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What is commonly called Noether’s theorem or Noether’s first theorem is a theorem due to Emmy Noether (Noether 1918) which makes precise and asserts that to every continuous symmetry of a Lagrangian physical system (prequantum field theory) there is naturally associated a conservation law stating the conservation of a charge (conserved current) when the equations of motion hold.
For instance the time-translation invariance of a physical system equivalently means that the quantity of energy is conserved, and the space-translation invariant of a physical system means that momentum is preserved.
The original and still most common formulation of the theorem is in terms of variational calculus applied to a local action functional. A modern version of this formulated properly in terms of the variational bicomplex we discuss below in
There is another formulation of the same physical content, but using the formalism of symplectic geometry of phase spaces. In this formulation of physics the relation between symmetries and charges/conserved currents happens to be built deep into the formalism in terms of Hamiltonian flows generated by the Poisson bracket with a Hamiltonian function. Accordingly, in this powerful formalism Noether’s theorem becomes almost a tautology. This we discuss in
Here we formulate Noether’s theorem for local action functional in terms of the variational bicomplex and the covariant phase space.
Before coming to the precise and general formulation, we indicate here schematically the simple idea which underlies Noether’s first theorem (in its original Lagrangian version).
Consider a local Lagrangian $L$ and assume for simplicity that it depends only on the first derivatives $\nabla \phi$ (the gradient) of the fields $\phi$, hence
(We write $\nabla \cdot (-)$ in the following for the divergence.)
Then the variational derivative of $L$ by the fields is
where in the second step the total derivative was introduced via the product rule of differentiation $f (\nabla g) = -(\nabla f) g + \nabla (f g)$.
From this law for the variation of the Lagrangian, one derives both the Euler-Lagrange equations of motion as well as Noether’s theorem by making different assumptions and setting different terms to zero:
Demanding that the variation $\delta \phi$ vanishes on some boundary of spacetime implies that the rightmost term in the above equation disappears in the variation $\delta S = \delta \int L$ of the action functional (by the Stokes theorem) and hence demanding that $\delta S = 0$ under variation that vanishes on the boundary is equivalent to demanding the Euler-Lagrange equation
On the other hand, assuming that for given $\delta \phi$ the variation $\delta L$ vanishes when these equations of motion hold – hence assuming that $\delta \phi$ is an on-shell symmetry of $L$ – is equivalent to assuming that the above expression is zero even without the left term, hence that
This is the statement of Noether’s theorem. The object
(here $p_\phi$ is the canonical momentum of the field $\phi$) is called the Noether current and the above says that this is (on-shell) a conserved current precisely if $\delta \phi$ is a symmetry of the Lagrangian.
This is at least the way that Noether’s theorem has been introduced and is often considered. But this formulation is more restrictive than is natural. Namely it is unnatural to demand of a symmetry that it leaves the Lagrangian entirely invariant, $\delta L = 0$:
More generally for the symmetry to be a symmetry of the action functional $\int L$ over a closed manifold it is sufficient that the Lagrangian changes by a divergence, $\delta L = \nabla \cdot \sigma$, for some term $\sigma$.
(This is really a sign of a higher gauge symmetry, where the symmetry holds only up to a homotopy $\sigma$. It happens for instance for the gauge-coupling term in the Wess-Zumino-Witten model because the WZW term is not strictly invariant under gauge transformations, but instead transforms by a total derivative. See at conserved current – In higher prequantum geometry).
In this more general case the above conservation law induced by the “weak” symmetry becomes
(This may be regarded as the Legendre transform of $\sigma$.)
Let $X$ be a spacetime of dimension $n$, $E \to X$ a field bundle, $Jet(E) \to X$ its jet bundle and write
the corresponding variational bicomplex with $d_V$ being the vertical and $d_H$ the horizontal de Rham differential.
For $\mathbf{L} \in \Omega^{n,0}(Jet(E))$ a local Lagrangian we have a unique decomposition of its de Rham differential
such that $\mathbf{E}$ is a source form – the Euler-Lagrange form of $\mathbf{L}$ – and for some $\Theta \in \Omega^{n-1,0}(j_\infty E)$.
The dynamical shell $\mathcal{E} \hookrightarrow Jet(E)$ is the zero locus of $\mathbf{E}$ together with its differential consequences.
The covariant phase space of the Lagrangian is the zero locus
that solves the Euler-Lagrange equations of motion.
