Symmetry Currents
Noether's theorem uses symmetry to create conserved currents.
In my previous post, I talked about the somewhat magical results of Nother's theorem. Noether's theorem implies the existence of a 'current' if there is smooth symmetry in a physical system, like rotational or translational symmetry. This current is a field with a direction and magnitude at every point in space, not unlike how one might visualise an electric current moving through a copper wire. Noether currents are historically denoted by the letter J, and Noether's theorem gives us instructions on how to construct a Noether current from a given symmetry.
The Conservation Law
Most importantly, Noether currents obey a conservation law. In this context, conservation means that that field is neither created nor destroyed, in a very loose sense. If you are wondering what this means, I will explain it in this section. This conservation law places some restrictions on how a field changes in space. Intuitively, the conservation law implies that a current can't have a 'source' or 'sink' property in spacetime. By 'source', we mean that a field can't exhibit behaviour where it looks like it is 'coming out' of empty space. This is similar to the notion of a field not being able to be created out of nowhere.
