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…
Lagrange theorem |G| = [G:H] |H|; Z_6 / Z_3 index 2
Lagrange theorem framework for finite groups (Lagrange 1771; Cauchy 1844). Setup: for a finite group G with subgroup H subseteq G, the…
Schur character orthogonality; Z_3 sum chi_0 = 3
Schur character orthogonality framework (Schur 1905 Sitzungsber Preuss Akad Wiss; Frobenius 1896). Setup: irreducible characters chi_i of…
Galois extension tower [E:F] = [E:K][K:F]; Q(sqrt 2, sqrt 3)/Q
Galois extension-tower framework (Galois 1832 Memoire sur les conditions de resolubilite des equations par radicaux). Setup: for a tower of…
Theorem: 6 mod 3 - 0 = 0 (Lagrange divisibility for Z_6 / Z_3)
Theorem (Lagrange-Z6/Z3 canonical): the cyclic group Z_6 = {0, 1, 2, 3, 4, 5} contains the subgroup Z_3 = {0, 2, 4} (multiples of 2 mod 6).…
Theorem: sum_{g in Z_3} 1 - 3 = 0 (trivial-character orthogonality on Z_3)
Theorem (Schur-Z3-trivial-character canonical): for Z_3 with trivial character chi_0(g) = 1 for all g, the sum sum_{j=0}^{2} chi_0(j)…
Theorem: 4 - 2 * 2 = 0 ([Q(sqrt 2, sqrt 3) : Q] tower factorisation)
Theorem (Galois-Q-sqrt-2-3-tower canonical): the field tower Q subset Q(sqrt 2) subset Q(sqrt 2, sqrt 3) has [Q(sqrt 2, sqrt 3) : Q] = 4 =…
Noether (1918)
E Noether 1918 continuous-symmetry yields conserved-current; modern foundation of all-physics conservation laws (energy / momentum / charge…
Wigner classification (1939)
E Wigner 1939 (Nobel 1963) Poincare irreps -> particles by mass + spin; foundation of relativistic quantum field theory.
Eightfold way (Gell-Mann 1961)
M Gell-Mann 1961 + Y Ne'eman 1961 SU(3) flavor; predicted Omega- (Nobel 1969); basis of quark-model.
Yang-Mills (1954)
C N Yang-R Mills 1954 non-abelian gauge theory; basis of Standard Model SU(3)xSU(2)xU(1); 'Mass Gap' Clay-Millennium open problem.
230 space groups
Fedorov-Schoenflies 1891 enumeration; ITC-A standard; modern Bilbao Crystallographic Server; 230 distinct 3D crystallographic groups.
Anyons (Frohlich 1989)
J Frohlich 1989 + Wilczek 1982 anyons in 2D; fractional statistics; modern non-abelian anyons + topological-quantum-computing prospects.
Noether theorem detail (1918)
E Noether 1918 conservation-laws ↔ symmetries; modern modern foundational text + Noether 2nd theorem + AdS/CFT-equivalence-class.
Wigner (1939)
E Wigner 1939 (Nobel 1963) Lorentz-group representations; modern modern foundational text + helicity / mass / spin classification particles.
SU(3) flavor (Gell-Mann 1964)
M Gell-Mann 1964 (Nobel 1969) eightfold-way SU(3); modern modern foundational text + quark-model + chiral-PT + lattice-QCD predictions.
Yang-Mills detail (1954)
C N Yang-R Mills 1954 SU(2) gauge-theory; modern modern foundational text + Standard-Model SU(3)×SU(2)×U(1) + Clay-mass-gap problem.
Fiber bundle (Ehresmann 1950)
C Ehresmann 1950 fiber-bundle; modern modern foundational text + principal-G-bundle + connection + curvature + characteristic-classes.
SSB (Anderson 1958)
P Anderson 1958 + Nambu-Goldstone 1960 + Higgs 1964 SSB; modern modern foundational text + Englert-Brout-Higgs Nobel 2013 LHC.