In mathematics, Noether identities characterize the degeneracy of the Lagrangian system . If a Lagrangian system and its Lagrangian are given , Noether identities are defined as a differential operator whose kernel contains the image of the Euler - Lagrange Lagrangian operator . Every Euler-Lagrange operator satisfies Noether identities, which are thereby divided into trivial and nontrivial. Lagrangian is called degenerate if its Euler-Lagrange operator satisfies the nontrivial Noether identities. In this case, the Euler - Lagrange equations are not independent.
The Noether identities are also not required to be independent and satisfy the Noether identities of the first rank, which, in turn, obey the Noether identities of the second rank, etc. Higher-ranking Noether identities are also subdivided into trivial and non-trivial. A degenerate Lagrangian is called reduced if there exist nontrivial Noether identities of higher rank. The Young – Mills gauge theory and the gravity gauge theory are examples of unreduced Lagrangian field models.
Various versions of the second Noether theorem establish a one-to-one correspondence between nontrivial reduced Noether identities and non-trivial reduced gauge symmetries . Formulated in its most general form, the second Noether theorem associates with a chain complex of reduced Noether identities indexed by antipoles, the BRST complex of reduced gauge symmetries parameterized by spirits , as is the case in classical field theory and Lagrangian BRST theory .
See also
- Noether's theorem
- Emmy Noether
- Lagrange system
- Gauge symmetry (math)
- Variation Bicomplex
Literature
- Gomis, G., Paris, J., Samuel, S., Antibracket, antifields and gauge theory quantization, Phys. Rep. 259 (1995) 1.
- Fulp, R., Lada, T., Stasheff, J., Noether variational theorem II and the BV formalism, arXiv: math / 0204079
- Bashkirov, D., Giachetta, G., Mangiarotti, L., Sardanashvily, G. , The KT-BRST complex of a degenerate Lagrangian system, Lett. Math. Phys. 83 (2008) 237; arXiv: math-ph / 0702097 .