Applications of finite- and Lie-group representation theory to the classification of physical states, selection rules, and conservation laws. Clebsch–Gordan decomposition — the tensor product of two irreducible representations of SU(2)…
group-theory-physics
SU(2) CG: (½)⊗(½) = 1 ⊕ 0
Clebsch–Gordan decomposition of tensor products of SU(2) irreducibles. General theorem: j₁⊗j₂ = ⊕_{j=|j₁−j₂|}^{j₁+j₂} j (Wigner 1931). …
Character orthogonality: (1/|G|)Σ χ_i(g) χ_j(g)* = δ_{ij}
First orthogonality relation for irreducible characters of a finite group (Schur 1905, Burnside 1911). For any finite group G with…
Pauli algebra: tr(σ_iσ_j) = 2δ_{ij}, [σ_i,σ_j] = 2iε_{ijk}σ_k
Pauli matrices σ_x, σ_y, σ_z span the three-dimensional real vector space of 2×2 traceless Hermitian matrices and generate the Lie algebra…
Singlet |0,0⟩: norm=1, ⟨singlet|triplet⟩=0 (exact)
Closed-form unit-normalisation and orthogonality check of the SU(2) Clebsch–Gordan decomposition (½)⊗(½) = 1 ⊕ 0 on the canonical basis. …
S₃ characters: ⟨χ_i,χ_j⟩ = 6·δ_{ij} (exact)
Closed-form verification of character orthogonality for the symmetric group S₃ (|G|=6). Conjugacy classes: {e} size 1, {(12),(13),(23)}…
tr(σ_iσ_j) = 2δ_{ij}, [σ_x,σ_y] = 2iσ_z (exact)
Closed-form matrix-algebra verification of the Pauli algebra. Using sympy Matrix objects: (i) tr(σ_xσ_x) = tr(σ_yσ_y) = tr(σ_zσ_z) = 2…