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Basis Conventions

Clebsch-Gordan Coefficients

The SU(2) reference implementation uses the Condon-Shortley convention for Clebsch-Gordan (CG) coefficients:

\[C^{j_3 m_3}_{j_1 m_1,\, j_2 m_2} = \langle j_1 m_1,\, j_2 m_2 \mid j_3 m_3 \rangle\]

Their values are computed using the Racah formula, which expresses each CG coefficient as an explicit algebraic sum over factorials of angular momentum quantum numbers.

Required properties:

  1. Real-valued: All CG coefficients are real numbers.
  2. Orthonormality: \(\displaystyle\sum_{m_1, m_2} C^{j_3 m_3}_{j_1 m_1, j_2 m_2}\, C^{j_3' m_3'}_{j_1 m_1, j_2 m_2} = \delta_{j_3 j_3'}\,\delta_{m_3 m_3'}\)
  3. Phase choice: The Condon-Shortley phase convention fixes the relative signs of CG coefficients so that the \(m_1 = j_1\), \(m_2 = j_2 - j_3 + j_1\) coefficient (highest available) is positive.

For SU(N) with \(N > 2\), the analogous reduced matrix elements (Wigner coefficients) must be defined with an analogous real, orthonormal, and positivity-compatible phase convention. The precise generalization is group-specific and must be documented in the conforming implementation.

Canonical Basis Normalization

The canonical basis tensor \(B_\mu\) for OM index \(\mu\) must be normalized so that the set \(\{B_\mu\}\) forms an orthonormal basis of the CG tensor space:

\[\langle B_\mu | B_\nu \rangle \equiv \sum_{m_1, \ldots, m_n} B_\mu(m_1, \ldots, m_n)\, B_\nu(m_1, \ldots, m_n) = \delta_{\mu\nu}\]

3rd-order normalization: For a 3rd-order tensor (\(n = 3\)), the OM dimension is always 1. The single basis element \(B_0\) acquires a normalization factor relative to the bare CG coefficient \(C^{j_3 m_3}_{j_1 m_1, j_2 m_2}\):

\[B_0(m_1, m_2, m_3) = \frac{1}{\sqrt{2j_3 + 1}}\, C^{j_3 m_3}_{j_1 m_1, j_2 m_2}\]

This factor ensures that the squared norm over all magnetic indices is 1.

Canonical Basis Direction

For an \(n\)th-order tensor we define:

  • Leading edges: the first \(n - 1\) external edges
  • Terminal edge: the last external edge

The SU(2) reference implementation builds each basis tensor from a fixed canonical direction: all leading edges incoming and the terminal edge outgoing. This is the reference configuration for which the fusion-tree CG contraction is performed directly.

When canonical_basis is called with a spec whose edge directions deviate from this canonical pattern, the implementation derives the result from the canonical basis by contracting each deviating edge \(k\) (with spin \(j_k\)) with the invariant metric \(g^{(j_k)}\) (see Arrow Conventions — Invariant Metric):

\[B^S_\mu = \left(\prod_{k \,\in\, \text{deviating}(S)} g^{(j_k)}\right) \cdot B^{\text{canonical}}_\mu\]

Each metric contraction on a deviating edge contributes a factor of \(\text{FS}(j_k) = (-1)^{2j_k}\) when the direction is subsequently inverted (bond inversion). The returned tensor already incorporates these metric contractions, so callers receive the correct basis for the given spec regardless of direction.