and
Renat ZHDANOV
Department of Applied Research,
Institute of Mathematics of NAS of Ukraine,
3 Tereshchenkivs'ka Str.,
01601 Kyiv-4, UKRAINE
E-mail: renat@imath.kiev.ua

Nonlinear Dirac equations through nonlinear gauge tranformations

Abstract:
A method proposed by H.D.D and G. A. Goldin to derive from a linear Schrodinger equation a nonlinear extension through a group of physically motivated nonlinear transformations (nonlinear gauge group) is generalised to the Dirac equation. The nonlinear transformations N on the corresponding Hilbert space are assumed to be local, separable (in connection with a tensor product Hilbert space) and Poincare invariant. Furthermore N is choosen such that the positional density is invariant. This group yields a family F of nonlinear Dirac equations, which represents a nonlinear symmetry of F and which describes systems physical equivalent to the linear Dirac system. If one breaks this symmetry partly, e.g. if one changes the coefficients and the functions which label N, one finds physically inequivalent i.e. 'new' nonlinear Dirac equations which are physically motivated as building blicks in a framework of a nonlinear extension of a quantum theory which could be interesting in the future.