https://doi.org/10.1140/epjc/s10052-026-15850-6
Regular Article - Theoretical Physics
Covariant conservation laws, local invariance and Noether’s second theorem
1
Faculdade de Ciências e Tecnologia, Universidade dos Açores, 9500-321, Ponta Delgada, Portugal
2
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C8, 1749-016, Lisbon, Portugal
3
Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C8, 1749-016, Lisbon, Portugal
4
William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, 21218, Baltimore, MD, USA
5
Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C8, 1749-016, Lisbon, Portugal
a
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Received:
23
January
2026
Accepted:
14
May
2026
Published online:
6
June
2026
Abstract
We lay down a set of requirements for a field theory to produce a covariant conservation law out of Noether’s second theorem, and show that neither local invariance implies a covariant conservation law, nor the existence of a covariant conservation law necessarily stems from the local invariance of the theory. We illustrate our results with the examples of well-known theories.
The authors acknowledge the support provided by their respective institutions and funding bodies.
© The Author(s) 2026
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