Title 
On Conformal Infinity and Compactifications of the Minkowski Space


Published in 
Advances in Applied Clifford Algebras, March 2011

DOI  10.1007/s0000601102855 
Authors 
Arkadiusz Jadczyk 
Abstract 
Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the "cone at infinity" but also the 2sphere that is at the base of this cone. We represent this 2sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge * operator twistors (i.e. vectors of the pseudoHermitian space H_{2,2}) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H_{p,q} are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail. 
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