In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations.
[1] The theorem is named after its discoverer, Emmy Noether.
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 action.
Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by k arbitrary functions and their derivatives up to order m, then the functional derivatives of L satisfy a system of k differential equations.
Noether's second theorem is sometimes used in gauge theory.
Gauge theories are the basic elements of all modern field theories of physics, such as the prevailing Standard Model.
Suppose that we have a dynamical system specified in terms of
th order partial derivatives of the dependent variables.
Multiindex notation for the latin indices is also introduced as follows.
and applying the inverse product rule of differentiation we get
are the Euler-Lagrange expressions of the Lagrangian, and the coefficients
It should be remarked that it is possible to extend infinitesimal (quasi-)symmetries by including variations with
However such symmetries can always be rewritten so that they act only on the dependent variables.
Therefore, in the sequel we restrict to so-called vertical variations where
For Noether's second theorem, we consider those variational symmetries (called gauge symmetries) which are parametrized linearly by a set of arbitrary functions and their derivatives.
are arbitrarily specifiable functions of the independent variables, and the latin indices
For these variations to be (exact, i.e. not quasi-) gauge symmetries of the Lagrangian, it is necessary that
If the variations are quasi-symmetries, it is then necessary that the current also depends linearly and differentially on the arbitrary functions, i.e. then
For simplicity, we will assume that all gauge symmetries are exact symmetries, but the general case is handled similarly.
The statement of Noether's second theorem is that whenever given a Lagrangian
linear differential relations between the Euler-Lagrange equations of
where on the first term proportional to the Euler-Lagrange expressions, further integrations by parts can be performed as
This relation is valid for any choice of the gauge parameters
Choosing them to be compactly supported, and integrating the relation over the manifold of independent variables, the integral total divergence terms vanishes due to Stokes' theorem.
Then from the fundamental lemma of the calculus of variations, we obtain that
are linear in the Euler-Lagrange expressions, they necessarily vanish on-shell).
Inserting this back into the initial equation, we also obtain the off-shell conservation law
The coefficients of the adjoint operator are obtained through integration by parts as before, specifically
, the value of the adjoint on the functions when contracted with the Euler-Lagrange expressions is a total divergence, viz.
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