# Spinors - Groups and Double Cover of Lorentz Group

As part of a project, I keep coming across a small nit-picking area regarding the spinor group $$SU(2)$$. The Lorentz group can be thought of as the group of rotations in $$SO(1,3)$$. I am under the impression that the spinors are a double cover of this Lorentz group. I keep getting confused as to whether we have the relation:

$$SU(2) \times SU(2) = SO(1,3)$$ or $$SU(2) \times SU(2) = SO(3)$$

Which case is correct and why?

Both of your isomorphisms are false. For instance, $$SU(2)\times SU(2)$$ is a compact group, while $$SO(1,3)$$ is a noncompact group. But, at least the two groups share the dimension (six). In contrast, $$SO(3)$$ is 3-dimensional, hence, it is not isomorphic to $$SU(2)\times SU(2)$$ either.
There are different spin groups for different Lie groups; thus you cannot talk about the spinor group, but different spinor groups. The setup is that you have a connected Lie group $$G$$ and a distinguished index 2 subgroup in its fundamental group $$\pi_1(G)$$. This data determines a connected Lie group $$\tilde{G}$$ which is admits a 2-fold nontrivial covering $$\tilde{G}\to G$$; by a slight abuse of notation, one says that $$\tilde{G}$$ is the spin group of $$G$$. The choice of the index 2 subgroup is usually canonical, so this notion of a spin-group of $$G$$ is well-defined.
1. The group $$SO_+(1,3)$$ is noncompact, its fundamental group is the same as $$\pi_1(SO(3))\cong {\mathbb Z}_2$$ since $$SO(3)$$ is the maximal compact subgroup of $$SO_+(1,3)$$. Hence, $$SO(1,3)$$ has a unique nontrivial 2-fold covering group called $$Spin(1,3)$$. The more common way to describe this groups is as follows: $$SO_+(1,3)\cong PSL(2, {\mathbb C}),$$ hence, the spin-cover is $$SL(2, {\mathbb C})\cong Spin(1,3)$$.
2. The group $$SU(2)=Spin(3)$$, is the spinor group of $$SO(3)$$.
For other examples: $$G=SO_+(2,1)$$ has infinite cyclic fundamental group; hence, it again has a canonical index 2 subgroup of $$\pi_1(G)$$; the corresponding spin group is $$SL(2, {\mathbb R})$$.