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

- 17 description of the real sixdimensional spacetime CRK

G. First let us discuss first the case Ci) One can consider the complex D=6 twistors: T=Co°

),i?

a> <5 C® Ca=l

,...,4) as the norm of the spinors for eight dimensional complex orthogonal group 0C8,<C) is: CT, T' ) = + rr o'

a = 0 CIS) a 3 the points of the complex D=ó Minkowski space <CM

e are represented by a complex 4x4 antisymmetric matrix z

a

b= —z

bc

i. The Penrose—incidence equation takes the form = z

a

bTi

b a., b = 1, . . . , 4 <16) This equation has a nontrivial solution if the twistors T are pure (simple) i.e. CT,T) = C17) in other words they have vanishing 0C8;O norm. The points of the real six dimensional Minkowski space EFM'' are represented by a 4x4 complex, antisymmetric matrix Z satisfying the reality condition in the form ( 0 1 Z = -Z~ where = B~

1Z"B , B -10 0 O 0 1 U -1 0 C18) and Z denotes the hermitean conjugated matrix. This reality condition for matrix Z is equivalent to the following condition for twistors o*

0

1« + 7i*«

a = 0 where o*

Q = o*

bCB~

1) * a a b cig:

) 71* = TT*CB)

b a b a. )ft and means the complex conjugation. The equation CIO) is in fact the condition of vanishing the VC4,4) norm. Therefore, D=6 twistors describe the points of the real Minkowski space if the following two norms are zero: 0C

8;O - norm: o

an = 0 C20a) a. UC

4,4) - norm: + - 0 C20b) ' a. a. It means that D=6 twistors describing the points of ÍRM

Ö are