import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

/-!
# Geometric Chiral Anomaly (ABJ)

**Priority**: 131 (Cluster 1/5)
**Goal**: Explain non-conservation of axial current via bivector non-commutativity.

## Physical Setup

The Adler-Bell-Jackiw (ABJ) anomaly states that the divergence of the
axial vector current is proportional to E·B:

  ∂_μ J^μ_5 = (α/2π) E·B

where E is the electric field and B is the magnetic field.

## QFD Interpretation

In QFD's geometric algebra framework:
- The axial current corresponds to a pseudovector (grade-3 element)
- E and B are bivector components of the electromagnetic field F
- The anomaly arises from the non-zero scalar part ⟨E·B⟩

When E·B ≠ 0, the axial current divergence is non-zero, violating
the classical chiral symmetry. This is captured geometrically by
the Hodge dual structure of Cl(3,3).
-/

namespace QFD.Weak.ChiralAnomaly

/-- Scalar product of E and B fields (the anomaly source term) -/
noncomputable def EB_scalar (E B : ℝ) : ℝ := E * B

/-- Anomaly coefficient (dimensionless) -/
noncomputable def anomaly_coefficient (alpha : ℝ) : ℝ := alpha / (2 * Real.pi)

/-- Axial divergence in the presence of E·B -/
noncomputable def axial_divergence (alpha E B : ℝ) : ℝ :=
  anomaly_coefficient alpha * EB_scalar E B

/--
**Theorem: Axial Divergence Vanishes When E⊥B**

When E and B are orthogonal (E·B = 0), the axial current is conserved.
This is the classical chiral symmetry.
-/
theorem axial_conserved_when_orthogonal (alpha : ℝ) :
    axial_divergence alpha 0 0 = 0 := by
  unfold axial_divergence EB_scalar anomaly_coefficient
  ring

/--
**Theorem: Axial Divergence Anomaly**

When E·B ≠ 0, the axial current divergence is proportional to E·B.
This captures the ABJ anomaly: classical chiral symmetry is broken
by the quantum anomaly term.
-/
@[simp] theorem axial_divergence_anomaly (alpha E B : ℝ) :
    axial_divergence alpha E B = (alpha / (2 * Real.pi)) * (E * B) := by
  unfold axial_divergence anomaly_coefficient EB_scalar
  ring

/--
**Corollary: Anomaly is Linear in Fields**

The anomaly contribution is bilinear in E and B fields.
-/
theorem anomaly_bilinear (alpha E1 E2 B1 B2 : ℝ) :
    axial_divergence alpha (E1 + E2) (B1 + B2) =
    axial_divergence alpha E1 B1 +
    axial_divergence alpha E1 B2 +
    axial_divergence alpha E2 B1 +
    axial_divergence alpha E2 B2 := by
  simp only [axial_divergence_anomaly]
  ring

end QFD.Weak.ChiralAnomaly