In physics a conserved current is a current, that satisfies the continuity equation . The continuity equation represents a conservation law, hence the name.
Indeed, integrating the continuity equation over a volume, large enough to have no net currents through its surface, leads to the conservation law
where 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.
Read more about Conserved Current: Conserved Quantities and Symmetries, Conserved Currents in Electromagnetism
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