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
Abstract:
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.
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