Noether's theorems for first and second order field theories in flat spacetimes

RESUMO

Technical derivations of the Noether identity (that is, showing and explaining all steps using methods of the calculus of variations), as well as of Noether's first and second theorems, are typically presented in explicit detail in the mathematics literature for scalar functions with first order derivatives in the action. This presentation has drawbacks for physicists wishing to learn these techniques to apply to physical field theories that have higher rank tensor fields (potentials) or higher orders of derivatives in the action. Motivated by this, and heavily influenced by the calculus of variations presentation of Gelfand and Fomin, we detail complete derivations of the Noether identity and Noether's theorems for first and second order field theories with tensor fields of arbitrary rank in flat (e.g., Minkowski) spacetimes, showing all steps explicitly. We then give some physical examples of how to apply the first and second theorems to field theories in physics, and detail the origin of conventional spacetime symmetries from the Killing and conformal Killing equations. We conclude by briefly reviewing the many areas of physics where the theorems have been discussed and applied.