Representations of SO(3) and O(3)

The orthogonal group

\[ \mathrm{O}(3) = \left\{ Q\in\mathbb R^{3\times3}:Q^{\mathsf T}Q=I \right\} \]

contains all orthogonal transformations of three-dimensional Euclidean space. Its subgroup

\[ \mathrm{SO}(3) = \left\{ R\in \mathrm{O}(3):\det R=+1 \right\} \]

contains the proper rotations. This chapter describes their finite-dimensional irreducible representations without yet selecting a complex or real spherical basis.

Irreducible Representations of SO(3)

An angular representation is irreducible when it has no proper nonzero subspace preserved by every rotation. An irreducible space is therefore one angular type that cannot be split into smaller angular types.

Theorem (classification of angular types). Up to equivalence, the finite-dimensional continuous irreducible representations of \(\mathrm{SO}(3)\) are indexed by

\[ \ell=0,1,2,\ldots. \]

The representation space \(V_\ell\) has dimension

\[ \dim V_\ell=2\ell+1. \]

We write its action as

\[ \rho_\ell:\mathrm{SO}(3)\longrightarrow\operatorname{GL}(V_\ell). \]

For an abstract state \(|\psi_\ell\rangle\in V_\ell\), an active rotation acts as

\[ |\psi_\ell'\rangle = \rho_\ell(R)|\psi_\ell\rangle. \]

The ket \(|\psi_\ell\rangle\) denotes a vector in \(V_\ell\) and does not select a basis.

These are the ordinary, single-valued spatial representations of \(\mathrm{SO}(3)\). Half-integer angular momenta are representations of the double cover \(\mathrm{SU}(2)\), not ordinary representations of \(\mathrm{SO}(3)\), and are outside the spatial-tensor discussion here.

Wigner \(\mathcal D\) Matrices

Choose an orthonormal basis

\[ B_\ell = \left\{ |\ell m\rangle \right\}_{m=-\ell}^{\ell} \]

of \(V_\ell\). The Wigner-D matrix is the matrix of the abstract rotation operator in this basis:

\[ \boxed{ \mathcal D^\ell_{m'm}(R) = \langle\ell m'| \rho_\ell(R) |\ell m\rangle }. \]

Inserting the resolution of the identity gives

\[ \boxed{ \rho_\ell(R)|\ell m\rangle = \sum_{m'} |\ell m'\rangle \mathcal D^\ell_{m'm}(R) }. \]

For an abstract state \(|\psi_\ell\rangle\), define its components by

\[ \psi_{\ell m} = \langle\ell m|\psi_\ell\rangle. \]

They transform as

\[ \psi_{\ell m'}' = \sum_m \mathcal D^\ell_{m'm}(R) \psi_{\ell m}. \]

The matrices obey

\[ \mathcal D^\ell(I)=I_{2\ell+1}, \qquad \mathcal D^\ell(R_1R_2) = \mathcal D^\ell(R_1)\mathcal D^\ell(R_2). \]

Thus \(\rho_\ell(R)\) is the basis-independent operator, while \(\mathcal D^\ell_{m'm}(R)\) is its basis-dependent matrix element. The prime on \(m'\) distinguishes the output component from the input component; it does not denote a different basis.

To compare two basis conventions \(A\) and \(B\), write their basis kets as \(|\ell m\rangle_A\) and \(|\ell m\rangle_B\). The overlap matrix

\[ \left(T_\ell^{B\leftarrow A}\right)_{m'm} = {}_B\langle \ell m' | \ell m \rangle_A \]

converts components from basis \(A\) to basis \(B\). The component columns and Wigner matrices are related by

\[ \boldsymbol\psi_\ell^B = T_\ell^{B\leftarrow A} \boldsymbol\psi_\ell^A, \qquad \mathcal D^{\ell,B}(R) = T_\ell^{B\leftarrow A} \mathcal D^{\ell,A}(R) \left( T_\ell^{B\leftarrow A} \right)^{-1} . \]

Consequently, different real-harmonic orderings or phases do not define different angular physics. They define different coordinate forms of the same action. Euler angles likewise parameterize \(R\); they do not define a second rotation law.

The label \(m\) always runs from \(-\ell\) to \(\ell\) in this section. In the conventional complex angular-momentum basis it is the magnetic quantum number. A real basis retains the same labels for its \(2\ell+1\) components, although individual real components are not eigenstates of the same angular momentum generator. The basis, ordering, and phases must still be specified.

