HOM Effect as S₂ Symmetry on Root System A₁ — E8 Intelligence Research

FINDING: The Hong-Ou-Mandel (HOM) effect is the bosonic-exchange manifestation of the symmetric group S₂ acting on the root system A₁, yielding perfect destructive interference for indistinguishable photons (bunching) and a characteristic dip in coincidence counts. | MATH: For two photons entering a 50:50 beam splitter, the output state transforms as \( |1_a,1_b → {1}{√2}(|2_c - |2_d) \) — the antisymmetric combination vanishes due to bosonic symmetry. The coincidence probability \( Pcc = 1/2(1 - Vcos(Δφ)) \) where \( V \) is visibility. The underlying group: S₂ has irreducible representations of dimensions 1 (trivial, symmetric) and 1 (sign, antisymmetric). The root system A₁ has two roots \( \{α, -α\} \) with Cartan matrix [2]; the Weyl group is S₂. The exchange phase is \( (-1)^P \) for permutation \( P ∈ S_2 \), giving +1 for bosons (symmetric) and −1 for fermions (antisymmetric). | CONNECTION: The HOM dip is the phy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com