Symmetries

When we think of symmetry, we usually think of how a face or animal is symmetrical from left to right. We could also say that a snowflake is symmetrical under a 60-degree rotation, or that a cube is symmetric over 90-degree rotations. These objects stay the same (or at least appear the same) under some sort of translation. These are examples of discrete symmetries.

​

However, there are also continuous symmetries. An example of continuous symmetry is that a sphere is symmetric under all rotational translations. We can also say that rotational translations are a degree of freedom.

​

Noether's theorem states that for every continuous symmetry, there is a conservation law. If you don't know what a conservation law is, it simply means that some property stays the same no matter what. For example, conservation. For example, electromagnetic charge is a conserved quantity.

​

In Noether's theorem, the thing that stays the same under transformation is the laws of motion. Let's look at the example of a flat, straight, road. Because gravity has the same force under each point of of the road, we can say that gravity is symmetric under spacial translation. As a result, the quantity that is conserved is momentum. If two cars collided on that road, their total momentum would be the same.

 

However, this conservation isn’t present on a hilly road. Because momentum would be lost or gained to the gravitational field. It is not symmetric to translations on that road. However, because the gravitational field doesn't change from one time to another, It is symmetric to time translation It doesn't matter when a collision happens, the results would be the same. This gives another conserved quantity, energy.

​

Symmetries, and conserved quantities, are the cornerstones of physics as they seem to peer into the most fundamental parts of our universe.