nLab
conserved current

The 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 $d=d_{\mathrm{dR}}$ 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\} \,.

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

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 \,.

Definition

For $\Sigma \hookrightarrow X$ a submanifold of dimension $n-1$, the charge of the conserved current $j$ with respect to $\Sigma$ is the integral

$$Q_{\Sigma }:={\int }_{\Sigma }j.$$Q_\Sigma := \int_\Sigma j \,.

Properties

Proposition

If $\Sigma ,\Sigma \prime \subset X$ are homolous, the associated charge is the same

$$Q_{\Sigma }=Q_{\Sigma \prime }.$$Q_{\Sigma} = Q_{\Sigma'} \,.

Proof

By Stokes' theorem.

Theorem

Every symmetry induces a conserved current.

This is Noether's theorem. See there for more details.

Related concepts

• conservation law

• classical observable

• Noether theorem

• charge

References

A general discussion as above is around definition 9 of

• G. J. Zuckerman, Action Principles and Global Geometry , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259–284. (pdf)

The relation of conserved currents to moment maps in symplectic geometry is highlighted for instance in

• Huijun Fan, Lecture 8, Moment map and symplectic reduction (pdf)