Conserved current

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In physics a conserved current is a current, j^\mu, that satisfies the continuity equation \partial_\mu j^\mu=0. The continuity equation represents a conservation law, hence the name.

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

 {\partial\over\partial t}Q=0\;,

where Q=\int_V j^0dV 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[edit]

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 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.

Conserved currents in electromagnetism[edit]

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


\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:

J =   \rho 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.