## On Conformal Infinity and Compactifications of the Minkowski Space

**Authors:** Arkadiusz Jadczyk

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 2-sphere that is at the base of
this cone. We represent this 2-sphere 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 pseudo-Hermitian space H2;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.

**Comments:** 38 pages

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### Submission history

[v1] 18 Sep 2010

[v2] 1 Dec 2010

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