Gravity Caldeira Leggett Debye Spectral Nonneg

The Debye spectral density

The spectral density of an oscillator bath is a function that describes how strongly the bath's many oscillators respond at each frequency. In the Caldeira-Leggett formalism, a quantum system is coupled to a bath of harmonic oscillators; when the bath is traced out, the system experiences damping and fluctuations related by the fluctuation-dissipation theorem. The spectral density J(Ω) must be nonnegative for all positive frequencies Ω, a condition called passivity, which ensures the bath does not amplify energy rather than dissipate it.

The Debye, or single-pole, spectral density is a specific choice of this function, written J_Debye(Ω) = (2λγ/π) · Ω/(γ² + Ω²). Here γ = 1/τ★ is a cutoff frequency (the inverse of a memory timescale) and λ is a coupling strength. This form appears in the gravitational adaptation of the Caldeira-Leggett construction, where a system coordinate couples to a bath of oscillators and the bath is integrated out to produce a frequency-dependent response with exponential memory.

The declaration debye_spectral_nonneg is a machine-checked proof of a simple fact: for positive coupling strength λ, positive cutoff γ, and positive frequency ω, the Debye spectral density is nonnegative. The proof is direct: all factors in the expression are positive, so their product is positive. The declaration also builds a full SpectralDensity structure, debye_density, which packages this nonnegativity as the required passivity condition for use in the larger formalism.

In Recognition Science, the framework's machine-checked library of formal theorems provides this lemma as part of a larger formalization of the Caldeira-Leggett action for gravity. The framework models dissipative gravitational response by coupling a baryon potential to an auxiliary field and a bath of oscillators, then integrating out the bath to obtain a transfer function. The nonnegativity lemma is a structural guarantee within that model: it ensures the chosen Debye spectral density is admissible as a passivity-respecting bath.

What the lemma does not claim is just as important as what it proves. It does not prove that the Caldeira-Leggett action actually gives rise to the claimed transfer function; that derivation is explicitly left as future work, with a placeholder theorem marked True. It does not prove that the Debye spectral density is the unique or physically correct choice for gravity. And it does not establish any empirical prediction about gravitational damping; it only verifies an algebraic inequality for a definitional choice.

The practical consequence is a clean separation of concerns. The framework's library can rely on the Debye spectral density being a valid passivity-respecting bath, without pretending that the harder question of deriving the transfer function from the action has been settled. The nonnegativity lemma is a small but necessary step, and the formalization is honest about what remains open.

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

def debye_density (lam γ : ℝ) (hlam : 0 < lam) (hγ : 0 < γ) : SpectralDensity where
  J := debye_spectral lam γ
  nonneg := fun ω hω => debye_spectral_nonneg lam γ ω hlam hγ hω

What this page does not claim

The Caldeira-Leggett action has been shown to produce the claimed transfer function. The Debye spectral density is the unique or physically correct choice for gravitational baths. The nonnegativity lemma makes any empirical prediction about gravitational damping.

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)

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