In mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its
Lagrangian ''L'', Noether identities can be defined as a
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
whose kernel contains a range of the
Euler–Lagrange operator of ''L''. Any
Euler–Lagrange operator obeys Noether identities which therefore are separated into the trivial and non-trivial ones. A
Lagrangian ''L'' is called degenerate if the
Euler–Lagrange operator of ''L'' satisfies non-trivial Noether identities. In this case
Euler–Lagrange equations 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 and
gauge gravitation theory
In quantum field theory, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity.
''Gauge gravitation theory'' should not be confused with th ...
exemplify irreducible Lagrangian field theories.
Different variants of
second Noether’s theorem state the one-to-one correspondence between the non-trivial reducible Noether identities and the non-trivial reducible
gauge symmetries. Formulated in a very general setting,
second Noether’s theorem associates to the Koszul–Tate complex of reducible Noether identities, parameterized by
antifields, the BRST complex of reducible gauge symmetries parameterized by
ghosts
In folklore, a ghost is the soul or Spirit (supernatural entity), spirit of a dead Human, person or non-human animal that is believed by some people to be able to appear to the living. In ghostlore, descriptions of ghosts vary widely, from a ...
. This is the case of
covariant classical field theory In mathematical physics, covariant classical field theory represents classical field theory, classical fields by Section (fiber bundle), sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of field ( ...
and Lagrangian
BRST theory.
See also
*
Noether's second theorem
*
Emmy Noether
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
*
Lagrangian system
In mathematics, a Lagrangian system is a pair , consisting of a smooth fiber bundle and a Lagrangian density , which yields the Euler–Lagrange differential operator acting on sections of .
In classical mechanics, many dynamical systems are L ...
*
Variational bicomplex
*
Gauge symmetry (mathematics)
In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field t ...
References
* Gomis, J., 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,
* Bashkirov, D., Giachetta, G., Mangiarotti, L.,
Sardanashvily, G., The KT-BRST complex of a degenerate Lagrangian system, Lett. Math. Phys. 83 (2008) 237; .
*
Sardanashvily, G., Noether theorems in a general setting, .
Calculus of variations
Theoretical physics
Mathematical identities
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