Group theory and the Standard Model gauge group - 4.4.1
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In this video we will take a look at group theory which is a cornerstone in building fundamental theories in physics. This might be a concept you are unfamiliar with, but the essence is not very complicated. It is based on symmetries, and as it turns out symmetry is very important in particle physics. These symmetries form symmetry groups, and actually the structure of the standard model comes from a certain collection of symmetry groups. In this video we will take a look at these symmetry groups and discuss what a group is. To do this we will firstly discuss some simple groups, before considering the slightly more complicated groups of the standard model. But before we start talking about groups, we should first take a look at what symmetries are.
The first step for us would be to recall what symmetries are. To do this let us consider and equilateral triangle. This is just a triangle where all the sides are equally long, and likewise this means each corner has the same angle. This triangle has 3 symmetry lines. These lines run from each corner to the opposite midpoint. Thus, if you put a mirror along these symmetry lines, the mirror imagine would complete the triangle correctly. Similarly, if you rotated the triangle 120 degrees you would end up with the same triangle. This is only possible when the object has symmetry. Thus, if you took this triangle and rotated it 120 degrees, nothing would change. Only if you labeled the corners would you realize that something changed. This is what group theory is about, and this equilateral triangle is described by a so-called group called the D_3 group. A group is formed by two main things. Firstly, the different ways we can configure our object. Thus, in the case of the triangle it would mean all the ways we could mirror and rotate the triangle to obtain a similarly looking triangle. Secondly, we need an operation to get from one configuration to another, and in the case of this triangle it could be rotations or mirroring. There are also some more details to what a group is.