Gravity Caldeira Leggett Coupling From Spectral

The coupling definition

The Caldeira-Leggett model is a standard tool in physics for describing how a quantum system loses energy. It couples the system's coordinate to a large collection of harmonic oscillators, a bath. When the bath's details are integrated out, the system experiences both damping and fluctuations, connected by the fluctuation-dissipation theorem. In the framework's machine-checked library of formal theorems, this model is adapted to gravity, and the declaration coupling_from_spectral is its mathematical core.

The declaration is a definition. It states that the coupling strength c(Ω) at a given frequency Ω is the square root of 2 J(Ω) Ω / π, where J(Ω) is the spectral density of the bath. The spectral density is a function that describes how the bath's oscillators are distributed across frequencies, and the framework requires it to be non-negative for all positive frequencies, a property called passivity. The formula is the standard inversion of the relation J(Ω) = (π/2) c(Ω)² / Ω, which appears in the Caldeira-Leggett action.

The library also defines a specific spectral density, the Debye or single-pole form, and proves that it is non-negative when its parameters are positive. It then defines a transfer function, which describes how the system responds to an external force. Two theorems are proved: at zero frequency the response equals one plus a DC enhancement, and at high frequency the response tends to one, the Newtonian limit. A third theorem shows that if the enhancement is positive, the response is always greater than one, meaning the bath amplifies rather than suppresses the system's motion.

What the declaration does not claim is as important as what it does. It does not prove that integrating out the bath yields the transfer function. The docstring states this derivation is pending. The theorem cl_action_gives_transfer_function is a placeholder, its proof is the trivial True. The framework's library provides the structure and the definitions, but the full action-based derivation, showing that the Caldeira-Leggett action gives rise to the causal transfer function, remains future work.

/-- Coupling strength function \(c(\Omega)\) derived from spectral density.
    \(J(\Omega) = \frac{\pi}{2} c(\Omega)^2 / \Omega\)
    implies \(c(\Omega) = \sqrt{2 J(\Omega) \Omega / \pi}\) -/
def coupling_from_spectral (J : ℝ → ℝ) (ω : ℝ) : ℝ :=
  sqrt (2 * J ω * ω / Real.pi)

/-- A spectral density function \(J(\Omega)\) for the oscillator bath.
    Must satisfy \(J(\Omega) \geq 0\) for all \(\Omega > 0\) (passivity). -/
structure SpectralDensity where
  J : ℝ → ℝ
  nonneg : ∀ ω, 0 < ω → 0 ≤ J ω

lemma debye_spectral_nonneg (lam γ ω : ℝ) (hlam : 0 < lam) (hγ : 0 < γ) (hω : 0 < ω) :
    0 ≤ debye_spectral lam γ ω := by
  unfold debye_spectral
  -- All factors are positive, so the product is positive
  positivity

/-- At zero frequency, the response equals \(1 + \Delta = w\). -/
theorem response_at_zero (H : TransferFunction) :
    response_function H 0 = 1 + H.Δ := by
  unfold response_function
  simp

/-- Passivity (enhancement, not suppression): if \(\Delta > 0\), then \(C(\omega) > 1\). -/
theorem response_enhancement (H : TransferFunction) (hΔ : 0 < H.Δ) (ω : ℝ) :
    1 < response_function H ω := by
  unfold response_function
  have h : 0 < H.Δ / (1 + (ω * H.τ)^2) := by
    apply div_pos hΔ
    have : 0 ≤ (ω * H.τ)^2 := sq_nonneg _
    linarith
  linarith

What this page does not claim

The declaration does not prove that the Caldeira-Leggett action gives rise to the transfer function; that derivation is pending. The declaration does not claim that the Debye spectral density is the only valid choice for a gravitational bath. The declaration does not establish that the coupling formula is derived from the framework's forcing chain; it is a definitional choice.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/CaldeiraLeggett.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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