What is amazing is that it has been proven mathematically that symmetry is connected to laws of conservation. This has been proven by Emmy Noether.
These are the things she proved:
| • | Time symmetry implies conservation of energy and vice versa. |
| • | Space symmetry implies conservation of momentum and vice versa. |
| • | Rotation symmetry implies conservation of angular momentum and vice versa. |
| • | Rotation symmetry means that if we do an experiment, and turn around and do the experiment, it should be the same. Look at a round ball. It is the same no matter how we turn the ball. This is rotation symmetry. |
When we study electric circuits, we specify the ground. We measure voltages with respect to this ground. If we add a potential to the entire circuit, nothing would change. What matters are the voltages, which are potential differences. Birds can sit on electrical wires without harm, yet if a person on the ground touches the wire he may be killed. The reason is that an electric current passes through the person's body due to the potential difference between the wire and the ground. The bird suffers no harm because the potential difference between the wire for one foot and the other foot is almost zero.
This symmetry, namely, the actual value of the potential, is called gauge symmetry, and it leads to conservation of charge. Also vice versa: conservation of charge implies gauge symmetry.
SEE THE BOOK FOR MORE DETAILS.