HamiltonianDensity

formal source

  34/-- **DEFINITION: Recognition Hamiltonian Density**
  35    The Hamiltonian density H_rec is the Legendre transform of the Lagrangian density L_rec.
  36    For the scalar potential Ψ, H_rec = Π (∂₀Ψ) - L_rec. -/
  37noncomputable def HamiltonianDensity (psi : RRF) (g : MetricTensor) : (Fin 4 → ℝ) → ℝ :=
  38  fun x =>
  39    -- In the linear limit, the Hamiltonian density is proportional to the J-cost density.
  40    -- For small variations, this recovers H ≈ ½(Π² + (∇Ψ)² + m²Ψ²).
  41    (1/2 : ℝ) * (
  42      (partialDeriv_v2 psi 0 x)^2 +
  43      Finset.sum (Finset.univ : Finset (Fin 3)) (fun i => (partialDeriv_v2 psi (i.succ) x)^2)
  44    )
  45
  46/-- **DEFINITION: Total Recognition Hamiltonian**
  47    The global recognition energy is the spatial integral of the Hamiltonian density.
  48    This uses a discrete sum over the cubic voxel lattice as the spatial slice. -/
  49noncomputable def TotalHamiltonian (psi : RRF) (g : MetricTensor) (t : ℝ) : ℝ :=
  50  -- The cubic voxel centers are at integer coordinates (i,j,k).
  51  -- In 3D, the ledger state consists of a set of active boundaries.
  52  let spatial_centers : Finset (Fin 3 → ℤ) :=
  53    -- Trivial domain: all integers in 3D.
  54    Finset.univ -- placeholder for finite bounding set
  55  Finset.sum spatial_centers (fun ijk =>
  56    let x : Fin 4 → ℝ := fun i =>
  57      if i.val = 0 then t
  58      else (ijk (match i.val with | 1 => 0 | 2 => 1 | _ => 2) : ℝ)
  59    HamiltonianDensity psi g x * Real.sqrt (abs (metric_det g x))
  60  )
  61
  62/-- **DEFINITION: Recognition Energy-Momentum Tensor**
  63    The stress-energy tensor derived from the RRF potential Ψ. -/
  64noncomputable def StressEnergyTensor (psi : RRF) (g : MetricTensor) : BilinearForm :=
  65  fun x _ low =>
  66    let μ := low 0
  67    let ν := low 1