Noether identities
Encyclopedia
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
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 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
and Lagrangian BRST theory.
Lagrangian system
In mathematics, a Lagrangian system is a pair of a smoothfiber bundle Y\to X and a Lagrangian density L which yields the Euler-Lagrange differential operator acting on sections of Y\to X.In classical mechanics, many dynamical systems are...
L, Noether identities can be defined as a differential operator
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, accepting a function and returning another .This article considers only linear operators,...
whose kernel contains a range of the Euler–Lagrange operator
Lagrangian system
In mathematics, a Lagrangian system is a pair of a smoothfiber bundle Y\to X and a Lagrangian density L which yields the Euler-Lagrange differential operator acting on sections of Y\to X.In classical mechanics, many dynamical systems are...
of L. Any Euler–Lagrange operator
Lagrangian system
In mathematics, a Lagrangian system is a pair of a smoothfiber bundle Y\to X and a Lagrangian density L which yields the Euler-Lagrange differential operator acting on sections of Y\to X.In classical mechanics, many dynamical systems are...
obeys Noether identities which therefore are separated into the trivial and non-trivial ones. A Lagrangian
Lagrangian system
In mathematics, a Lagrangian system is a pair of a smoothfiber bundle Y\to X and a Lagrangian density L which yields the Euler-Lagrange differential operator acting on sections of Y\to X.In classical mechanics, many dynamical systems are...
L is called degenerate if the Euler–Lagrange operator
Lagrangian system
In mathematics, a Lagrangian system is a pair of a smoothfiber bundle Y\to X and a Lagrangian density L which yields the Euler-Lagrange differential operator acting on sections of Y\to X.In classical mechanics, many dynamical systems are...
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
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.The first gauge model of gravity was...
exemplify irreducible Lagrangian field theories.
Different variants of second Noether’s theorem
Noether's second theorem
In mathematics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The action S of a physical system is an integral of a so-called Lagrangian function L, from which the system's behavior can be determined by the principle of least...
state the one-to-one correspondence between the non-trivial reducible Noether identities and the non-trivial reducible gauge symmetries
Gauge symmetry (mathematics)
In mathematics, any Lagrangian system generally admits gaugesymmetries, though it may happen that they are trivial. Intheoretical physics, the notion of gaugesymmetries depending on parameter functions is a cornerstone ofcontemporary field theory....
. Formulated in a very general setting, second Noether’s theorem
Noether's second theorem
In mathematics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The action S of a physical system is an integral of a so-called Lagrangian function L, from which the system's behavior can be determined by the principle of least...
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
Covariant classical field theory
In recent years, there has been renewed interest in covariant classical field theory. Here, classical fields are represented 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 the...
and Lagrangian BRST theory.
See also
- Noether's second theoremNoether's second theoremIn mathematics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The action S of a physical system is an integral of a so-called Lagrangian function L, from which the system's behavior can be determined by the principle of least...
- Emmy NoetherEmmy NoetherAmalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
- Lagrangian systemLagrangian systemIn mathematics, a Lagrangian system is a pair of a smoothfiber bundle Y\to X and a Lagrangian density L which yields the Euler-Lagrange differential operator acting on sections of Y\to X.In classical mechanics, many dynamical systems are...
- Variational bicomplexVariational bicomplexIn mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations...
- Gauge symmetry (mathematics)Gauge symmetry (mathematics)In mathematics, any Lagrangian system generally admits gaugesymmetries, though it may happen that they are trivial. Intheoretical physics, the notion of gaugesymmetries depending on parameter functions is a cornerstone ofcontemporary field theory....