The double Lagrangian with U(1, J) symmetry and its applications to the cosmology

By introducing the double complex scalar field, this paper constructs the double Lagrangian with the potential V(\it\Phi(J), \it\Phi^*(J))=V(|\it\Phi(J)|), which not only can describe the evolution of quintom Universe, but also can naturally give the spintessence and hessence Universe. Furthermore, the U(1,J) symmetry of the double complex Lagrangian is verified, and the total conserved charge within the physical volume is derived by means of the No?ther theorem. Moreover, it can point out that the `imaginary motion' of the angular parameter in Ref.14, in fact, is only a real phase displacement in our model.