On the Röle of the Group CU of Local Complex Orthogonal Transformations in a Nonlinear Theory of Elementary Particles

It is supposed that there exists a system O' (intrinsic system) in which the field equation for a spin i representation has the simple form yf* dyj'/dxf*' =0. This system is related to the physical system (in which all measurements are performed) by an affine connection which is induced by a certain group of local transformations. The investigation given here deals with the group of local four-dimensional complex orthogonal transformations. Subjecting xp' to such a transformation Q one gets with xp' (x') = Q (x) xp (x) the following equation y^dxp/dx^ + y^-dQ/dx^-xp = 0 . The interaction term splits up into a vector and a pseudovector part: yI düßx^ = y y 5 P^. The special cases of real local orthogonal ( L o r e n t z -) transformations ( £ ^ / t= — ; £ k l real, £ 4 1 imaginary; xp—>x) and special complex local orthogonal transformations blXu= uX:> Vkl imagi­ nary, rj4i real; xp-*-cp) are first separately considered. It is required that V̂ and P are to be built up from the fundamental covariants of the field. In order that certain conservation laws hold at least approximately, the following assumptions are made: Im {V k} = ±k2 <PYk<P , R e{V4} = ± k 2yyi (p , Im {Pk} = ± I2 Xyk ys X » Re iPi} = ± XYx Ya X