What is a Noether charge?

A Noether charge refers to a physical quantity that is conserved due to the presence of continuous symmetries in a physical system, as defined by Noether's theorem. This theorem establishes a profound connection between symmetries and conservation laws in physics. For example, a system that is invariant under time translation (meaning the laws of physics remain the same over time) will conserve energy. Similarly, spatial translational symmetry leads to the conservation of momentum, while rotational symmetry results in the conservation of angular momentum.

The significance of a Noether charge lies in its role in identifying conserved quantities in various physical theories, particularly in classical mechanics and field theories. Each continuous symmetry corresponds to a different conservation law, making Noether charges extremely important in understanding the fundamental behaviors of physical systems.