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In modern terminology, the conserved quantity is called the Noether charge, while the flow carrying that charge is called the Noether current. The Noether current is defined up to a solenoidal (divergenceless) vector field. In the context of gravitation, Felix Klein's statement of Noether's theorem for action I stipulates for the invariants: [8]
In physics a conserved current is a current, , that satisfies the continuity equation =.The continuity equation represents a conservation law, hence the name. Indeed, integrating the continuity equation over a volume , large enough to have no net currents through its surface, leads to the conservation law =, where = is the conserved quantity.
The axial current resulting from the motion of an axially charged moving particle is formally defined as = ¯, where is the particle field represented by Dirac spinor (since the particle is typically a spin-1/2 fermion) and and are the Dirac gamma matrices.
Associated to Lorentz invariance is a conserved Noether current, or rather a tensor of conserved Noether currents (). Similarly, since the equation is invariant under translations, there is a tensor of conserved Noether currents T μ ν {\displaystyle T^{\mu \nu }} , which can be identified as the stress-energy tensor of the theory.
The net current into a volume is = where S = ∂V is the boundary of V oriented by outward-pointing normals, and dS is shorthand for NdS, the outward pointing normal of the boundary ∂V. Here J is the current density (charge per unit area per unit time) at the surface of the volume. The vector points in the direction of the current.
Noether's theorem implies that there is a conserved current associated with translations through space and time; for details see the section above on the stress–energy tensor in special relativity. This is called the canonical stress–energy tensor.
More generally, a Ward–Takahashi identity is the quantum version of classical current conservation associated to a continuous symmetry by Noether's theorem. Such symmetries in quantum field theory (almost) always give rise to these generalized Ward–Takahashi identities which impose the symmetry on the level of the quantum mechanical amplitudes.
The thing that "flows" in the current is the "charge", the charge is the generator of the (local) symmetry group. This charge is sometimes called the Noether charge. Thus, for example, the electric charge is the generator of the U(1) symmetry of electromagnetism. The conserved current is the electric current.