul. 20th, 2010 08:24 pm (UTC)
I dunno; I've convinced myself there is some wholly symmetric example, but I may just be smoking the good weed today. (Okay, so I don't do any drugs, but that doesn't stop me from having special, special thoughts.)
vaelynphi
Jul. 20th, 2010 10:17 pm (UTC)
Also, in reading up, I've been reminded of the annoying habit physicists have of conflating 'invariant' and 'covariant', nevermind the frustrating neglect of contravariant versus covariant.
( 17 comments — Leave a comment )

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