Noether's Theorem

Noether's theorem states that for every continuous symmetry of a system there is an associated conserved quantity - called a conserved current.

Symmetries

A symmetry of a system is some transformation that we can perform on the system which leaves the dynamics of the system unchanged. This means that if we transform the system under some symmetry transformation and then allow it to evolve under it's own dynamics we should arrive at the same final state as if we had let the original system evolve for the same amount of time and then apply the symmetry transformation. Importantly, this property should hold for the system between all times - arbitrarily small or arbitrarily large.

Conserved Quantities

Noether's Theorem

Noether's theorem simply states that whenever there exists a continuous symmetry of a system there will also be a conserved quantity - the 2 come in pairs.

One of the first observations we made about classical physics was that there is a distinction between those quantities that are objectively measureable and those that are only measureable relative to some reference. In classical physics, the latter of these are position/momentum and the former are the higher order derivatives of position - such as acceleration.

You can feel acceleration. If I were to put you in a perfectly insulating black box, you would still be able to tell me if the box were accelerating or not as you would feel the acceleration inside the box. You would not however be able to tell me: (A) Where the box is or (B) how fast the box is moving. Such questions bear no meaning in our universe; it's only possible to define the position and velocity of something relative to something else. Acceleration, on the other hand, is objective.

Sometimes, when we consider some small part of the universe in isolation, these 2 symmetries are broken. For example, when considering the orbit of the Earth around the Sun, the symmetries of linear position and momentum are broken by virtue of the fact that the Sun gives us a natural reference point by which to compute positions and velocities relative to - namely the position and velocity of the Sun respectively.

However, this symmetry breaking gives rise to a new set of symmetries. While we no longer have a symmetry of linear position, we have a new symmetry in rotational position. As the Earth orbits the sun, there is no objective reference point which we should consider as the zero position of the Earth.