special relativity - Idea of Covering Group

1. 
   

   $SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence?

   
   


2. 
   

   I heard that this fact is related to the description of bosons and fermions. But how does it follow from the fact that $SU(2)$ is the double cover of $SO(3)$?

   
   

Answer

Great, important question. Here's the basic logic:

1. 
   

   We start with Wigner's Theorem which tells us that a symmetry transformation on a quantum system can be written, up to phase, as either a unitary or anti-unitary operator on the Hilbert space $\mathcal H$ of the system.

   
   


2. 
   

   It follows that if we want to represent a Lie group $G$ of symmetries of a system via transformations on the Hilbert space, then we must do so with a projective unitary representation of the Lie group $G$. The projective part comes from the fact that the transformations are unitary or anti-unitary "up to phase," namely we represent such symmetries with a mapping $U:G\to \mathscr U(\mathcal H)$ such that for each $g_1,g_2\in G $, there exists a phase $c(g_1, g_2)$ such that \begin{align} U(g_1g_2) = c(g_1, g_2) U(g_1) U(g_2) \end{align} where $\mathscr U(\mathcal H)$ is the group of unitary operators on $\mathcal H$. In other words, a projective unitary representation is just an ordinary unitary representation with an extra phase factor that prevents it from being an honest homomorphism.

   
   


3. 
   

   Working with projective representations isn't as easy as working with ordinary representations since they have the pesky phase factor $c$, so we try to look for ways of avoiding them. In some cases, this can be achieved by noting that the projective representations of a group $G$ are equivalent to the ordinary representations of $G'$ its universal covering group, and in this case, we therefore elect to examine the representations of the universal cover instead.

   
   


4. 
   
   

   In the case of $\mathrm{SO}(3)$, the group of rotations, we notice that its universal cover, which is often called $\mathrm{Spin}(3)$, is isomorphic to $\mathrm{SU}(2)$, and that the projective representations of $\mathrm{SO}(3)$ match the ordinary representations of $\mathrm{SU}(2)$, so we elect to examine the ordinary representations of $\mathrm{SU}(2)$ since it's more convenient.

   
   

This is all very physically important. If we had only considered the ordinary representations of $\mathrm{SO}(3)$, then we would have missed the "half-integer spin" representations, namely those that arise when considering rotations on fermionic systems. So, we must be careful to consider projective representations, and this naturally leads to looking for the universal cover.

Note: The same sort of thing happens with the Lorentz group in relativistic quantum theories. We consider projective representations of $\mathrm{SO}(3,1)$ because Wigner says we ought to, and this naturally leads us to consider its universal cover $\mathrm{SL}(2,\mathbb C)$.