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arxiv: 2605.25936 · v1 · pith:N5ZHKITAnew · submitted 2026-05-25 · 🌀 gr-qc · quant-ph

Generalising gravitationally induced decoherence beyond linear environmental interactions in a microscopic quantum mechanical toy model

Max Joseph Fahn , Renata Ferrero , Kristina Giesel , Roman Kemper This is my paper

Pith reviewed 2026-06-29 20:24 UTC · model grok-4.3

classification 🌀 gr-qc quant-ph
keywords gravitationally induced decoherenceWeyl elementspolymer quantum mechanicsmaster equationMarkov approximationquantum toy modelenvironmental correlation functions
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The pith

Gravitationally induced decoherence is generalized to sinus-like couplings using Weyl elements of environmental position operators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends a quantum mechanical toy model of gravitationally induced decoherence by replacing the linear coupling between the system Hamiltonian and environmental position operators with an interaction based on Weyl elements. This allows for a sinus-like coupling that remains in the Schrödinger representation and reduces to the linear case for small Weyl parameter. Two methods are developed to compute the environmental correlation functions analytically, one using Wick's theorem and another using short-time Fourier transform. Numerical analysis confirms rapid decay of correlations, validating the Markov approximation, and the renormalised master equation is solved after showing reproduction of prior models at first order.

Core claim

Using a Taylor expansion in the Weyl parameter, the first-order term reproduces the decoherence model of Xu, Blencowe (2022) and Domi et al. (2024); the renormalised master equation is solved for the generalised interaction based on sinus-like Weyl coupling.

What carries the argument

The sinus-like coupling formulated in terms of Weyl elements of the environment's position operators, which generates non-linear interactions and reduces to linear coupling in the small Weyl parameter limit.

If this is right

  • The first-order term in the Weyl parameter expansion reproduces the previous linear decoherence models.
  • The two complementary methods for calculating correlation functions yield identical results and one is generalisable to polymer quantisation.
  • The environmental correlation functions decay rapidly, supporting the Markov approximation in the master equation derivation.
  • The spectral density is generalised for the exponential coupling structure.
  • The solution to the renormalised master equation is derived for the generalised interaction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach opens the door to incorporating polymer quantum mechanics effects into gravitational decoherence calculations.
  • If the sinus-like coupling captures real non-linear gravitational interactions, it could lead to observable differences in decoherence rates at higher orders.
  • Extensions to other non-linear couplings via truncations of the exponential Weyl elements might be explored in future toy models.

Load-bearing premise

The sinus-like coupling can be quantised using the Schrödinger representation and the Markov approximation holds due to rapid decay of the numerical correlation functions.

What would settle it

A calculation or measurement showing that the correlation functions do not decay rapidly enough to justify the Markov approximation, or that the higher-order terms in the Weyl expansion produce decoherence rates inconsistent with the renormalised master equation solution.

Figures

Figures reproduced from arXiv: 2605.25936 by Kristina Giesel, Max Joseph Fahn, Renata Ferrero, Roman Kemper.

Figure 1
Figure 1. Figure 1: FIG. 1. Numerical evaluation of the real and imaginary part with parameters [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Numerical evaluation of the real and imaginary decoherence factors with parameters [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The time evolution of Re[ [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
read the original abstract

We generalise the quantum mechanical toy model for gravitationally induced decoherence presented in Xu, Blencowe (2022) and Domi et al. (2024). In contrast to earlier formulations, in which the Hamiltonian of the system of interest is linearly coupled to the position operators of the oscillators in the environment, we consider an interaction formulated in terms of Weyl elements of the environment's position operators. This extension is motivated by polymer quantum mechanics, in which Weyl elements are fundamental operators, as well as by the possibility of generating non-linear interactions through suitable truncations of the exponential Weyl elements. Here we focus on a sinus-like coupling that is still quantised using the Schr\"odinger representation and, in the limit of a small Weyl parameter, reproduces the conventional linear interaction. To derive the corresponding master equation, we developed two complementary methods for the analytical calculation of the environmental correlation functions. The first utilises Wick's theorem for thermal expectation values in conjunction with annihilation and creation operators, while the second is based on the short-time Fourier transform and completely avoids the use of annihilation and creation operators, making it more readily transferable and generalisable to a polymer quantisation. Both approaches yield identical results. We further generalise the spectral density required for the exponential coupling structure. A numerical analysis shows that the environmental correlation functions decay rapidly with time, which supports the validity of the Markov approximation. Using a Taylor expansion in the Weyl parameter, we show that the first-order term reproduces the decoherence model of Xu, Blencowe (2022) and Domi et al. (2024). Finally, we derive the solution to the renormalised master equation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper generalizes a microscopic toy model of gravitationally induced decoherence by replacing linear system-environment coupling with an interaction based on Weyl elements of the environmental position operators. It focuses on a sinus-like coupling (quantized in the Schrödinger representation) that reduces to the linear case for small Weyl parameter. Two independent analytic methods for the environmental correlation functions are derived and shown to agree; the spectral density is generalized accordingly. Numerical evaluation demonstrates rapid decay of the correlations, which is used to justify the Markov approximation. A Taylor expansion in the Weyl parameter recovers the master equation of Xu, Blencowe (2022) and Domi et al. (2024) at first order, and the renormalised master equation for the generalized interaction is solved.

