Gravity Caldeira Leggett

Caldeira-Leggett and gravity

The Caldeira-Leggett model is a standard way to describe a small quantum system that loses energy to its surroundings. It couples the system to a bath of harmonic oscillators, simple vibrating modes that carry energy away. When the bath is traced out of the equations, the system feels both damping, a gradual loss of motion, and random fluctuations, and the two are linked by the fluctuation-dissipation theorem. That theorem is the workhorse that connects how strongly a system is pushed around to how readily it absorbs energy.

The construction dates to 1983, when Amir Caldeira and Anthony Leggett used it to study quantum tunneling with friction. The same machinery appears across physics, from superconducting circuits to the motion of atoms in a crystal. The idea is always the same: replace the messy environment with a manageable set of oscillators, do the bookkeeping, and extract the effective behavior of the system you care about.

In Recognition Science, the framework adapts this formalism to gravity. The module defines a gravitational action that couples a baryon potential, an auxiliary field, and a bath of oscillators. The action reads S = ∫ dt d³x [ (1/8πG)|∇Φ_baryon|² + (κ/2)X² + gXΦ_baryon + ∫₀^∞ dΩ (1/2)(q̇_Ω² - Ω² q_Ω²) + X ∫₀^∞ dΩ c(Ω) q_Ω ]. The spectral density J(Ω) = (π/2)c(Ω)²/Ω is required to be nonnegative, which encodes passivity: the bath can absorb energy but not amplify it. A Debye spectral density, J_Debye(Ω) = (2λγ/π) · Ω/(γ² + Ω²), gives a single-pole response with a memory timescale τ = 1/γ.

The module proves three structural facts about the resulting transfer function H(iω) = 1 + Δ/(1 + iωτ). At zero frequency, the response equals 1 + Δ, a DC enhancement. At high frequency, it tends to 1, recovering the Newtonian limit. And if Δ > 0, the response is always greater than 1, meaning the enhancement is real, not a suppression. These are the kernel-checked results of the module.

What this establishes in plain language is a scaffold. The framework shows that a conservative action with an oscillator bath can produce a causal, frequency-dependent gravitational response with memory. The full derivation, integrating out the bath and the auxiliary field to obtain the transfer function from the action, remains to be formalized. The module provides the structure and the definitions, and the key theorems about the response function, but the bridge from action to transfer function is still open.

/-- 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

/-- 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

/-- 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)

/-- 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

/-- At infinite frequency, the response approaches 1 (Newtonian limit). -/
theorem response_limit_high_freq (H : TransferFunction) (hΔ : H.Δ ≠ 0) :
    Filter.Tendsto (response_function H) Filter.atTop (nhds 1) := by
  -- As ω → ∞, Δ/(1 + (ωτ)²) → 0
  unfold response_function
  -- Show the denominator tends to `+∞` as ω → +∞.
  have hmul : Filter.Tendsto (fun ω : ℝ => ω * H.τ) Filter.atTop Filter.atTop := by
    simpa using ((Filter.tendsto_id).atTop_mul_const H.τ_pos)
  have hsq : Filter.Tendsto (fun ω : ℝ => (ω * H.τ) ^ (2 : ℕ)) Filter.atTop Filter.atTop :=
    (Filter.tendsto_pow_atTop (α := ℝ) (n := 2) (by decide)).comp hmul
  have hmono :
      (fun ω : ℝ => (ω * H.τ) ^ (2 : ℕ))
        ≤ᶠ[Filter.atTop] (fun ω : ℝ => 1 + (ω * H.τ) ^ (2 : ℕ)) :=
    Filter.Eventually.of_forall (fun _ω => by linarith)
  have hden :
      Filter.Tendsto (fun ω : ℝ => 1 + (ω * H.τ) ^ (2 : ℕ)) Filter.atTop Filter.atTop :=
    Filter.tendsto_atTop_mono' Filter.atTop hmono hsq

  have hinv :
      Filter.Tendsto (fun ω : ℝ => (1 + (ω * H.τ) ^ (2 : ℕ))⁻¹) Filter.atTop (nhds 0) :=
    (tendsto_inv_atTop_zero).comp hden

  have hfrac_mul :
      Filter.Tendsto (fun ω : ℝ => H.Δ * (1 + (ω * H.τ) ^ (2 : ℕ))⁻¹) Filter.atTop (nhds 0) := by
    have hΔconst : Filter.Tendsto (fun _ω : ℝ => H.Δ) Filter.atTop (nhds H.Δ) := by
      simpa using (tendsto_const_nhds : Filter.Tendsto (fun _ω : ℝ => H.Δ) Filter.atTop (nhds H.Δ))
    -- H.Δ * (denom)⁻¹ → H.Δ * 0 = 0
    simpa using (hΔconst.mul hinv)

  have hfrac :
      Filter.Tendsto (fun ω : ℝ => H.Δ / (1 + (ω * H.τ) ^ (2 : ℕ))) Filter.atTop (nhds 0) := by
    simpa [div_eq_mul_inv] using hfrac_mul

  -- Add back the constant `1`.
  have hone : Filter.Tendsto (fun _ω : ℝ => (1 : ℝ)) Filter.atTop (nhds 1) := by
    simpa using (tendsto_const_nhds : Filter.Tendsto (fun _ω : ℝ => (1 : ℝ)) Filter.atTop (nhds 1))
  simpa using hone.add hfrac

/-- 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 full derivation from the action to the transfer function is not yet formalized. The module does not predict a specific value for the enhancement Δ or the memory time τ. The Caldeira-Leggett adaptation is a model, not a proved consequence of the forcing chain.

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.

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