Gauge Freedom

At each point x on the manifold, the propagating field carries an internal state in ℋ_EW ≅ ℂ². Is the choice of basis physical?

The two basis vectors |ℒ⟩ and |𝒫⟩ label the break’s relationship to each sector. The involution σ maps ℒ ↔ 𝒫, so the labelling is ambiguous: σ swaps the basis. This is a discrete (ℤ₂) ambiguity. But the full gauge redundancy is larger than ℤ₂.

All physical observables on ℋ_EW depend on inner products: transition amplitudes are |⟨φ|ψ⟩|² (Born rule, AP25), and expectation values are ⟨ψ|A|ψ⟩. Any transformation on ℋ_EW that preserves all inner products is physically undetectable. The group of inner-product-preserving transformations on ℂ² is U(2). The involution σ is one element of U(2). But U(2) contains all inner-product-preserving transformations, not just the swap.

Axiom S gives the two sectors equal standing — the 1:1 — and no axiom introduces a preferred basis or subgroup of U(2) on ℋ_EW.

Lemma 2 (Electroweak Gauge Freedom). The choice of basis on ℋ_EW at each point of the manifold is unphysical. Demanding consistency under local basis changes forces a gauge connection with structure group U(2).

Proof. (i) A local basis change on ℋ_EW at point x preserves the inner product on ℂ² (the inner product is physical: Born rule, AP25). The group of such transformations is U(2). The discrete involution σ ∈ U(2) motivates the basis ambiguity; the Born rule promotes it from discrete to continuous by requiring invariance under all inner-product-preserving transformations.

(ii) No additional structure from the axioms reduces U(2) to a proper subgroup. The physical observables on ℋ_EW are exhausted by inner-product structure. No further invariant structure is introduced by the axioms.

(iii) If the basis choice varies from point to point — if U(x) ∈ U(2) depends on position — then partial derivatives of the field pick up extra terms. A compensating gauge connection A_μ(x) must be introduced. This is the standard construction of a non-Abelian gauge theory.

(iv) The logic is identical to AP15 (phase → U(1)) and AP19 (orientation → SU(3)). Here: sector-relationship freedom → local U(2) invariance → electroweak gauge connection.

Note: U(2) has four generators (the 2×2 Hermitian matrices form a 4-dimensional real vector space). Four gauge bosons. The count matches the Standard Model’s electroweak sector (W¹, W², W³, B before symmetry breaking).

You have watched the same mechanism three times now. Identify a freedom. Demand local invariance. Force a gauge connection. Phase → U(1). Orientation → SU(3). Sector-relationship → U(2). The Standard Model’s gauge structure is not arbitrary. It is the axioms’ internal geometry, projected onto the manifold one freedom at a time.