In mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
''L'', Noether identities can be defined as a
differential operator 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
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
''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 the trivial and non-trivial once. A degenerate Lagrangian is called reducible if there exist non-trivial higher-stage Noether identities.
Yang–Mills gauge theory and
gauge gravitation theory 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. This is the case of
covariant classical field theory In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and ...
and Lagrangian
BRST theory.
See also
*
Noether's second theorem
In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations.
:Translated in The action ''S'' of a physical system is an integral of a so-called Lagrangian f ...
*
Emmy Noether
Amalie Emmy NoetherEmmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
*
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 Lagr ...
*
Variational bicomplex
In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber b ...
*
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
Differential equations
Theoretical physics
Mathematical identities
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