Symmetry and exact solutions of nonlinear spinor equations

Abstract

This review is devoted to the application of algebraic-theoretical methods to the problem of constructing exact solutions of the many-dimensional nonlinear systems of partial differential equations for spinor, vector and scalar fields widely used in quantum field theory. Large classes of nonlinear spinor equations invariant under the Poincaré group P(1, 3), Weyl group (i.e. Poincaré group supplemented by a group of scale transformations), and the vonformal group C(1, 3) are described. Ansätze invariant under the Poincaré and the Weyl groups are constructed. Using these we reduce the Poincaré-invariant nonlinear Dirac equations to systems of ordinary differential equations and construct large families of exact solutions of the nonlinear Dirac-Heisenberg equation depending on arbitrary parameters and functions. In a similar way we have obtained new families of exact solutions of the nonlinear Maxwell-Dirac and Klein-Gordon-Dirac equations. The obtained solutions can be used for quantization of nonlinear equations.