The Electroweak Decomposition

The gauge group U(2) is not simple. It decomposes into two factors.

Proposition 1 (Electroweak Decomposition). U(2) on ℋ_EW decomposes as U(2) ≅ SU(2) × U(1)_Y. SU(2) acts on the relative orientation of the two sector-components (weak isospin). U(1)_Y acts on the overall phase of the doublet (hypercharge).

Proof. Every unitary matrix U ∈ U(2) can be uniquely written as U = e^{iα} · V, where α ∈ ℝ and V ∈ SU(2) (det(V) = 1). The factor e^{iα} is the determinant of U: det(U) = e^{2iα}. For the Lie algebra: u(2) ≅ su(2) ⊕ u(1).

The SU(2) factor acts on the relative orientation of the two components in ℋ_EW. It mixes |ℒ⟩ and |𝒫⟩ while preserving their total norm and the determinant. Three generators: the Pauli matrices σ₁, σ₂, σ₃ (equivalently, the weak isospin operators T¹, T², T³).

The U(1) factor is the overall phase: multiplication by e^{iα} on both components simultaneously. One generator: hypercharge Y.

This U(1)_Y is NOT the electromagnetic U(1)_EM of AP15. The electromagnetic charge is a combination: Q = T³ + Y/2. AP15 derived the post-breaking remnant. This paper derives the pre-breaking structure from which it emerges after electroweak symmetry breaking. ∎

Note on global structure: the Lie group isomorphism is U(2) ≅ (SU(2) × U(1)) / ℤ₂. This global structure constrains allowed representations and is part of hypercharge quantisation. This paper derives only the local gauge algebra. Global structure, allowed representations, and anomaly constraints are open debts.

Note on hypercharge assignments: the decomposition identifies U(1)_Y as hypercharge, but the specific hypercharge values for each fermion species are not derived here. These must come from the detailed representation content of ε’s internal structure and anomaly cancellation. See KS-65.

Note on AP15 consistency: AP15’s U(1) is the electromagnetic remnant after symmetry breaking. AP27’s U(1)_Y is hypercharge before breaking. Their consistency requires a symmetry-breaking mechanism the axioms have not yet derived. This is part of debt (iii) in the scope statement, and is the work of AP16. See KS-67.