TotalHamiltonian

formal source

  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
  68    (partialDeriv_v2 psi μ x) * (partialDeriv_v2 psi ν x) -
  69    (1/2 : ℝ) * g.g x (fun _ => 0) low * (
  70      Finset.sum (Finset.univ : Finset (Fin 4)) (fun α =>
  71        Finset.sum (Finset.univ : Finset (Fin 4)) (fun β =>
  72          inverse_metric g x (fun i => if i.val = 0 then α else β) (fun _ => 0) *
  73          (partialDeriv_v2 psi α x) * (partialDeriv_v2 psi β x)
  74        )
  75      )
  76    )
  77
  78/-- **DEFINITION: Time-Translation Invariance**
  79    A property of the metric and potential where they are independent of coordinate time x₀. -/