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 the 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

• 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 of the Lagrangian 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

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

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

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

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

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

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

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:

1. 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

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

2. On the other hand, assuming that for given $\delta \phi$ 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

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

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

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

(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

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

(This may be regarded as the Legendre transform of $\sigma$.)

Formulation via the variational bicomplex

Let $X$ be a spacetime of dimension $n$, $E \to X$ a field bundle, $Jet(E) \to X$ its jet bundle and write

$$\Omega^{\bullet,\bullet}(Jet(E)), (d = d_H + d_V)$$

the corresponding variational bicomplex with $d_V$ being the vertical and $d_H$ the horizontal de Rham differential.

Proposition

For $\mathbf{L} \in \Omega^{n,0}(Jet(E))$ a local Lagrangian we have a unique decomposition of its de Rham differential

$$d L = d_V L = \mathbf{E} - d_H \Theta$$

such that $\mathbf{E}$ is a source form – the Euler-Lagrange form of $\mathbf{L}$ – and for some $\Theta \in \Omega^{n-1,0}(j_\infty E)$.

Definition

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

$$\{\phi \in \Gamma(E) | \mathbf{E}(j_\infty \phi) = 0\}$$

that solves the Euler-Lagrange equations of motion.

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

$$\delta \theta \coloneqq \delta \int_\Sigma \Theta$$

is the presymplectic structure on covariant phase space.

Definition

An infinitesimal variational symmetry of $\mathbf{L}$ is a vertical vector field $v$ such that

$$\mathcal{L}_v \mathbf{L} = d_H \sigma_v$$

(with $\mathcal{L}_v$ denoting the Lie derivative) for some

$$\sigma_v \in \Omega^{n-1,0}(Jet(E)) \,.$$

Remark

By Cartan's magic formula and since $v$ is assumed vertical while $L$ is horizontal, this is equivalent to

$$\iota_{v} d_V \mathbf{L} = d_H \sigma_v \,.$$

Definition

A conserved current is an element

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

which is horizontally closed on the dynamical shell

$$(d_H\, j)|_{\mathcal{E}} = 0 \,.$$

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

Theorem

For $v \in T_v(j_\infty E)$ an infinitesimal variational symmetry according to def. , then

$$j_v \coloneqq \sigma_v - \iota_v \Theta$$

is a conserved current, def. .

Proof

By prop. and def. we have

$$\begin{aligned} d_H (\sigma_v - \iota_v \Theta) & = \iota_v d_V \mathbf{L} + \iota_v d_H \Theta \\ & = \iota_v \mathbf{E} \end{aligned} \,.$$

Remark

A symmetry of the Lepage form

$$\rho \coloneqq \mathbf{L} + \Theta \,.$$

would be defined with the full differential $d = d_V + d_H$:

$$\mathcal{L}_v \rho = d (\sigma_v + \kappa_v) \,,$$

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

$$\left\{ \array{ \iota_v \mathbf{E} & = d_H (\sigma_v - \iota_v \Theta) \\ \iota_v \omega & = d_H \kappa_v + d_V \sigma_v } \right.$$

The first one expresses the conserved current corresponding to $v$ as in theorem , the second constrains $v$ to be a Hamiltonian vector field with respect to the presymplectic current.

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 preserves 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,

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

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

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

In this symplectic formulation this is immediate, because

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

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

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

The term on the right is in the form in which the conserved quantity obtained from Noether’s theorem is traditionally written (using that, given a Lagrangian $L$, we have $p_i = \frac{\delta L}{\delta (\dot q^i)}$).

Examples

Energy-momentum current

• energy-momentum tensor– see at geometry of physics – A first idea of quantum field theory this example

Dirac current

• Dirac current– see at geometry of physics – A first idea of quantum field theory this example

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). (gdz)

Lecture notes include

• Chapter 7, Noether’s theorem (pdf)

• Máximo Bañados, Ignacio A. Reyes, A short review on Noether’s theorems, gauge symmetries and boundary terms, for students (arXiv:1601.03616)

• A first idea of quantum field theory – Symmetries (PF Insights version)

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, in Physical Theory and its Interpretation,

  The Western Ontario Series in Philosophy of Science Volume 72, 2006, pp 43-100(2006) (pdf)

Monographs:

• 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, vol. 182 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1999. (pdf)

• Gennadi Sardanashvily, Noether’s Theorems: Applications in Mechanics and Field Theory, Studies in Variational Geometry (2016) [doi:10.2991/978-94-6239-171-0]

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 Hamiltonian 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)

Discussion in the context of the variational bicomplex includes

• Glenn Barnich, Friedemann Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl.Phys.B633:3-82,2002 (arXiv:hep-th/0111246)

Discussion of a generalization to discrete groups of symmetries includes

• Anthony Ashton, Conservation laws and non-Lie symmetries for linear PDEs, Journal of Non-linear Mathematical Physics, 2013 (web)

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)

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

• Urs Schreiber, Classical field theory via Cohesive homotopy types (2013)

A formalization of invariance of Lagrangians in parametric dependent type theory is discussed in

• Robert Atkey, From Parametricity to Conservation Laws, via Noether’s Theorem, talk at Principles of Programming Languages (POPL) 2014 (pdf article, web slides, pdf slides)

Discussion in the convenient context of smooth sets:

• Grigorios Giotopoulos, Hisham Sati, §6 in: Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]