Compactified Minkowski Space Research Articles

Properties of dilatations and conformal transformations in Minkowski space

The geometric structure of systems which are invariant under the 15-parameter conformal group of the Minkowski space is investigated. In particular, we analyse the action of the group on the compactified Minkowski space M 4 c and the 5-dimensional manifold K 5 ( E, b ) of events E and measuring rods b . The properties of these manifolds as homogeneous spaces and the structure of their tangent and cotangent spaces are disussed, too. Finally, we investigate the notion of causality in the context of a conformally invariant world: we show that it is possible to introduce a conformally invariant local causal structure which is just the same as that of a Poincaré-invariant one in Minkowski space. Globally, this world is highly acausal, because any two points can be connected by timelike curves.

Twistors and the conformal group

The connected symmetry group SU(2, 2) of twistor space (T), a four-dimensional complex manifold with metric dsT2≡dX0 dX2¯+dX2 dX0¯+dX1 dX3¯+dX3 dX1¯,and the connected symmetry group O0(2, 4) of conformal space (C), a six-dimensional real manifold with metric dsC2≡(dX0)2−(dX1)2−(dX2)2−(dX3)2−(dX4)2+(dX5)2,are 4:1 and 2:1 homomorphic, respectively, to the restricted conformal group in compactified Minkowski space M. We obtain explicit realizations for these homomorphisms and explore the invariant geometrical relationships they imply between T, C, and M. As an application of the twistor formalism, we show that every continuous conformal transformation has a unique decomposition as the product of a Lorentz transformation, a translation, an acceleration, a dilation, and one of four special conformal transformations.