nLab
Noether's theorem

Idea

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) corresponds equivalently 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

• Lagrangian version – In terms of the variational bicomplex

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

• Hamiltonian/symplectic version – In terms of moment maps.

Lagrangian version – In terms of the variational bicomplex

Here we formulate Noether’s theorem for local action functional in terms of the variational bicomplex and the covariant phase space.

Simple schematic idea

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 $\nabla \varphi$ (the gradient) of the fields $\varphi$, hence

$$L:\varphi \mapsto L(\varphi ,\nabla \varphi ).$$L \colon \phi \mapsto L(\phi, \nabla \phi) \,.

(We write $\nabla \cdot (-)$ in the following for the divergence.)

Then the variational derivative of $L$ by the fields is

$$\begin{array}{rl}\delta L& =\left(\frac{\delta }{\delta \varphi }L\right)\delta \varphi +\left(\frac{\delta }{\delta \nabla \varphi }L\right)\cdot \nabla \delta \varphi \\ & =(\frac{\delta }{\delta \varphi }L-\nabla \cdot \left(\frac{\delta }{\delta \nabla \varphi }L\right))\cdot (\delta \varphi )+\nabla \cdot \left(\left(\frac{\delta }{\delta \nabla \varphi }L\right)\delta \varphi \right)\end{array},$$\begin{aligned} \delta L & = \left(\frac{\delta}{\delta \phi} L\right) \delta \phi + \left(\frac{\delta}{\delta \nabla \phi} L\right) \cdot \nabla \delta \phi \\ & = \left( \frac{\delta}{\delta \phi} L - \nabla \cdot \left(\frac{\delta}{\delta \nabla\phi}L\right) \right) \cdot (\delta \phi) + \nabla \cdot \left( \left(\frac{\delta }{\delta \nabla \phi} L\right) \delta \phi \right) \end{aligned} \,,

where in the second step the total derivative was introduced via the product rule of differentiation $f(\nabla g)=-(\nabla f)g+\nabla (fg)$.

From this law for the variation of the Lagrangian, one derives both the Equler-Lagrange equations of motion as well as Noether’s theorem by making different assumptions and setting different terms to zero:

1. Demanding that the variation $\delta \varphi$ 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

   $$\frac{\delta }{\delta \varphi }L-\nabla \cdot \left(\frac{\delta }{\delta \nabla \varphi }L\right)=0.$$\frac{\delta}{\delta \phi} L - \nabla \cdot \left(\frac{\delta}{\delta \nabla\phi}L\right) = 0 \,.

2. On the other hand, assuming that for given $\delta \varphi$ the variation $\delta L$ vanishes when these equations of motion hold – hense assuming that $\delta \varphi$ is an on-shell symmetry of $L$ – is equivalent to assuming that the above expression is zero even without the left term, hence that

$$\nabla \cdot \left(\left(\frac{\delta }{\delta \nabla \varphi }L\right)\delta \varphi \right)=0.$$\nabla \cdot \left(\left(\frac{\delta }{\delta \nabla\phi} L\right) \delta \phi\right) = 0 \,.

This is the statement of Noether’s theorem. The object

$$p_{\varphi }\delta \varphi ≔\left(\frac{\delta }{\delta \nabla \varphi }L\right)\delta \varphi$$p_\phi \delta \phi \coloneqq \left(\frac{\delta }{\delta \nabla \phi} L\right) \delta \phi

(here $p_{\varphi }$ is the canonical momentum of the field $\varphi$) is called the Noether current and the above says that this is (on-shell) a conserved current precisely if $\delta \varphi$ 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.)

In this more general case the above conservation law induced by the “weak” symmetry becomes

$$\nabla \cdot (p_{\varphi }\delta \varphi -\sigma )=0.$$\nabla \cdot \left( p_\phi \delta \phi - \sigma \right) = 0 \,.

The formal context

Let $X$ be a spacetime of dimension $n$, $E\to X$ a bundle, $j_{\mathrm{\infty }}E\to X$ its jet bundle and

$${\Omega }^{•,•}(j_{\mathrm{\infty }}E),(D=\delta +d)$$\Omega^{\bullet,\bullet}(j_\infty E), (D = \delta + d)

the corresponding variational bicomplex with $\delta$ being the vertical and $d$ the horizontal differential.

Proposition

For $L\in {\Omega }^{n,0}(j_{\mathrm{\infty }}E)$ a Lagrangian we have that

$$\delta L=E(L)+d\Theta$$\delta L = E(L) + d \Theta

for $E$ the Euler-Lagrange operator.

The covariant phase space of the Lagrangian is the locus

$$\{\varphi \in \Gamma (E)\mid E(L)(j_{\mathrm{\infty }}\varphi )=0\}$$\{\phi \in \Gamma(E) | E(L)(j_\infty \phi) = 0\}

that solves the Euler-Lagrange equations of motion.

For $\Sigma \subset X$ any $(n-1)$-dimensional submanifold,

$$\delta \theta :=\delta {\int }_{\Sigma }\Theta$$\delta \theta := \delta \int_\Sigma \Theta

is the presymplectic structure on covariant phase space.