For $\Sigma \subset X$ any compact $(n-1)$-dimensional submanifold,
is the presymplectic structure on covariant phase space.
An infinitesimal variational symmetry of $\mathbf{L}$ is a vertical vector field $v$ such that
(with $\mathcal{L}_v$ denoting the Lie derivative) for some
By Cartan's magic formula and since $v$ is assumed vertical while $L$ is horizontal, this is equivalent to
A conserved current is an element
which is horizontally closed on the dynamical shell
With the above notions and notation, Noether’s theorem states:
For $v \in T_v(j_\infty E)$ an infinitesimal variational symmetry according to def. 2, then
is a conserved current, def. 3.
A symmetry of the Lepage form
would be defined with the full differential $d = d_V + d_H$:
where $\sigma$ is horizontal and $\kappa$ is vertical. Decomposing the result into horizontal and vertical components, then for vertical $v$ this is equivalent to the pair of equations
The first one expresses the conserved current corresponding to $v$ as in theorem 1, the second constrains $v$ to be a Hamiltonian vector field with respect to the presymplectic current.
In symplectic geometry the analog of Noether’s theorem is the statement that the moment map of a Hamiltonian action which preserbes a given time evolution is itself conserved by this time evolution.
Souriau called this the symplectic Noether theorem, sometimes it is called the Hamiltonian Noether theorem. A review is for instance in (Butterfield 06).
Let $(X,\omega)$ be a symplectic manifold and let $\mathbb{R} \to \mathfrak{Poisson}(X,\omega)$ be a Hamiltonian action with Hamiltonian $H \in C^\infty(X)$, thought of as the time evolution of a physical system with phase space $(X,\omega)$.
Then let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\mathfrak{g} \to \mathfrak{Poisson}(X,\omega)$ be a Hamiltonian action with Hamiltonian/moment map $\Phi \in C^\infty(X,\mathfrak{g}^\ast)$. We say this preserves the (time evolution-)Hamiltonian $H$ if for all $\xi \in \mathfrak{g}$ the Poisson bracket between the two vanishes,
In this situation now the statement of Noether’s theorem is that the generators $\Phi(\xi)$ of the symmetry are preserved by the time evolution
In this symplectic formulation this is immediate, because
by the above assumtion that $H$ is preserved. Hence the “Hamiltonian Noether theorem” is all captured already by the very notion of Hamiltonian action and the statement that the Poisson bracket is skew-symmetric (is a Lie algebra bracket).
Specifically, if one has a global polarization of $(X,\omega)$ with canonical coordinates $\{q^i\}$ and canonical momenta $\{p_i\}$ and if the symmetry action is on the canonical coordinates (on configuration space), then for $v_\xi$ the vector field corresponding to the generator $\xi$ the moment map is
On the right this is the term in the form in which the conserved quantity obtained from the Nother theorem is traditionally written (using that given a Lagrangian $L$ we have $p_i = \frac{\delta L}{\delta (\dot q^i)}$).
The original article is
Basic lecture notes include
A comprehensive exposition of both the Lagrangian and the Hamiltonian version of the theorem is in
Textbook accounts include
Yvette Kosmann-Schwarzbach, Les théorèmes de Noether: invariance et lois de conservation au XXe siècle : avec une traduction de l’article original, “Invariante Variationsprobleme”, Editions Ecole Polytechnique, 2004 (pdf)
Alexandre Vinogradov, I. S. Krasilshchik (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics (pdf)
For 1-parameter groups of symmetries in classical mechanics, the formulation and the proof of Noether's theorem can be found in the monograph
For more general case see for instance the books by Peter Olver.
The Hamitlonian Noether theorem is also reviewed in a broader context of mathematical physics as theorem 7.3.2 in
Discussion in the context of the variational bicomplex includes
Discussion of a generalization to discrete groups of symmetries includes
The example of conserved currents in Chern-Simons theory is discussed around (5.381) on p. 925 of
and also in
and for higher dimensional Chern-Simons theory in
A formalization of Noether’s theorem in cohesive homotopy type theory is discussed in sections “2.7 Noether symmetries and equivariant structure” and “3.2 Local observables, conserved currents and higher Poisson bracket homotopy Lie algebras” of
A formalization of invariance of Lagrangians in parametric dependent type theory is discussed in