# Conserved current

In physics a conserved current is a current, ${\displaystyle j^{\mu }}$, that satisfies the continuity equation ${\displaystyle \partial _{\mu }j^{\mu }=0}$. The continuity equation represents a conservation law, hence the name.

Indeed, integrating the continuity equation over a volume ${\displaystyle V}$, large enough to have no net currents through its surface, leads to the conservation law

${\displaystyle {\frac {\partial }{\partial t}}Q=0\;,}$
where ${\textstyle Q=\int _{V}j^{0}dV}$ is the conserved quantity.

In gauge theories the gauge fields couple to conserved currents. For example, the electromagnetic field couples to the conserved electric current.

## Conserved quantities and symmetries

Conserved current is the flow of the canonical conjugate of a quantity possessing a continuous translational symmetry. The continuity equation for the conserved current is a statement of a conservation law. Examples of canonical conjugate quantities are:

Conserved currents play an extremely important role in theoretical physics, because Noether's theorem connects the existence of a conserved current to the existence of a symmetry of some quantity in the system under study. In practical terms, all conserved currents are the Noether currents, as the existence of a conserved current implies the existence of a symmetry. Conserved currents play an important role in the theory of partial differential equations, as the existence of a conserved current points to the existence of constants of motion, which are required to define a foliation and thus an integrable system. The conservation law is expressed as the vanishing of a 4-divergence, where the Noether charge forms the zeroth component of the 4-current.

## Examples

### Electromagnetism

The conservation of charge, for example, in the notation of Maxwell's equations,

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0}$

where

• ρ is the free electric charge density (in units of C/m3)
• J is the current density
${\displaystyle \mathbf {J} =\rho \mathbf {v} }$
with v as the velocity of the charges.

The equation would apply equally to masses (or other conserved quantities), where the word mass is substituted for the words electric charge above.

### Complex scalar field

The Lagrangian density

${\displaystyle {\mathcal {L}}=\partial _{\mu }\phi ^{*}\,\partial ^{\mu }\phi +V(\phi ^{*}\,\phi )}$
of a complex scalar field $\phi :\mathbb {R} ^{n+1}\mapsto \mathbb {C}$ is invariant under the symmetry transformation
${\displaystyle \phi \mapsto \phi '=\phi \,e^{i\alpha }\,.}$
Defining $\delta \phi =\phi '-\phi$ we find the Noether current
${\displaystyle j^{\mu }:={\frac {d{\mathcal {L}}}{d(\partial _{\mu })\phi }}\,{\frac {d(\delta \phi )}{d\alpha }}{\bigg |}_{\alpha =0}+{\frac {d{\mathcal {L}}}{d(\partial _{\mu })\phi ^{*}}}\,{\frac {d(\delta \phi ^{*})}{d\alpha }}{\bigg |}_{\alpha =0}=i\,\phi \,(\partial ^{\mu }\phi ^{*})-i\,\phi ^{*}\,(\partial ^{\mu }\phi )}$
which satisfies the continuity equation.