8 Representations of SU(2) and SO(3)

For the case of $G=U(1)$ , in chapter 2 we were able to classify all complex irreducible representations by an element of Z and explicitly construct each irreducible representation. We would like to do the same thing here for representations of $SU(2)$ and $SO(3)$ . The end result will be that irreducible representations of $SU(2)$ are classified by a non-negative integer $n=0,1,2,3,\cdots$ and have dimension $n+1,$ , so we’ll (hoping for no confusion with the irreducible representations $({\pi }_n,\mathbf{C})$ of $U(1))$ denote them $({\pi }_n,{\mathbf{C}}^{n+1})$ ). For even n these will correspond to an irreducible representation ${\rho }_n$ of $SO(3)$ in the sense that

$${\pi }_n={\rho }_n\circ \Phi$$

but this will not be true for odd n. It is common in physics to label these representations by $s=\frac{n}{2}=0,\frac{1}{2},1,\cdot \cdot \cdot$ and call the representation labeled by $s$ the “spin $s$ representation”. We already know the first three examples:

• Spin 0: ${\pi }_0$ or ${\rho }_0$ is the trivial representation for $SU(2)$ or $SO(3)$ . In physics this is sometimes called the “scalar representation”. Saying that states transform under rotations as the scalar representation just means that they are invariant under rotations.

• Spin $\frac{1}{2}:$ : Taking

$${\pi }_1(g)=g\in SU(2)\subset U(2)$$

gives the defining representation on ${\mathbf{C}}^2$ . This is the spinor representation discussed in chapter 7. It does not correspond to a representation of $SO(3)$

• Spin 1: Since $SO(3)$ is a group of 3 by 3 matrices, it acts on vectors in ${\mathbf{R}}^3$ This is just the standard action on vectors by rotation. In other words, the representation is $({\rho }_2,{\mathbf{R}}^3)$ , with ${\rho }_2$ the identity homomorphism

$$g\in SO(3)\to {\rho }_2(g)=g\in SO(3)$$

This is sometimes called the “vector representation”, and we saw in chapter 6 that it is isomorphic to the adjoint representation.

Composing the homomorphisms $\Phi$ and $\rho :$

$${\pi }_2=\rho \circ \Phi :SU(2)\to SO(3)\subset GL(3,\mathbf{R})$$

gives a representation $({\pi }_2,{\mathbf{R}}^3)$ of $SU(2)$ , the adjoint representation. Complexifying gives a representation on ${\mathbf{C}}^3$ , which in this case is just the action with $SO(3)$ matrices on complex column vectors, replacing the real coordinates of vectors by complex coordinates.

中文目录

Chapter contents

• 8.1 Representations of SU(2) : classification
  • 8.1.1 Weight decomposition
  • 8.1.2 Lie algebra representations: raising and lowering operators
• 8.2 Representations of SU(2) : construction
• 8.3 Representations of SO(3) and spherical harmonics
• 8.4 The Casimir operator
• 8.5 For further reading

Previous chapter: 7 Rotations and the Spin 1/2 Particle in a Magnetic Field

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