Irreducible representation and observables

Question

Can any one explain why all observables can be associated with irreducible representation? I do not understand what is the relation between these two.

Answer 1

There is a priori no relation between observables and irreducible representations: an observable is just an observable, and it can connect states in different representations of a group. An example would be the $\hat z$-component of the dipole moment, which can connect states of different angular momentum. The radial position $r$ is also a observable, and it can connect states of different energies.

It is often the cases that a collection of observable transform irreducibly under a group. The three components of the dipole moment transform under rotation as angular momentum $L=1$ operators. In terms of commutation relations, the components of the dipole operators are linear combinations of operators $T^{1}_m\sim r Y^1_m(\theta,\phi)$ with commutation relations \begin{align} [\hat L_\pm, T^1_m]=\sqrt{(1\mp m)(2\pm m)}T^1_{m\pm 1}\, ,\qquad [\hat L_z,T^1_m]=mT^1_m\, . \end{align} The quadrupole moments are a set of operators transforming irreducibly under rotation, this time as $L=2$ operators.

In a finite dimensional space, any observable can be expanded in terms of irreducible tensor operators. For instance, acting on states of total spion $S$, the operator $S_z$ is proportional to \begin{align} S_z\propto \sqrt{\frac{3}{2S+1}} \sum_{m} C^{Sm}_{Sm;10}\vert Sm\rangle \langle Sm\vert\, , \end{align} where $C^{Sm'}_{Sm;LM}$ is a Clebsch-Gordan coefficient.

More generally, one can show that any operator of the type $\vert a\rangle\langle b\vert$ can be expanded in terms of tensor operators, so that any observable acting on states of total spin $S$ built as (Hermitian) combination of the type \begin{align} \sum_{ab} c_{ab} \vert a\rangle\langle b\vert &=\sum_{LM} t_{LM}T^{L}_M \qquad \hbox{where}\\ T^{L}_M&= \sqrt{\frac{2L+1}{2S+1}}\sum_{mm'} C^{Sm'}_{Sm;LM}\vert Sm'\rangle \langle Sm\vert\, . \tag{1} \end{align} The expansion coefficients $t_{LM}$ can be found using the trace orthogonality of the tensors defined in Eq.(1). Details can be found in multiple sources, such as

Varshalovich, Dmitriĭ Aleksandrovich, Anatolï Nikolaevitch Moskalev, and Valerii Kel'manovich Khersonskii. Quantum theory of angular momentum. 1988,

or (for a less complete but more focused treatment)

Klimov, Andrei B., and Sergei M. Chumakov. A group-theoretical approach to quantum optics: models of atom-field interactions. John Wiley & Sons, 2009.

In this case, the matrix elements of the components of a tensor operators are related through the Wigner-Eckart theorem: \begin{align} \langle \alpha \ell ' m'\vert T^L_M \vert\beta \ell m\rangle = \frac{\langle \alpha \ell '\Vert T^L \Vert\beta \ell \rangle}{\sqrt{2\ell'+1}} C^{\ell' m'}_{LM;\ell m} \tag{2} \end{align} where the last factor is a Clebsch-Gordan coefficient, and where $\alpha$ and $\beta$ are whatever other labels are needed to fully label the initial and final states. The reduced matrix element $\langle \alpha \ell '\Vert T^L \Vert\beta \ell \rangle$ is the same for all the components of the tensor.

Eq.(2) states that the ratio of matrix elements of tensor operators between given initial and final states is just a ratio of Clebsch-Gordan coefficients. The selection rules are deduced either from the possible non-zero CGs or from the possible non-zero balues of the reduced matrix element.

The Wigner-Eckart theorem is not limited to angular momentum and associated tensor operators, but in fact can be formulated quite generally.