Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1990. Sectio Physicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 20)
Anatol Nowicki: Composite spacetime from twistors and its extensions
- 10 1.e. the complex fourvector in the fundamental representation of a covering conformal group SUC2,2) . A correspondence between the twistors and the spacetime points is given by the incidence equation — Penrose relation. The twistor formalism formulated originally by Penrose for the four — dimensional CD=d.) spacetime can be extended in two ways: i ) extending the Penrose—relation in a supersymmetric way one obtains a correspondence between the supertwistors and the points of D=d superspaces t2,3], ii) replacing the complex numbers by quaternios in the Penrose relation one can bring the quaternionic twistors into connection with thepoints of the D=6 spacetime C4J. Furtheron, one can extend this quaternionic twistor formalism supersymmetrically introducing quaternionic fermionic degrees of freedom. 2. Composite Dn4 spacetime from twistors. Let us consider the fundamental steps in a more didactical way leading to the formulation of the Penrose—relation. It is well known that any spacetime point described by the fourvector x=Cx°,x 1 ,x 2 ,x 3> can be brought into connection with a hermitean 2x2 dimensional matrix, using the Pauli matrices o : fJl X * X • fc'lix* this correspondence is one to one. One can also consider the complex fourvector z=Cz°, z 4 , z^ , z 3) instead of the real one x. The complex fourvector z describes a point of the complexified Minkowski Space Ol'*. A similar relation to CI 3 gives us the correspondence between the points of Ö1 4" and two dimensional complex matrices:
