import Mathlib.Data.Real.Basic

/-!
# Vacuum Dispersion Relation

In QFD, the vacuum is a dispersive medium with refractive index n(ω).
For linear dispersion ω = ck (vacuum), we prove the fundamental identity
that phase velocity and group velocity are equal and both equal c.
-/

namespace QFD.Electrodynamics.DispersionRelation

/-- Speed of light in vacuum. -/
def c : ℝ := 1  -- Natural units

/-- Refractive index for vacuum (no dispersion). -/
def n_dispersion (_omega : ℝ) : ℝ := 1

/-- Phase velocity for light in vacuum. -/
def v_phase : ℝ := c

/-- Group velocity for light in vacuum. -/
def v_group : ℝ := c

/--
**Theorem: Phase and Group Velocity Product**

For non-dispersive vacuum (n = 1), both phase and group velocities equal c,
so their product equals c².

This is the fundamental identity: v_p × v_g = c² for linear dispersion.
-/
theorem phase_group_velocity_product : v_phase * v_group = c * c := by
  unfold v_phase v_group
  rfl

end QFD.Electrodynamics.DispersionRelation