Editing Conserved current (section)

==Conserved quantities and symmetries==
Conserved current is the flow of the [[canonical conjugate]] of a quantity possessing a [[continuous function|continuous]] [[translational symmetry]]. The [[continuity equation]] for the conserved current is a statement of a ''[[Conservation law (physics)|conservation law]]''. Examples of canonical conjugate quantities are:
*[[Time]] and [[energy]] - the continuous translational symmetry of time implies the [[conservation of energy]]
*[[Space]] and [[momentum]] - the continuous translational symmetry of space implies the [[conservation of momentum]]
*Space and [[angular momentum]] - the continuous ''rotational'' symmetry of space implies the [[conservation of angular momentum]]
*[[Wave function]] [[Phase (waves)|phase]] and [[electric charge]] - the continuous phase angle symmetry of the wave function implies the [[conservation of electric charge]]

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 current]]s, 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 equation]]s, 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 (physics)|charge]] forms the zeroth component of the [[four-current|4-current]].