Local invariance and covariant conservation laws are independent in field theories; the paper gives requirements for Noether's second theorem to produce the latter and shows the two are not equivalent.
(...) Noether’s second theorem, applied to the matter part of the action, gives the covariant analogue of the energy and momentum conservation equation (...)
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Covariant conservation laws, local invariance and Noether's second theorem
Local invariance and covariant conservation laws are independent in field theories; the paper gives requirements for Noether's second theorem to produce the latter and shows the two are not equivalent.