Extending from SO(3) to O(3)

Let \(P=-I\) denote spatial inversion. It commutes with every proper rotation, and every \(Q\in\mathrm{O}(3)\) has a unique factorization

\[ Q=P^\epsilon R, \qquad R\in\mathrm{SO}(3), \qquad \epsilon\in\{0,1\}. \]

Here \(\epsilon=0\) for \(\det Q=+1\) and \(\epsilon=1\) for \(\det Q=-1\).

Because inversion is central and satisfies \(P^2=I\), it acts by a sign on an irreducible representation.

Theorem (irreps of \(\mathrm{O}(3)\)). The finite-dimensional continuous irreducible representations of \(\mathrm{O}(3)\) are labeled by

\[ (\ell,p), \qquad \ell=0,1,2,\ldots, \qquad p\in\{+1,-1\}. \]

Their action is

\[ \rho_{\ell,p}(P^\epsilon R) = p^\epsilon\rho_\ell(R) . \]

For a state \(|\psi_{\ell,p}\rangle\) of this type,

\[ |\psi_{\ell,p}'\rangle = p^\epsilon \rho_\ell(R) |\psi_{\ell,p}\rangle. \]

The sign \(p\) is the parity: \(p=+1\) is even under inversion and \(p=-1\) is odd.

Some familiar \(\mathrm{O}(3)\) types are:

Object	\(\mathrm{O}(3)\) type	Inversion
scalar	\((0,+1)\)	unchanged
pseudoscalar	\((0,-1)\)	changes sign
polar vector	\((1,-1)\)	changes sign
axial vector	\((1,+1)\)	unchanged

Thus \(\ell\) alone does not specify an \(\mathrm{O}(3)\) type. Polar and axial vectors carry the same \(\mathrm{SO}(3)\) irrep but different parity.

A reflection is a proper rotation composed with inversion. Its action therefore contains both the Wigner rotation and the parity sign; it is not described by multiplying every component by \(p\) alone.

Spherical Tensors and Irrep Features

Definition (spherical tensor). A spherical tensor of degree \(\ell\) is a quantity \(T^{(\ell)}\in V_\ell\). Under a proper rotation,

\[ {T'}^{(\ell)} = \rho_\ell(R)T^{(\ell)}. \]

After choosing an angular basis, its components

\[ T_m^{(\ell)} = \langle\ell m|T^{(\ell)}\rangle \]

transform as

\[ {T'}_{m'}^{(\ell)} = \sum_{m=-\ell}^{\ell} \mathcal D^\ell_{m'm}(R)T_m^{(\ell)}. \]

The word spherical refers to this transformation law. A spherical tensor need not be a function on the sphere, and its degree \(\ell\) is not the rank of a Cartesian tensor. Spherical harmonics provide one realization of the same irreducible transformation types.

For \(\mathrm{O}(3)\), the angular degree must be supplemented by parity.

Definition (irrep feature). An irrep feature of type \((\ell,p)\) is a value \(x^{(\ell,p)}\in V_{\ell,p}\), the carrier space of that \(\mathrm{O}(3)\) irrep. For \(Q=P^\epsilon R\), its components transform as

\[ \boxed{ {x'}_{m'}^{(\ell,p)} = p^\epsilon \sum_{m=-\ell}^{\ell} \mathcal D^\ell_{m'm}(R)x_m^{(\ell,p)} }. \]

Thus \((\ell,p)\) is the transformation type, \(x^{(\ell,p)}\) is a value carrying that type, and \(m\) labels its \(2\ell+1\) components in a chosen basis. If parity is ignored, an irrep feature of angular degree \(\ell\) is the same object called a spherical tensor above.

A collection may contain several copies of each type:

\[ V = \bigoplus_{\ell,p} \bigoplus_{c=1}^{n_{\ell p}} V_{\ell,p}. \]

The index \(c\) is a copy or channel index, while \(m\) is an angular component. The \(\mathrm{O}(3)\) action applies the same \((\ell,p)\) transformation to every copy and does not mix \(c\). A single \(x_c^{(\ell,p)}\) is one irrep feature; the entire direct sum is a collection of irrep features.

In e3nn, Irrep describes one \((\ell,p)\) type and Irreps describes such a direct sum with multiplicities. For example, 3x0e + 2x1o + 1x2e denotes three even scalars, two odd vectors, and one even degree-\(2\) feature. These descriptors specify how data transform; they do not themselves contain the feature values.