20.4 Representations of N × K , N commutative

20.4 Representations of N × K , N commutative

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · irreducible representation · Lie group

The representation theory of semi-direct products $N⋊K$ will in general be rather complicated. However, when $N$ is commutative things simplify considerably, and in this section we’ll survey some of the general features of this case. The special cases of the Euclidean groups in 2 and 3 dimensions were covered in chapter 19 and the Poincar´e group case will be discussed in chapter 42.

For a general commutative group $N$, one does not have the simplifying feature of the Heisenberg group, the uniqueness of its irreducible representation. On the other hand, while N will have many irreducible representations, they are all one dimensional. As a result, the set of representations of N acquires its own group structure, also commutative, and one can define:

Definition (Character group)

For N a commutative group, let $\hat{N}$ be the set of characters of N, $i$.e., functions

$$\alpha :N\to \mathbf{C}$$

that satisfy the homomorphism property

$$\alpha (n_1{n}_2)=\alpha (n_1)\alpha (n_2)$$

The elements of $\hat{N}$ form a group, with multiplication

$$({\alpha }_1{\alpha }_2)(n)={\alpha }_1(n){\alpha }_2(n)$$

When $N$ is a Lie group, we will restrict attention to characters that are diferentiable functions on $N$. We only will actually need the case $N={\mathbf{R}}^d.$ , where we have already seen that the diferentiable irreducible representations are one dimensional and given by

$${\alpha }_{\mathbf{p}}(\mathbf{a})=e^{i\mathbf{p}\cdot \mathbf{a}}$$

where $\mathbf{a}\in N$ . So the character group in this case is $\hat{N}={\mathbf{R}}^d$ , with elements labeled by the vector $p$.

For a semi-direct product $N\times K$ , we will have an automorphism ${\Phi }_k$ of $N$ for each $k\in K$ . From this action on $N.$ , we get an induced action on functions on $N_{:}$ , in particular on elements of $\hat{N}$ , by

$${\hat{\Phi }}_k:\alpha \in \hat{N}\to {\hat{\Phi }}_k(\alpha )\in \hat{N}$$

where ${\hat{\Phi }}_k(\alpha )$ is the element of $\hat{N}$ satisfying

$${\hat{\Phi }}_k(\alpha )(n)=\alpha ({\Phi }_k^{-1}(n))$$

For the case of $N={\mathbf{R}}^d$ , we have

$${\hat{\Phi }}_k({\alpha }_{\mathbf{p}})(\mathbf{a})=e^{i\mathbf{p}\cdot {\Phi }_k^{-1}(\mathbf{a})}=e^{i({\Phi }_k^{-1}{)}^T(\mathbf{p})\cdot \mathbf{a}}$$

so

$${\hat{\Phi }}_k({\alpha }_{\mathbf{p}})={\alpha }_{({\Phi }_k^{-1}{)}^T(\mathbf{p})}$$

When $K$ acts by orthogonal transformations on $N={\mathbf{R}}^d,{\Phi }_k^T={\Phi }_k^{-1}$ so

$${\hat{\Phi }}_k({\alpha }_{\mathbf{p}})={\alpha }_{{\Phi }_k(\mathbf{p})}$$

To analyze representations $(\pi ,V)$ of $N\times K$ , one can begin by restricting attention to the N action, decomposing V into subspaces $V_{\alpha }$ where $N$ acts according to $\alpha$. $v\in V$ is in the subspace $V_{\alpha }$ when

$$\pi (n,1)v=\alpha (n)v$$

Acting by $K$ will take this subspace to another one according to

Theorem

Theorem.

$$v\in V_{\alpha }⟹\pi (0,k)v\in V_{{\hat{\Phi }}_k(\alpha )}$$

Proof. Using the definition of the semi-direct product in chapter 18 one can show that the group multiplication satisfies

$$(0,k^{-1})(n,1)(0,k)=({\Phi }_{k^{-1}}(n),1)$$

Using this, one has

$$\begin{array}{rl}\pi (n,1)\pi (0,k)v& =\pi (0,k)\pi (0,k^{-1})\pi (n,1)\pi (0,k)v\\ & =\pi (0,k)\pi ({\Phi }_{k^{-1}}(n),1)v\\ & =\pi (0,k)\alpha ({\Phi }_{k^{-1}}(n))v\\ & ={\hat{\Phi }}_k(\alpha )(n)\pi (0,k)v\end{array}$$

For each $\alpha \in \hat{N}$ one can look at its orbit under the action of K by ${\hat{\Phi }}_k$ , which will give a subset ${\mathcal{O}}_{\alpha }\subset \hat{N}$ . From the above theorem, we see that if $V_{\alpha }\ne 0.$ then we will also have $V_{\beta }\ne 0$ for $\beta \in {\mathcal{O}}_{\alpha },$ so one piece of information that characterizes a representation $V$ is the set of orbits one gets in this way.

$\alpha$ also defines a subgroup $K_{\alpha }\subset K$ consisting of group elements whose action on $\hat{N}$ leaves $\alpha$ invariant:

Definition (Stabilizer group or little group)

The subgroup $K_{\alpha }\subset K$ of elements $k\in K$ such that

$${\hat{\Phi }}_k(\alpha )=\alpha$$

for a given $\alpha \in \hat{N}$ is called the stabilizer subgroup (by mathematicians) or little subgroup (by physicists).

The group $K_{\alpha }$ will act on the subspace $V_{\alpha }$ , and this representation of $K_{\alpha }$ is a second piece of information that can be used to characterize a representation.

In the case of the Euclidean group $E(2)$ we found that the non-zero orbits ${\mathcal{O}}_{\alpha }$ were circles and the groups $K_{\alpha }$ were trivial. For $E(3)$ , the non-zero orbits were spheres, with $K_{\alpha }$ an $SO(2)$ subgroup of $SO(3)$ (one that varies with $\alpha$). In these cases we found that our construction of representations of $E(2)$ or $E(3)$ on spaces of solutions of the single-component Schr¨odinger equation corresponded under Fourier transform to a representation on functions on the orbits ${\mathcal{O}}_{\alpha }.$ We also found in the $E(3)$ case that using multiple-component wavefunctions gave new representations corresponding to a choice of orbit ${\mathcal{O}}_{\alpha }$ and a choice of irreducible representation of $K_{\alpha }=SO(2)$ . We did not show this, but this construction gives an irreducible representation when a single orbit ${\mathcal{O}}_{\alpha }$ occurs (with a transitive $K$ action), with an irreducible representation of $K_{\alpha }$ on $V_{\alpha }$

We will not further pursue the general theory here, but one can show that distinct irreducible representations of $N\times K$ will occur for each choice of an orbit ${\mathcal{O}}_{\alpha }$ and an irreducible representation of $K_{\alpha }$ . One way to construct these representations is as the solution space of an appropriate wave equation, with the wave equation corresponding to the eigenvalue equation for a Casimir operator. In general, other “subsidiary conditions” then must be imposed to pick out a subspace of solutions that gives an irreducible representation of $N⋊K;$ this corresponds to the existence of other Casimir operators. Another part of the general theory has to do with the question of the unitarity of representations produced in this way, which will require that one starts with an irreducible representation of $K_{\alpha }$ that is unitary.

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