Clebsch–Gordan Coefficients as SU(2) Tensor Product Unitary Transformations — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Clebsch–Gordan coefficients are the unitary transformation matrix elements between uncoupled and coupled angular momentum bases, directly encoding SU(2) tensor product decomposition into irreducible representations. | MATH: For j1 ⊗ j2 = ⊕j (j), the coefficients ⟨ j1 m1 j2 m2 | J M⟩ satisfy orthogonality: ∑m1,m2 ⟨ j1 m1 j2 m2 | J M⟩ ⟨ j1 m1 j2 m2 | J' M'⟩ = δJJ'δMM'. Recursion via ladder operators: √((J∓ M)(J± M+1))⟨ j1 m1 j2 m2 | J, M± 1⟩ = √((j1± m1)(j1∓ m1+1))⟨ j1, m1∓ 1, j2 m2 | J M⟩ + √((j2± m2)(j2∓ m2+1))⟨ j1 m1, j2, m2∓ 1 | J M⟩. Racah formula gives closed form as sum over integers z: \(\langle j_1 m_1 j_2 m_2 | J M\rangle = \delta_{M,m_1+m_2} \sqrt{(2J+1)\frac{(J+j_1-j_2)!(J-j_1+j_2)!(j_1+j_2-J)!}{(J+j_1+j_2+1)!}} \times \sqrt{(J+M)!(J-M)!(j_1+m_1)!(j_1-m_1)!(j_2+m_2)!(j_2-m_2)!}
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026clebschgordan,
title = {{Clebsch–Gordan Coefficients as SU(2) Tensor Product Unitary Transformations — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23616458},
url = {https://zenodo.org/records/23616458},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Clebsch–Gordan Coefficients as SU(2) Tensor Product Unitary Transformations — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23616458
UR - https://zenodo.org/records/23616458
AB - FINDING: Clebsch–Gordan coefficients are the unitary transformation matrix elements between uncoupled and coupled angular momentum bases, directly encoding SU(2) tensor product decomposition into irreducible representations. | MATH: For j_1 ⊗ j_2 = ⊕_j (j), the coefficients ⟨ j_1 m_1 j_2 m_2 | J M⟩ satisfy orthogonality: ∑_m_1,m_2 ⟨ j_1 m_1 j_2 m_2 | J M⟩ ⟨ j_1 m_1 j_2 m_2 | J' M'⟩ = δ_JJ'δ_MM'. Recursion via ladder operators: √((J∓ M)(J± M+1))⟨ j_1 m_1 j_2 m_2 | J, M± 1⟩ = √((j_1± m_1)(j_1∓ m_1+1))⟨ j_1, m_1∓ 1, j_2 m_2 | J M⟩ + √((j_2± m_2)(j_2∓ m_2+1))⟨ j_1 m_1, j_2, m_2∓ 1 | J M⟩. Racah formula gives closed form as sum over integers z: \(\langle j_1 m_1 j_2 m_2 | J M\rangle = \delta_{M,m_1+m_2} \sqrt{(2J+1)\frac{(J+j_1-j_2)!(J-j_1+j_2)!(j_1+j_2-J)!}{(J+j_1+j_2+1)!}} \times \sqrt{(J+M)!(J-M)!(j_1+m_1)!(j_1-m_1)!(j_2+m_2)!(j_2-m_2)!} Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -