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

• classical observable

References

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)