IndisputableMonolith.Gravity.CaldeiraLeggett

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Caldeira-Leggett Action Formalization
   6
   7This module formalizes the Caldeira-Leggett construction, providing a
   8conservative action-based realization of causal-response dynamics.
   9
  10## Background
  11
  12The Caldeira-Leggett formalism describes dissipative quantum systems by coupling
  13a system coordinate to a bath of harmonic oscillators. When the bath is traced out,
  14the system experiences damping and fluctuations related by the fluctuation-dissipation
  15theorem.
  16
  17## Gravitational Adaptation
  18
  19The Caldeira-Leggett formalism can be adapted to gravity:
  20
  211. **Action**:
  22   \[
  23   S = \int dt\,d^3x \Big[
  24     \frac{1}{8\pi G}|\nabla\Phi_{\rm baryon}|^2
  25     + \frac{\kappa}{2}X^2 + gX\Phi_{\rm baryon}
  26     + \int_0^\infty d\Omega \frac{1}{2}(\dot q_\Omega^2 - \Omega^2 q_\Omega^2)
  27     + X \int_0^\infty d\Omega\, c(\Omega) q_\Omega
  28   \Big]
  29   \]
  30
  312. **Spectral Density**: \(J(\Omega) = \frac{\pi}{2} c(\Omega)^2 / \Omega \geq 0\)
  32
  333. **Transfer Function**: Integrating out the bath gives a frequency-dependent response
  34   \(H(i\omega)\) with the single-pole form for exponential memory.
  35
  36## Status
  37
  38This module provides the **structure and definitions** for the Caldeira-Leggett
  39realization. Full proofs (e.g., that integrating out the bath gives the claimed
  40transfer function) are pending.
  41
  42-/
  43
  44namespace IndisputableMonolith
  45namespace Gravity
  46namespace CaldeiraLeggett
  47
  48open Real
  49
  50noncomputable section
  51
  52/-! ## Spectral Density -/
  53
  54/-- A spectral density function \(J(\Omega)\) for the oscillator bath.
  55    Must satisfy \(J(\Omega) \geq 0\) for all \(\Omega > 0\) (passivity). -/
  56structure SpectralDensity where
  57  J : ℝ → ℝ
  58  nonneg : ∀ ω, 0 < ω → 0 ≤ J ω
  59
  60/-- Coupling strength function \(c(\Omega)\) derived from spectral density.
  61    \(J(\Omega) = \frac{\pi}{2} c(\Omega)^2 / \Omega\)
  62    implies \(c(\Omega) = \sqrt{2 J(\Omega) \Omega / \pi}\) -/
  63def coupling_from_spectral (J : ℝ → ℝ) (ω : ℝ) : ℝ :=
  64  sqrt (2 * J ω * ω / Real.pi)
  65
  66/-! ## Single-Pole (Debye) Spectral Density -/
  67
  68/-- The Debye (single-pole) spectral density:
  69    \(J_{\rm Debye}(\Omega) = \frac{2\lambda\gamma}{\pi} \cdot \frac{\Omega}{\gamma^2 + \Omega^2}\)
  70
  71    where \(\gamma = 1/\tau_\star\) is the cutoff frequency (inverse memory time)
  72    and \(\lambda\) is the coupling strength. -/
  73def debye_spectral (lam γ : ℝ) (ω : ℝ) : ℝ :=
  74  2 * lam * γ / Real.pi * ω / (γ^2 + ω^2)
  75
  76lemma debye_spectral_nonneg (lam γ ω : ℝ) (hlam : 0 < lam) (hγ : 0 < γ) (hω : 0 < ω) :
  77    0 ≤ debye_spectral lam γ ω := by
  78  unfold debye_spectral
  79  -- All factors are positive, so the product is positive
  80  positivity
  81
  82def debye_density (lam γ : ℝ) (hlam : 0 < lam) (hγ : 0 < γ) : SpectralDensity where
  83  J := debye_spectral lam γ
  84  nonneg := fun ω hω => debye_spectral_nonneg lam γ ω hlam hγ hω
  85
  86/-! ## Transfer Function -/
  87
  88/-- The complex transfer function \(H(i\omega)\) for a single-pole response.
  89    \(H(i\omega) = 1 + \frac{\Delta}{1 + i\omega\tau_\star}\)
  90
  91    where \(\Delta = w - 1\) is the DC enhancement. -/
  92structure TransferFunction where
  93  Δ : ℝ  -- DC enhancement (w - 1)
  94  τ : ℝ  -- Memory timescale
  95  τ_pos : 0 < τ
  96
  97/-- The real part of the transfer function (response function).
  98    \(C(\omega) = \text{Re}[H(i\omega)] = 1 + \frac{\Delta}{1 + (\omega\tau)^2}\) -/
  99def response_function (H : TransferFunction) (ω : ℝ) : ℝ :=
 100  1 + H.Δ / (1 + (ω * H.τ)^2)
 101
 102/-- The imaginary part of the transfer function (quadrature function).
