  A new representation of Lagrangians of 4D nonlinear electrodynamics is
considered. In this new formulation, in parallel with the standard Maxwell
field strength F, an auxiliary bispinor (tensor) field V is introduced. The
gauge field strength appears only in bilinear terms of the full Lagrangian,
while the interaction Lagrangian E depends on the auxiliary fields, E = E(V).
Two types of self-duality inherent in the nonlinear electrodynamics models
admit a simple characterization in terms of the function E. The continuous
SO(2) duality symmetry between nonlinear equations of motion and Bianchi
identities amounts to requiring E to be a function of the SO(2) invariant
quartic combination |V|^4. The discrete self-duality (or self-duality under
Legendre transformation) amounts to a weaker condition E(V)= E(iV). This
approach can be generalized to a system of n Abelian gauge fields exhibiting
U(n) duality. The corresponding interaction Lagrangian should be U(n) invariant
function of n bispinor auxiliary fields.
