# Conserved current

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{{Short description|Concept in physics and mathematics that satisfies the continuity equation}}
{{inline|date=December 2009}}
In [physics](/source/physics) a '''conserved current''' is a [current](/source/Current_(physics)), <math>j^\mu</math>, that satisfies the [continuity equation](/source/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 display="block"> \frac{\partial}{\partial t}Q = 0\;,</math>where <math display="inline">Q = \int_V j^0 dV</math> is the [conserved quantity](/source/charge_(physics)).

In [gauge theories](/source/gauge_theory) the gauge fields couple to conserved currents. For example, the [electromagnetic field](/source/electromagnetic_field) couples to the [conserved electric current](/source/charge_conservation).

==Conserved quantities and symmetries==
Conserved current is the flow of the [canonical conjugate](/source/canonical_conjugate) of a quantity possessing a [continuous](/source/continuous_function) [translational symmetry](/source/translational_symmetry). The [continuity equation](/source/continuity_equation) for the conserved current is a statement of a ''[conservation law](/source/Conservation_law_(physics))''. Examples of canonical conjugate quantities are:
*[Time](/source/Time) and [energy](/source/energy) - the continuous translational symmetry of time implies the [conservation of energy](/source/conservation_of_energy)
*[Space](/source/Space) and [momentum](/source/momentum) - the continuous translational symmetry of space implies the [conservation of momentum](/source/conservation_of_momentum)
*Space and [angular momentum](/source/angular_momentum) - the continuous ''rotational'' symmetry of space implies the [conservation of angular momentum](/source/conservation_of_angular_momentum)
*[Wave function](/source/Wave_function) [phase](/source/Phase_(waves)) and [electric charge](/source/electric_charge) - the continuous phase angle symmetry of the wave function implies the [conservation of electric charge](/source/conservation_of_electric_charge)

Conserved currents play an extremely important role in [theoretical physics](/source/theoretical_physics), because [Noether's theorem](/source/Noether's_theorem) connects the existence of a conserved current to the existence of a [symmetry](/source/symmetry) of some quantity in the system under study. In practical terms, all conserved currents are the [Noether current](/source/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](/source/partial_differential_equation)s, as the existence of a conserved current points to the existence of [constants of motion](/source/constants_of_motion), which are required to define a [foliation](/source/foliation) and thus an [integrable system](/source/integrable_system). The conservation law is expressed as the vanishing of a 4-[divergence](/source/divergence), where the Noether [charge](/source/Charge_(physics)) forms the zeroth component of the [4-current](/source/four-current).

==Examples==
===Electromagnetism===
The ''conservation of charge'', for example, in the notation of [Maxwell's equations](/source/Maxwell's_equations),<math display="block">\frac{\partial \rho} {\partial t} + \nabla \cdot \mathbf{J} = 0</math>

where
* ρ is the ''free'' electric charge density (in units of C/m<sup>3</sup>)
* '''J''' is the '''current density''' <math display="block"> \mathbf J = \rho \mathbf v </math> 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 [Klein-Gordon Lagrangian density](/source/Klein-Gordon_equation)
<math display="block"> \mathcal{L}=\partial_\mu\phi^*\,\partial^\mu\phi +V(\phi^*\,\phi)</math>
of a complex scalar field <math display> \phi:\mathbb{R}^{n+1}\mapsto\mathbb{C} </math> is invariant under the symmetry transformation<math display="block"> \phi\mapsto\phi'=\phi\,e^{i\alpha}\, . </math> 
Defining <math display> \delta\phi=\phi'-\phi </math> we find the Noether current
<math display="block"> j^\mu:= \frac{d\mathcal{L}}{d \dot{\mathbf{q}}} \cdot \mathbf{Q}_r = \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)</math>which satisfies the continuity equation. Here <math> \mathbf{Q}_r </math> is the generator of the symmetry, which is <math> \frac{d (\delta \mathbf{q})}{d \alpha_r} </math> in the case of a single parameter <math> \alpha </math>.

== See also ==
* [Conservation law (physics)](/source/Conservation_law_(physics))
* [Noether's theorem](/source/Noether's_theorem)

== References ==

*{{cite book |last=Goldstein |first=Herbert |author-link=Herbert Goldstein 
 |year=1980
 |title= [Classical Mechanics](/source/Classical_Mechanics_(Goldstein))
 |edition=2nd |publisher=Addison-Wesley
 |location=Reading, MA
 |isbn= 0-201-02918-9
 |pages=588–596}}

*{{cite book
 | author=David J Griffiths
 | title=Introduction to electrodynamics
 | year=1999
 | edition=Third
 | pages=[https://archive.org/details/introductiontoel00grif_0/page/356 356–357]
 | publisher=Prentice Hall
 | isbn=978-0-13-805326-0
 | url=https://archive.org/details/introductiontoel00grif_0/
 }}

*{{cite book
 |last1=Peskin |first1=Michael E.  
 |last2=Schroeder |first2=Daniel V.
 |year=1995
 |title=An Introduction to Quantum Field Theory
 |isbn= 978-0-201-50397-5 
 |chapter=Chapter I.2.2. Elements of Classical Field Theory
 |publisher=CRC Press
}}
{{DEFAULTSORT:Conserved Current}}
Category:Electromagnetism
Category:Theoretical physics
Category:Conservation equations
Category:Symmetry

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Adapted from the Wikipedia article [Conserved current](https://en.wikipedia.org/wiki/Conserved_current) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Conserved_current?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.