# What is the reason for the 'electromagnetism terminology' when discussing the conserved quantities found through Noether's theorem?

In (theoretical) physics, it is customary to describe the system under consideration in terms of the Lagrangian. One of the major advantages of this approach is that it allows us to analyze the symmetries of the system in a systematic way. Arguably the most important result in this context is what is known as Noether's theorem: For any continuous symmetry of the Lagrangian, we have what is known as a 'conserved current'. Mathematically speaking:

$$\delta \mathcal L=0 \Leftrightarrow \partial_\mu j^\mu=0$$

where $j^\mu$ is the conserved current associated with this symmetry of the Lagrangian, defined by

$$j^\mu= \frac{\partial \mathcal L}{\partial (\partial_\mu \phi)}\delta\phi$$

or some variation/generalization thereof. The zero-component $j^0$ is conventionally called the charge.

My question is: Where did this terminology (charge, current) have its origins? Can something meaningful be said about this, beyond the fact that we can use the often-cited example of a complex scalar field to demonstrate that something similar to classical (electromagnetic) current and charge is conserved due to Noether's theorem? As an aside, any historically oriented reviews of the development of Noether's theorem would also be much appreciated.

## 1 Answer

At the time Emmy Noether formulated her theorems (1910-s) electromagnetism was the only fundamental field theory with well developed terminology. But even in the electromagnetism itself the word current was borrowed earlier from analogies with fluids, and probably would have spread into other theories for the same reason. After Einstein's work on relativity matter fields (elasticity, fluids) were far from the center of attention, and for practical reasons no one studied currents in classical gravity. So charge, rather than mass, was a natural choice.

Noether herself was focused on conservation of energy and its apparent failure in general relativity discovered by Klein. Hilbert conjectured that there is a distinction between "proper" conservation laws as in electromagnetism, and "improper" ones as in general relativity. Noether not only proved this in great generality, but supplied a link to symmetry groups. Conservation is proper when the symmetry group has at most countably many independent infinitesimal generators, and "improper" when it has a continuum of them as in general relativity. Klein, by the way called energy current "Hilbert's energy vector", but Noether's theorems were general so this wasn't a good name.

According to Gursey "Before Noether's Theorem the principle of conservation of energy was shrouded in mystery, leading to the obscure physical systems of Mach and Ostwald. Noether's simple and profound mathematical formulation did much to demystify physics." More details on historical circumstances of Noether's work can be found here.