Noether’s theorem states that For every continuous symmetry of the action, there is a corresponding conserved quantity.
- Continuous symmetry → a transformation that can be done smoothly, not discrete.
- Action → integral of the Lagrangian over time.
| Symmetry | Conserved quantity |
|---|---|
| Time translation (Lagrangian doesn’t depend explicitly on time) | Energy |
| Space translation (Lagrangian doesn’t depend on position) | Momentum |
| Rotational symmetry (Lagrangian doesn’t change under rotations) | Angular momentum |
| Global phase rotation of a complex field (\psi \to e^{i\alpha} \psi) | Electric charge |
We know the first three clearly but not explore deep on the fourth: Charge conservation is a direct consequence of the invariance of the Lagrangian under global phase rotations of the electron field. They are all unified under the general Noether’s theorum, let’s look into how the general Noether formula itself is derived
So J is general term even we call it current, it’s current of conserved quantity, i.e. energy current, momentum current and charge current.
In Dirac field (electron is in), it’s pointing specific to electric current we usually refer to:
