Noether identities

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Noether identities describe relations among the equations of motion that arise from symmetries of a system described by a Lagrangian.

- Start with a Lagrangian L and its Euler–Lagrange operator EL(L), which gives the equations of motion.
- A Noether identity is a differential operator K such that K(EL(L)) = 0 identically. Some identities are trivial (always true); others are non-trivial and reflect genuine degeneracy.
- A Lagrangian is degenerate if its Euler–Lagrange equations satisfy non-trivial Noether identities, meaning the equations are not all independent.
- Identities can occur in stages: first-stage, second-stage, and so on. Some of these identities may be trivial, while non-trivial higher-stage identities make the Lagrangian reducible.

Irreducible theories have no non-trivial higher-stage identities; examples include Yang–Mills gauge theory and gauge gravity.

Second Noether’s theorem links non-trivial reducible Noether identities to non-trivial reducible gauge symmetries. In modern terms, this is expressed through the Koszul–Tate complex of reducible Noether identities (with antifields) and the BRST complex of reducible gauge symmetries (with ghosts) in covariant classical field theory and Lagrangian BRST theory.