Dimension-independence of the Lindblad-resolvent ratio for the d-mode damped quantum harmonic oscillator: a collective-coordinate reduction

Rongzhen Dai

Published Jun 8, 2026• 0 citations

Abstract

For the d-mode damped quantum harmonic oscillator at finite temperature, we ask whether the Lindblad-resolvent ratio Δ_q*/g_F derived for the 1D case (Dai 2026, DC paper / paper #52; with p → ∞ limit −2) generalises to a dimension-dependent limit −2d, as a naive "phase-space dimension" intuition might suggest. We establish that the limit is exactly −2, independent of d, for the collective polynomial perturbation A_p = (Σ_i (a_i + a_i†))^p. The argument rests on a collective-coordinate reduction theorem: under the symmetric d-mode thermal Lindblad (same n̄, γ, ω across modes), A_p depends only on the single bosonic mode b = (1/√d) Σ_i a_i, which is itself in a 1D thermal state at the same β under the original Lindblad. The d-mode problem reduces exactly to the 1D problem with rescaling A_p ↦ d^{p/2}(b+b†)^p. The dimensionless ratio Δ_q*/g_F is therefore strictly invariant under d. We further show: (i) For the asymmetric sum-of-powers observable B_p = Σ_i (a_i + a_i†)^p, the finite-p ratio has a closed-form d-correction DC(B_p) = DC_{1D}(p) − (d−1)⟨x^p⟩²_{1D}/g_{F,1D}(p), but the correction decays geometrically, so the limit is still −2. (ii) For anisotropic temperatures β_i ≠ β_j, the symmetry is broken at every p, but numerical evidence (d=2, p up to 12) is consistent with the limit again being −2, suggesting the −2 ceiling reflects a single-mode property of the thermal Lindblad eigenstructure, not phase-space geometry. We discuss why the "phase-space dim = 2d" intuition fails, and propose this as a precise mathematical instantiation of emergence: the d-mode system, in the collective-observable sector, behaves identically to a 1-mode system at the same β.

DOI: 10.5281/zenodo.20589319

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