# Summary Let's consider working in 3 spatial and 1 temporal dimension. To rotate quantum mechanical [[spin-half|spin-1/2 particles]], we use the [[Special Unitary Group in 2 Dimensions]] $SU(2)$, a subgroup of the [[Special Linear Group in Two Complex Dimensions]] $SL(2, \mathbb{C})$, which implements [[Lorentz transformation|Lorentz transformations]] on spin-1/2 particles. $SU(2)$ is the [[Relationship Between SO(3) and SU(2)|double cover]] of $SO(3)$, the [[Special Orthogonal Group in 3 Dimensions]], which implements proper rotations. $SO(3)$ itself is a subgroup of the [[Restricted Lorentz Group in 4 Dimensions]] $SO(3,1)_o$, which implements Lorentz transformations. Moreover, $SL(2, \mathbb{C})$ is the [[Relationship Between SO(3,1) and SL(2,C)|double cover]] of $SO(3,1)_o$, a subgroup of the [[Lorentz Group in 4 Dimensions]] $O(3,1)$ which implements improper Lorentz transformations. Finally, $SO(3)$ is a subgroup of the [[Orthogonal Group in 3 Dimensions]] $O(3)$ which implements improper [[Rotation|rotations]]. Finally, $O(3)$ itself is a subgroup of $O(3,1)$. This summarizes the relationship between some of the most common Lie Groups in physics. ![[Pasted image 20210704122804.png]]