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

- 19 quaternionic twistor space as follows subspace space by columns of dx2 quaternionic { matrix S2 ' } C2Ő3 By a similar procedure to eqs.C4,S,ó) we get the quaternionic Penrose—relation o

a = e

2Z

c^

3 ß cx, ß = 1,2 C27> where the quaternionic twistor has the form t = C<o~

x, • A real D=6 Minkowski spacetime point is described by a sixdimensional vector x = Cx

(

),x

l,. . . ,x

s) e KM® which can be mapped on a quaternionic Hermitean 2x2 matrix Cef.eqCl)): Ü? = k = 1,2,3 C28) -f* The reality condition Z = 2 C2 denotes a quaternionic conjugated and transposed matrix) is equivalent to the following condition for quaternionic twistor t <t,t> = S ae

2 a + ae

2o

a = 0 C29) therefore, twistors t describe a point of KM

G if their OC4^0-0 = = U^C

4 ; 0-0 norms vanish. Using the decomposition C2S) of quaternionic coordinates of twistor DH one can immediately show that eq. C24) is equivalent to the relations C20), so the descriptions of by the D=6 complex twistors and D=6 quaternionic twistors are equivalent. 5. Final remarks. It is worthwhile to notice that the two approaches above for the twistor description of D=6 spacetime are equivalent only for real space—time. This spacetime can be extended in two nonequivalent ways: by complexification or quaternionization