 103    \(S(\omega) = \text{Im}[H(i\omega)] = -\frac{\Delta \cdot \omega\tau}{1 + (\omega\tau)^2}\) -/
 104def quadrature_function (H : TransferFunction) (ω : ℝ) : ℝ :=
 105  -H.Δ * ω * H.τ / (1 + (ω * H.τ)^2)
 106
 107/-! ## Key Properties -/
 108
 109/-- At zero frequency, the response equals \(1 + \Delta = w\). -/
 110theorem response_at_zero (H : TransferFunction) :
 111    response_function H 0 = 1 + H.Δ := by
 112  unfold response_function
 113  simp
 114
 115/-- At infinite frequency, the response approaches 1 (Newtonian limit). -/
 116theorem response_limit_high_freq (H : TransferFunction) (hΔ : H.Δ ≠ 0) :
 117    Filter.Tendsto (response_function H) Filter.atTop (nhds 1) := by
 118  -- As ω → ∞, Δ/(1 + (ωτ)²) → 0
 119  unfold response_function
 120  -- Show the denominator tends to `+∞` as ω → +∞.
 121  have hmul : Filter.Tendsto (fun ω : ℝ => ω * H.τ) Filter.atTop Filter.atTop := by
 122    simpa using ((Filter.tendsto_id).atTop_mul_const H.τ_pos)
 123  have hsq : Filter.Tendsto (fun ω : ℝ => (ω * H.τ) ^ (2 : ℕ)) Filter.atTop Filter.atTop :=
 124    (Filter.tendsto_pow_atTop (α := ℝ) (n := 2) (by decide)).comp hmul
 125  have hmono :
 126      (fun ω : ℝ => (ω * H.τ) ^ (2 : ℕ))
 127        ≤ᶠ[Filter.atTop] (fun ω : ℝ => 1 + (ω * H.τ) ^ (2 : ℕ)) :=
 128    Filter.Eventually.of_forall (fun _ω => by linarith)
 129  have hden :
 130      Filter.Tendsto (fun ω : ℝ => 1 + (ω * H.τ) ^ (2 : ℕ)) Filter.atTop Filter.atTop :=
 131    Filter.tendsto_atTop_mono' Filter.atTop hmono hsq
 132
 133  have hinv :
 134      Filter.Tendsto (fun ω : ℝ => (1 + (ω * H.τ) ^ (2 : ℕ))⁻¹) Filter.atTop (nhds 0) :=
 135    (tendsto_inv_atTop_zero).comp hden
 136
 137  have hfrac_mul :
 138      Filter.Tendsto (fun ω : ℝ => H.Δ * (1 + (ω * H.τ) ^ (2 : ℕ))⁻¹) Filter.atTop (nhds 0) := by
 139    have hΔconst : Filter.Tendsto (fun _ω : ℝ => H.Δ) Filter.atTop (nhds H.Δ) := by
 140      simpa using (tendsto_const_nhds : Filter.Tendsto (fun _ω : ℝ => H.Δ) Filter.atTop (nhds H.Δ))
 141    -- H.Δ * (denom)⁻¹ → H.Δ * 0 = 0
 142    simpa using (hΔconst.mul hinv)
 143
 144  have hfrac :
 145      Filter.Tendsto (fun ω : ℝ => H.Δ / (1 + (ω * H.τ) ^ (2 : ℕ))) Filter.atTop (nhds 0) := by
 146    simpa [div_eq_mul_inv] using hfrac_mul
 147
 148  -- Add back the constant `1`.
 149  have hone : Filter.Tendsto (fun _ω : ℝ => (1 : ℝ)) Filter.atTop (nhds 1) := by
 150    simpa using (tendsto_const_nhds : Filter.Tendsto (fun _ω : ℝ => (1 : ℝ)) Filter.atTop (nhds 1))
 151  simpa using hone.add hfrac
 152
 153/-- Passivity (enhancement, not suppression): if \(\Delta > 0\), then \(C(\omega) > 1\). -/
 154theorem response_enhancement (H : TransferFunction) (hΔ : 0 < H.Δ) (ω : ℝ) :
 155    1 < response_function H ω := by
 156  unfold response_function
 157  have h : 0 < H.Δ / (1 + (ω * H.τ)^2) := by
 158    apply div_pos hΔ
 159    have : 0 ≤ (ω * H.τ)^2 := sq_nonneg _
 160    linarith
 161  linarith
 162
 163/-! ## Caldeira-Leggett Action (Sketch)
 164
 165The full action-based derivation would show:
 166
 1671. Define the action \(S[Φ, X, \{q_Ω\}]\) with Newtonian potential, auxiliary field,
 168   and oscillator bath.
 169
 1702. Integrate out the oscillators \(q_Ω\) to get an effective action for \((Φ, X)\).
 171
 1723. Integrate out the auxiliary field \(X\) to get the frequency-dependent
 173   response \(H(i\omega)\).
 174
 1754. Show that \(H(i\omega) = 1 + \Delta/(1 + i\omega\tau)\) for Debye spectral density.
 176
 177This is left for future formalization.
 178-/
 179
 180/-- The Caldeira-Leggett action gives rise to the causal transfer function. -/
 181theorem cl_action_gives_transfer_function :
 182    True := by  -- Placeholder for the full derivation
 183  trivial
 184
 185end
 186
 187end CaldeiraLeggett
 188end Gravity
 189end IndisputableMonolith
 190