Special Conformal Transformations

[Knecht]

Hello,

In conformal geometry there is a 15-parameter symmetry group.

I have an rough conceptual understanding of the 3 spatial translations, the 1 temporal translation, the 3 rotations, the 3 Lorentz "boosts", and the 1 dilation transformation.

I am having trouble conceptualizing the remaining 4 "special conformal transformations", which appear to be combinations of translations, rotations, and possibly something referred to as "inversion". Some have drawn a parallel to "accelerations"?

If there are any math aficionados out there who can give me a more intuitive, conceptual, visual understanding of what is going on physically with these special conformal transformations, I would be eternally grateful.

Maybe one approach might be to start with a homogeneous sphere and say: "Ok, when we subject the sphere to a SCT, these physical processes happen to the sphere.

Also, why are there specifically 4 SCTs?

Any help would be welcome and the more different perspectives, the better.

Yours in science
Knecht
www.amherst.edu/~rloldershaw

 

[jvicens]

Hello Knecht,

Thank you for your question about the special conformal transformations (SCTs). As you mentioned, there are 15 parameters in the symmetry group of conformal geometry. These 15 parameters can be broken down into the 10 parameters of the Poincaré group (3 translations, 3 rotations, and 4 Lorentz boosts) and the 5 parameters of the special conformal transformations.

The special conformal transformations are indeed combinations of translations, rotations, and inversions. In fact, they can be thought of as "inverted translations" or "reflections" about a point. This is where the term "inversion" comes from. These transformations can also be seen as "accelerations" because they involve changing the position and orientation of objects in space.

To better understand the physical meaning of these transformations, it may be helpful to think about how they affect a simple object, such as a sphere. When a sphere undergoes a special conformal transformation, it is essentially being compressed or stretched in certain directions. This can result in changes in the shape and size of the sphere, as well as its orientation in space.

As for why there are specifically 4 SCTs, this is related to the dimensionality of space. In three-dimensional space, there are three translations, three rotations, and one dilation transformation. This leaves four parameters for the special conformal transformations. In higher dimensions, there would be a different number of SCTs.

I hope this helps to provide a better understanding of the special conformal transformations. Remember, these transformations are simply mathematical tools that help us describe and understand the symmetries of the universe. Keep exploring and asking questions, and you will continue to deepen your understanding of this fascinating topic.

Best of luck in your studies,