VolumeElement

formal source

  16open Geometry
  17
  18/-- Volume element d⁴x with metric measure √(-g). -/
  19structure VolumeElement where
  20  grid_spacing : ℝ  -- Δx for discrete approximation
  21  grid_spacing_pos : 0 < grid_spacing
  22
  23/-- Sample points for discrete integration (uniform grid). -/
  24def sample_grid (vol : VolumeElement) (n_points : ℕ) : List (Fin 4 → ℝ) :=
  25  -- Simplified: n_points^4 grid over [0, L]^4
  26  -- Full version would use adaptive quadrature
  27  []  -- Placeholder
  28
  29/-- Square root of minus the metric determinant.
  30    STATUS: SCAFFOLD — Uses constant 1 as placeholder. -/
  31noncomputable def sqrt_minus_g (_g : MetricTensor) (_x : Fin 4 → ℝ) : ℝ := 1
  32
  33/-- Integrate a scalar function over spacetime volume (discrete approximation). -/
  34noncomputable def integrate_scalar
  35  (f : (Fin 4 → ℝ) → ℝ) (g : MetricTensor) (vol : VolumeElement) : ℝ :=
  36  let n := 10
  37  let Δx4 := vol.grid_spacing ^ 4
  38  Δx4 * Finset.sum (Finset.range n) (fun i =>
  39    sqrt_minus_g g (fun _ => (i : ℝ) * vol.grid_spacing) *
  40    f (fun _ => (i : ℝ) * vol.grid_spacing))
  41
  42/-- Kinetic action integral: (1/2) ∫ √(-g) g^{μν} (∂_μ ψ)(∂_ν ψ) d⁴x. -/
  43noncomputable def kinetic_action
  44  (φ : ScalarField) (g : MetricTensor) (vol : VolumeElement) : ℝ :=
  45  (1/2) * integrate_scalar (gradient_squared φ g) g vol
  46
  47/-- Potential action integral: (1/2) ∫ √(-g) m² ψ² d⁴x. -/
  48noncomputable def potential_action
  49  (φ : ScalarField) (m_squared : ℝ) (g : MetricTensor) (vol : VolumeElement) : ℝ :=