Significance. If the Markov approximation remains valid, the work supplies a controlled microscopic extension of gravitational decoherence models into the non-linear regime, with direct relevance to polymer quantization and to future experimental tests that could distinguish linear from non-linear gravitational effects. Credit is due for the explicit agreement between the two correlation-function derivations (Wick theorem versus short-time Fourier transform) and for the parameter-free recovery of the prior linear result via Taylor expansion rather than by assumption.

major comments (1)
  1. [Numerical analysis of environmental correlation functions] Numerical analysis section: the validity of the Born-Markov master equation for the generalized (sinus-like) interaction rests on the numerical observation that correlation functions decay rapidly. No analytic bound on the decay rate, no scaling with the Weyl parameter λ, and no explicit comparison to the system’s intrinsic timescales are supplied. If the decay slows for λ away from the linear limit, the time-local master equation and its renormalised solution cease to apply in the regime the paper presents as the central generalization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript and for the constructive comment on the justification of the Markov approximation. We address the major comment below and agree that the manuscript will benefit from additional analysis in this area.

read point-by-point responses

  1. Referee: Numerical analysis section: the validity of the Born-Markov master equation for the generalized (sinus-like) interaction rests on the numerical observation that correlation functions decay rapidly. No analytic bound on the decay rate, no scaling with the Weyl parameter λ, and no explicit comparison to the system’s intrinsic timescales are supplied. If the decay slows for λ away from the linear limit, the time-local master equation and its renormalised solution cease to apply in the regime the paper presents as the central generalization.

    Authors: We agree that the current justification rests primarily on numerical observation and that an explicit scaling analysis with λ together with a comparison to system timescales would strengthen the claim. The correlation functions are available in closed analytic form from both derivation methods, which in principle permits extraction of the decay envelope. In the revised manuscript we will add a dedicated paragraph (or short subsection) that (i) extracts the leading large-t decay from the analytic expressions, (ii) shows its dependence on λ for the range of interest, and (iii) compares the resulting environmental correlation time to the system’s natural frequency and to the gravitational decoherence timescale. We will also state the parameter domain in which the Markov approximation is thereby justified. A fully rigorous analytic bound valid for arbitrary λ is not presently available and would require further technical work beyond the scope of the present toy model; we will therefore qualify the regime of validity accordingly. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior linear model; central generalization from new Hamiltonian is independent

full rationale

The paper derives the master equation for the new sinus-like Weyl coupling from the interaction Hamiltonian using two analytical methods for correlation functions (Wick's theorem and short-time Fourier transform). The first-order Taylor expansion in the Weyl parameter is shown explicitly to recover the Xu/Blencowe/Domi linear model as a consistency check rather than an input assumption. The Markov approximation is justified by a separate numerical observation of rapid correlation decay, not by fitting or definitional reduction. Self-citations to the 2022/2024 works serve only to identify the model being generalized and are not load-bearing for the new derivation or its solution. No parameters are fitted to force the target result, and the derivation chain does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The work relies on standard quantum mechanics (Wick's theorem, thermal expectation values) and the validity of the Markov approximation justified numerically. No new free parameters beyond the Weyl parameter (treated as small) or invented entities are introduced in the abstract.

free parameters (1)
  • Weyl parameter
    Small parameter controlling the strength of the non-linear coupling; appears in the Taylor expansion that recovers the linear model.
axioms (2)
  • standard math Wick's theorem applies to thermal expectation values of the environmental oscillators
    Invoked in the first method for calculating correlation functions.
  • domain assumption Markov approximation is justified by rapid decay of environmental correlations
    Supported by the numerical analysis mentioned in the abstract.

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discussion (0)

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Reference graph

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