Definition

A symmetry of $L$ is a vertical vector field $v$ such that

$$v(L)=d{\sigma }_v$$v(L) = d \sigma_v

for some

$${\sigma }_v\in {\Omega }^{n-1,0}(j_{\mathrm{\infty }}E).$$\sigma_v \in \Omega^{n-1,0}(j_\infty E) \,.

Remark

By Cartan's magic formula this is equivalent to

$${\iota }_v\delta L=d{\sigma }_v.$$\iota_{v} \delta L = d \sigma_v \,.

Definition

A conserved current is an element

$$j\in {\Omega }^{n-1,0}(j_{\mathrm{\infty }}E)$$j \in \Omega^{n-1, 0}(j_\infty E)

which is horizontally closed on covariant phase space

$$dj{\mid }_{E(L)=0}=0.$$d j|_{E(L) = 0} = 0 \,.

Statement

With the above notions and notation, Noether’s theorem states:

Theorem

If $v\in T_v(j_{\mathrm{\infty }}E)$ is a symmetry, def. 1, then

$$j_v:={\sigma }_v-{\iota }_v\theta$$j_v := \sigma_v - \iota_v \theta

is a conserved current, def. 2.

Proof

By prop. 1 and def. 1 we have

$$\begin{array}{rl}d({\sigma }_v-{\iota }_v\theta )& ={\iota }_v\delta L+{\iota }_{v}d\theta \\ & ={\iota }_{v}E(L)\end{array}.$$\begin{aligned} d (\sigma_v - \iota_v \theta) & = \iota_v \delta L + \iota_v d \theta \\ & = \iota_v E(L) \end{aligned} \,.

Hamiltonian/symplectic version – In terms of moment maps

In traditional symplectic geometry

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 $ℝ\to \mathrm{𝔓𝔬𝔦𝔰𝔰𝔬𝔫}(X,\omega )$ be a Hamiltonian action with Hamiltonian $H\in C^{\mathrm{\infty }}(X)$, 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 $𝔤$ and let $𝔤\to \mathrm{𝔓𝔬𝔦𝔰𝔰𝔬𝔫}(X,\omega )$ be a Hamiltonian action with Hamiltonian/moment map $\Phi \in C^{\mathrm{\infty }}(X,{𝔤}^{*})$. We say this preserves the (time evolution-)Hamiltonian $H$ if for all $\xi \in 𝔤$ the Poisson bracket between the two vanishes,

$${\delta }_{\xi }H≔\{\Phi (\xi ),H\}=0.$$\delta_\xi H \coloneqq \{\Phi(\xi), H\} = 0 \,.

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

$$\frac{d}{dt}{\Phi }^{\xi }=0.$$\frac{d}{d t} \Phi^\xi = 0 \,.

In this symplectic formulation this is immediate, because

$$\frac{d}{dt}{\Phi }^{\xi }=\{H,{\Phi }^{\xi }\}=-\{{\Phi }^{\xi },H\}=0,$$\frac{d}{d t}\Phi^\xi = \{H,\Phi^\xi\} = - \{\Phi^\xi, H\} = 0 \,,

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

$${\Phi }^{\xi }=p_i(v_{\xi }{)}^i.$$\Phi^\xi = p_i (v_\xi)^i \,.

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 $p_i=\frac{\delta L}{\delta ({\dot{q}}^i)}$).

In higher symplectic geometry

We discuss aspects of the Noether theorem in higher symplectic geometry/higher prequantum field theory (…)

Related concepts

• conservation law

• conserved current

• charge

• moment map

References

The original article is

• Emmy Noether, Invariante Variationsprobleme Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Math. Phys. Kl, 235 (1918).

Basic lecture notes include

• Chapter 7, Noether’s theorem (pdf)

A comprehensive exposition of both the Lagrangian and the Hamiltonian version of the theorem is in

• Jeremy Butterfield, On symmetry and conserved quantities in classical mechanics (2006) (pdf)

A textbook account is in

• 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

For 1-parameter groups of symmetries in classical mechanics, the formulation and the proof of Noether's theorem can be found in the monograph

• Vladimir Arnold, Mathematical methods of classical mechanics

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

• Frédéric Paugam, Towards the mathematics of quantum field theory (pdf)

The example of conserved currents in Chern-Simons theory is discussed around (5.381) on p. 925 of

• Vladimir Ivancevic, Tijana Ivancevi, Applied differential geometry: a modern introduction

and also in

• M. Francaviglia, M. Palese, E. Winterroth, Locally variational invariant field equations and global currents: Chern-Simons theories, Communications in Matheamtical Physics 20 (2012) (pdf)

and for higher dimensional Chern-Simons theory in

• G.Giachetta, L.Mangiarotti, G.Sardanashvily, Noether conservation laws in higher-dimensional Chern-Simons theory, Mod. Phys. Lett. A18 (2003) 2645-2651 (arXiv:math-ph/0310067)