# Noether identities

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In mathematics, **Noether identities** characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its [Lagrangian](/source/Lagrangian_system) *L*, Noether identities can be defined as a [differential operator](/source/Differential_operator) whose kernel contains a range of the [Euler–Lagrange operator](/source/Lagrangian_system) of *L*. Any [Euler–Lagrange operator](/source/Lagrangian_system) obeys Noether identities which therefore are separated into the trivial and non-trivial ones. A [Lagrangian](/source/Lagrangian_system) *L* is called degenerate if the [Euler–Lagrange operator](/source/Lagrangian_system) of *L* satisfies non-trivial Noether identities. In this case [Euler–Lagrange equations](/source/Euler%E2%80%93Lagrange_equation) are not independent.

Noether identities need not be independent, but satisfy first-stage Noether identities, which are subject to the second-stage Noether identities and so on. Higher-stage Noether identities also are separated into trivial and non-trivial cases. A degenerate Lagrangian is called reducible if there exist non-trivial higher-stage Noether identities. [Yang–Mills gauge theory](/source/Yang%E2%80%93Mills_theory) and [gauge gravitation theory](/source/Gauge_gravitation_theory) exemplify irreducible Lagrangian field theories.

Different variants of [second Noether's theorem](/source/Noether's_second_theorem) state the one-to-one correspondence between the non-trivial reducible Noether identities and the non-trivial reducible [gauge symmetries](/source/Gauge_symmetry_(mathematics)). Formulated in a very general setting, [second Noether's theorem](/source/Noether's_second_theorem) associates to the Koszul–Tate complex of reducible Noether identities, parameterized by [antifields](/source/Batalin%E2%80%93Vilkovisky_formalism), the BRST complex of reducible gauge symmetries parameterized by [ghosts](/source/Faddeev%E2%80%93Popov_ghost). This is the case of [covariant classical field theory](/source/Covariant_classical_field_theory) and Lagrangian [BRST theory](/source/BRST_formalism).

## See also

- [Noether's second theorem](/source/Noether's_second_theorem)
- [Emmy Noether](/source/Emmy_Noether)
- [Lagrangian system](/source/Lagrangian_system)
- [Variational bicomplex](/source/Variational_bicomplex)
- [Gauge symmetry (mathematics)](/source/Gauge_symmetry_(mathematics))

## References

- Gomis, J., Paris, J., Samuel, S., Antibracket, antifields and gauge theory quantization, Phys. Rep. **259** (1995) 1.
- Fulp, R., Lada, T., [Stasheff, J.](/source/Jim_Stasheff) Noether variational theorem II and the BV formalism, [arXiv:math/0204079](https://arxiv.org/abs/math/0204079)
- Bashkirov, D., Giachetta, G., Mangiarotti, L., [Sardanashvily, G.](/source/Gennadi_Sardanashvily), The KT-BRST complex of a degenerate Lagrangian system, Lett. Math. Phys. **83** (2008) 237; [arXiv:math-ph/0702097](https://arxiv.org/abs/math-ph/0702097).
- [Sardanashvily, G.](/source/Gennadi_Sardanashvily), Noether theorems in a general setting, [arXiv:1411.2910](https://arxiv.org/abs/1411.2910).

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