Morley's trisector theorem: Difference between revisions

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In [[physics]] a '''conserved current''' is a current, <math>j^\mu</math>, that satisfies the [[continuity equation]] <math>\partial_\mu j^\mu=0</math>. The continuity equation represents a conservation law, hence the name.
 
Indeed, integrating the continuity equation over a volume <math>V</math>, large enough to have no net currents through its surface, leads to the conservation law
:<math> {\partial\over\partial t}Q=0\;,</math>
where <math>Q=\int_V j^0dV</math> is the conserved quantity.
 
In [[gauge theory|gauge theories]] the gauge fields couple to conserved currents. For example, the [[electromagnetic field]] couples to the [[charge conservation|conserved electric current]].
 
==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]]''.
 
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 [[Electric charge#Conservation of charge|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 [[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 forms the zeroth component of the [[four-current|4-current]].
 
==Conserved currents in electromagnetism==
The ''conservation of charge'', for example, in the notation of [[Maxwell's equations]],
 
:<math>
\frac{\partial \rho} {\partial t} + \nabla \cdot \mathbf{J} = 0
</math>
 
where:
 
ρ is the ''free'' electric charge density (in units of C/m³)
 
'''J''' is the '''current density''':
 
:'''J''' = <math>  \rho </math>'''v'''
 
'''v''' is 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.
 
{{DEFAULTSORT:Conserved Current}}
[[Category:Electromagnetism]]
[[Category:Theoretical physics]]
[[Category:Partial differential equations]]
[[Category:Conservation laws]]
[[Category:Symmetry]]

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