Conserved current

A conserved current has components $j^{\mu }=(\rho ,\mathbf{j})$ satisfying the continuity equation ${\partial }_t\rho +\nabla \cdot \mathbf{j}=0$. It relates local accumulation of a quantity to its transport.

For a fixed volume $V$, the divergence theorem givesThus the integrated charge density, not an arbitrary spatial integral of every current component, is constant when boundary flux vanishes. Electric charge provides one example; a continuous field symmetry can provide a Noether current for another. A measured local density can change while the global charge remains conserved, because current redistributes it across the sample.

An integral of the time component of a conserved current is a conserved charge when the spatial boundary flux vanishes. Charges need not come from an immediately apparent elementary symmetry; a field theory can have an infinite higher-derivative current hierarchy. A Noether charge is obtained from the symmetry current supplied by the Noether theorem.

In a massless four-dimensional theory, current conservation gives a transverse tensor structure, $\int d^{4}x{e}^{-ikx}⟨J_{\mu }(x)J_{\nu }(0)⟩=(k_{\mu }{k}_{\nu }-{\delta }_{\mu \nu }{k}^2)G(k^2)$. With current mass dimension three, the scalar coefficient has dimension zero.