Tensor Product Representations

Tensor products describe how two transformation types combine. For angular representations, their decomposition gives the familiar rules for adding angular momenta. Clebsch–Gordan coefficients express that decomposition after bases have been chosen.

Tensor-Product Representation

Let \(V_1\) and \(V_2\) carry representations \(\rho_1\) and \(\rho_2\) of the same group \(G\). The tensor-product representation is defined by

\[ (\rho_1\otimes\rho_2)(g) \left( |\psi_1\rangle\otimes|\psi_2\rangle \right) = \rho_1(g)|\psi_1\rangle \otimes \rho_2(g)|\psi_2\rangle, \]

and extended linearly to \(V_1\otimes V_2\).

After bases are selected, the same action is represented by the Kronecker product

\[ D_{1\otimes2}(g) = D_1(g)\otimes D_2(g). \]

Product Groups and Diagonal Restrictions

The definition above assumes that both factors carry representations of the same group and that the same element \(g\) acts on both. This condition is what later makes Clebsch–Gordan decomposition possible.

If \(V_\ell^{\mathrm{space}}\) carries a spatial rotation \(R\) and \(V_s^{\mathrm{spin}}\) independently carries a spin rotation \(S\), their tensor product instead represents the product group \(\mathrm{SO}(3)_{\mathrm{space}}\times\mathrm{SO}(3)_{\mathrm{spin}}\) through

\[ (R,S) \longmapsto \mathcal D^\ell(R)\otimes\mathcal D^s(S). \]

The resulting type \((\ell,s)\) does not decompose by ordinary angular-momentum addition: its two factors are acted on by independently chosen group elements. If the action is restricted to the diagonal subgroup \(S=R\), the same vector space becomes a representation of one \(\mathrm{SO}(3)\) and then decomposes as \(\ell\otimes s\). This distinction is developed physically in Spatial and Spin Symmetries.

Angular-Momentum Coupling

For two irreducible representations of \(\mathrm{SO}(3)\),

\[ V_{\ell_1}\otimes V_{\ell_2} \cong \bigoplus_{L=|\ell_1-\ell_2|}^{\ell_1+\ell_2} V_L . \]

Here \(\cong\) means isomorphic as \(\mathrm{SO}(3)\) representations: there is an invertible linear map between the two vector spaces that commutes with every rotation. After bases are chosen, the Clebsch–Gordan change of basis realizes this isomorphism.

Thus the allowed total angular momentum satisfies

\[ |\ell_1-\ell_2| \le L \le \ell_1+\ell_2. \]

Every allowed \(V_L\) occurs once.

For \(\mathrm{O}(3)\), parity multiplies:

\[ (\ell_1,p_1)\otimes(\ell_2,p_2) \cong \bigoplus_{L=|\ell_1-\ell_2|}^{\ell_1+\ell_2} (L,p_1p_2) . \]

The triangle rule and \(p=p_1p_2\) are the basis-independent selection rules.

Clebsch–Gordan Coefficients

Choose angular-momentum bases

\[ |\ell_1m_1\rangle, \qquad m_1=-\ell_1,\ldots,\ell_1, \]

\[ |\ell_2m_2\rangle, \qquad m_2=-\ell_2,\ldots,\ell_2. \]

Their tensor products form the uncoupled basis:

\[ |\ell_1m_1;\ell_2m_2\rangle \equiv |\ell_1m_1\rangle\otimes|\ell_2m_2\rangle. \]

The decomposition into irreps instead gives the coupled basis

\[ |LM\rangle, \qquad M=-L,\ldots,L. \]

The Clebsch–Gordan coefficients are the overlaps between these two bases:

\[ \boxed{ C^{LM}_{\ell_1m_1,\ell_2m_2} \equiv \langle \ell_1m_1;\ell_2m_2 | LM \rangle }. \]

They give the change of basis

\[ \boxed{ |LM\rangle = \sum_{m_1,m_2} C^{LM}_{\ell_1m_1,\ell_2m_2} |\ell_1m_1;\ell_2m_2\rangle }. \]

The inverse relation is

\[ |\ell_1m_1;\ell_2m_2\rangle = \sum_{L,M} \left( C^{LM}_{\ell_1m_1,\ell_2m_2} \right)^* |LM\rangle. \]

The first relation embeds the coupled basis into the tensor-product space. In the opposite direction, if

\[ |x\rangle = \sum_{m_1,m_2} x_{m_1m_2} |\ell_1m_1;\ell_2m_2\rangle, \]

then its coupled components are

\[ x_{LM} = \langle LM|x\rangle = \sum_{m_1,m_2} \left( C^{LM}_{\ell_1m_1,\ell_2m_2} \right)^* x_{m_1m_2}. \]

In a real orthonormal basis the CG coefficients are real.

In this conventional complex angular-momentum basis, the coefficient vanishes unless

\[ |\ell_1-\ell_2| \le L \le \ell_1+\ell_2, \qquad M=m_1+m_2. \]

Under a rotation, each fixed-\(L\) block transforms independently:

\[ \left[ \rho_{\ell_1}(R)\otimes\rho_{\ell_2}(R) \right] |LM\rangle = \sum_{M'} |LM'\rangle \mathcal D^L_{M'M}(R). \]

The numerical CG coefficients depend on the bases, phases, and normalization used for all three irreps. A Wigner \(\mathcal D\) matrix and a CG table can be combined only when their conventions agree. The corresponding changes of spherical basis are described in Spherical Harmonics.

Wigner \(3j\) symbols encode the same coupling information with a different arrangement, phase, and normalization. They should not be identified numerically with CG coefficients without specifying the convention.

Linear Maps and Operators

A linear map from \(V_{\mathrm{in}}\) to \(V_{\mathrm{out}}\) belongs to

\[ \operatorname{Hom} \left( V_{\mathrm{in}},V_{\mathrm{out}} \right) \cong V_{\mathrm{out}}\otimes V_{\mathrm{in}}^*. \]

Its induced transformation is

\[ L' = \rho_{\mathrm{out}}(g) L \rho_{\mathrm{in}}(g)^{-1} . \]

For \(L:V_\ell\to V_{\ell'}\), its matrix elements in angular-momentum bases are

\[ L_{\ell'm',\ell m} \equiv \langle \ell'm' |L| \ell m \rangle. \]

For an operator on one representation space,

\[ L'=\rho_V(g)L\rho_V(g)^{-1}. \]

If the representation is unitary, \(\rho_V(g)^{-1}=\rho_V(g)^\dagger\). This is the starting point for the transformation of Hamiltonians, density operators, and their angular blocks. The construction is applied to atom-centered matrix blocks in LCAO Operators.