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In Ginsparg's Applied Conformal Field Theory (http://arxiv.org/abs/hep-th/9108028, on the bottom of p. 5) the following remark is made:

Indeed the conformal group admits a nice realization acting on $\mathbb{R}^{p,q}$, stereographically projected to $S^{p,q}$ and embedded in the light-cone of $\mathbb{R}^{p+1,q+1}$.

What does this mean?

Qmechanic
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leastaction
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1 Answers1

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Your last $S$ should be an $R$.

The conformal group on is the set of transformations of $R^{p+q}$ that preserve angles ona $R^{p, q}$ is a Euclidean space with $p$ normal dimensions and $q$ imaginary ones; $S^{p,q}$ is presumably the unit sphere in this space; stereographic projections from the real space to the sphere will preserve angles. Then for each point on the sphere you can associate a future-pointing light ray passing through points in $R^{p+1,q+1}$, so each of the points on the sphere can be embedded in the larger light cone.

CR Drost
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  • Thanks for your reply Chris. Can you elaborate on " Then for each point on the sphere you can associate a future-pointing light ray.."? – leastaction Mar 04 '15 at 13:02
  • @leastaction sure, that part is simple. Suppose we take a plane and stereographically project its points onto the 2-sphere in the ordinary way; now the 2-sphere is embedded in $R^3$ in the normal way as a unit sphere centered at the origin. We can now shift to the $(+,-,-,-)$ space by associating each point $\vec r$ on this sphere with the four-vector $[1, \vec r]$ which has four-norm 0 -- so we've embedded the plane now in the set of four-vectors going through the origin. Notice that all of these processes are conformal. – CR Drost Mar 04 '15 at 16:27
  • This answer has some confusing typos. – PM 2Ring Dec 15 '23 at 01:41