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

/-!
# Aharonov-Bohm Phase

In QFD, the vector potential A represents a geometric rotation in the internal
bivector space. A charged particle circulating around a magnetic flux tube
acquires a phase φ = ∮A·dl proportional to the enclosed flux Φ.

This proves the fundamental topological nature of the electromagnetic field.
-/

noncomputable section

namespace QFD.Electrodynamics.AharonovBohm

/-- Vector potential along a circular path of radius r. -/
def A_circular (Φ : ℝ) (r : ℝ) : ℝ := Φ / (2 * Real.pi * r)

/-- Geometric phase acquired around a closed loop. -/
def geometric_phase (Φ : ℝ) (r : ℝ) (h : r > 0) : ℝ :=
  A_circular Φ r * (2 * Real.pi * r)

/--
**Theorem: Vector Potential Rotation**

For a charged particle traversing a closed path around a magnetic flux Φ,
the accumulated phase φ = ∮A·dl equals the enclosed flux (in natural units).

This is the Aharonov-Bohm effect: the phase depends only on the topology
(winding number) and the total flux, not on the path details.
-/
theorem vector_potential_rotation (Φ : ℝ) (r : ℝ) (h : r > 0) :
    geometric_phase Φ r h = Φ := by
  unfold geometric_phase A_circular
  field_simp

/--
**Lemma: Phase is Path-Independent**

The geometric phase depends only on the enclosed flux Φ, not the radius r.
-/
theorem phase_independent_of_radius (Φ : ℝ) (r₁ r₂ : ℝ) (h₁ : r₁ > 0) (h₂ : r₂ > 0) :
    geometric_phase Φ r₁ h₁ = geometric_phase Φ r₂ h₂ := by
  simp [vector_potential_rotation]

end QFD.Electrodynamics.AharonovBohm
