Noether analysis of zilch conservation laws and their generalization for the electromagnetic field. II. Use of Poincaré-invariant formulation of the principle of least action

The Noether analysis of conservation laws for the electromagnetic field is carried out basing on the Lagrange function in terms of field strengths $\mathbf{E,H}$ which is scalar with respect to the total Poincare group $\tilde {\mathrm P}(1,3)$. It is shown that the $\tilde {\mathrm P}$-scalar Lagrange function differs from the other Lagrange functions discussed before in such a way that it is exactly conservation law for the energy momentum $P_\mu$ of the electromagnetic field which this function puts into correspondence with the generators $\partial_\mu$ of space-time translations according to the Noether theorem; moreover, this function makes it possible to establish an adequate connection between the zilch conservation laws and symmetries of the Maxwell equations and also to introduce the minimal and local $\tilde {\mathrm P}$-scalar interaction of the electromagnetic field $\mathbf{(E, H)}$ and spinor field. Analysis of the Noether correspondence between symmetry operators and conservation laws, together with other criteria, makes it possible to single out a suitable Lagrange function for the tensor electromagnetic field $F=\mathbf{(E, H)}$ in the set of $s$-equivalent Lagrangians.