# Why are covering groups more 'fundamental'?

So I understand that the Lie algebra of $$SO(1,3)$$ is isomorphic to the Lie algebra of $$SU(2)\oplus SU(2)$$, and the Lie algebra of $$SO(3)$$ is isomorphic to one copy of $$SU(2)$$ (at the group level we have $$SO(3)\cong SU(2)/Z_2$$).

I guess my question is in two parts:

1. In ”Physics from Symmetry” by Schwichtenberg, he says that the covering group is more fundamental - i.e. by considering $$su(2)$$ instead of $$SO(3)$$ we get spin.

2. As far as I am aware, if we take the exponential map of a Lie algebra homomorphism/representation then we get a representation of the group. This would imply $$SO(3)$$ is a representation of $$SU(2)$$, but how can there exist a homomorphism from $$SU(2)$$ to $$SO(3)$$, if each element in $$SU(2)$$ gets identified with two elements of $$SO(3)$$?

• The homomorphism (isomorphism) only exists around the unit element of the group. Dec 26 '19 at 9:48
• This Q&A of mine discusses in technical detail how quantum mechanics leads to the usage of covering groups. Dec 26 '19 at 12:18
• You want to be careful here. The complexification of $SO(3,1)$ is isomorphic to $SU(2)\oplus SU(2)$. Dec 26 '19 at 14:38

$$SO(3)$$ irreps (irreducible representations) naturally only contain spin $$0,1,2,\dots$$

$$SU(2)$$ introduces half integer irreps.

The fundamental representation of $$SO(3)$$ is a $$\bf 3$$, consisting of the rotations around the three axes:

$$\left(\begin{array} {c c c} \cos(\phi)& -\sin(\phi) &0\\ \sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1\end{array}\right)$$

$$\left(\begin{array} {c c c} \cos(\psi)& 0 & \sin(\psi) \\ 0 & 1 & 0 \\-\sin(\psi) & 0 & \cos(\psi) \end{array}\right)$$

$$\left(\begin{array} {c c c} 1 & 0 & 0 \\ 0 & \cos(\chi)& -\sin(\chi) \\ 0 & \sin(\chi) & \cos(\chi) \end{array}\right)$$

It corresponds to $$j=1$$, the spin-1-representation. Tensor products generate all other representations with $$j\in \mathbb{N}$$.

$$SU(2)$$ is more fundamental, since it contains besides representations with $$j\in \mathbb{N}$$ also half-integer spins $$j\in \frac{\mathbb{N}}{2}$$.

The fundamental representation $$\bf 2$$ of $$SU(2)$$ is represented by the $$2\times 2$$-complex-valued matrices, obtained by exponentiating the Pauli-spin matrices. It corresponds to $$j=\frac{1}{2}$$. From it, all integer and half-integer representations can be generated.

• With help of those three matrices we can safely recalculate a given vector of any length from one reference frame to another. In order to speak of spin, we must deal with wave functions, not just with vectors in 3D space. Dec 29 '19 at 17:45

For myself, I answered part 1 of your question empirically. There are objects (eg: electrons) in the world which only return to their original state after rotation by $$4\pi$$ (not $$2\pi$$). These "half integer spin" objects also have an even number of states (ie: z projections of angular momentum). No representations of $$SO(3)$$ provide these properties, but representations of $$SU(2)$$ do and also provide representations to rotate all the integer spin particles too. Thus $$SU(2)$$ more completely represents nature (ie: is more fundamental).