A symmetry is a physical or mathematical property of a system that is preserved under a certain transformation. They can be of two types:
- Discrete symmetries describe discontinuous changes in the system and are associated with multiplicative operators. For example, a square possesses rotational symmetry. This symmetry is discrete because it only holds when you rotate by 90° and is therefore discontinuous.
- Continuous symmetries describe continuous changes and are associated with continuous operators. For example, a circle has continuous rotational symmetry because no matter how much or little you rotate it, it's still going to be the same circle.
Nöther's theorem#
The greatest value of symmetry in fundamental physics is its connection to one of the most important theorems in physics: Nöther's theorem. In its most general form, it states that all symmetries have a one-to-one relation with a conserved quantity (and its law of conservation). This is a extremely strong statement since conservation laws are by far the most important tools in physics, as they whenever they hold, they allow discriminating between possible and impossible phenomena simply by checking if the conservation is violated.
In particle physics in particular, conservation laws are everything. Particle processes are allowed and disallowed on behalf of these laws. But since Nöther proved that these are all connected to a symmetry, then symmetries themselves must be of paramount importance.
While there are numerous symmetries, there are three which are critical to the cosmology of the Universe: parity, charge conjugation and time reversal.