{"id":"401278ba-d503-4ffd-899d-17012ebd93c0","arxiv_id":"2608.02054","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A polar-form decomposition of the Caldeira-Leggett density matrix yields a residual effective potential that survives the epsilon to 0 limit and gives deterministic, temperature-dependent Bohmian trajectories.","lead":"This paper describes a scaled version of the Caldeira-Leggett model in which a quantumness parameter smoothly turns off quantum effects in an open quantum system. It claims that the resulting classical-limit trajectories are deterministic and temperature-dependent, unlike the stochastic trajectories of the usual Langevin description.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed classical limit is not a limit of the CL model: Qres is extracted from an ad hoc scaled equation whose epsilon->0 solution is singular, so Eq. (15) is unsupported.","rationale":"The reader's weakest-assumption analysis correctly identifies the scaled CL equation (19) and the epsilon->0 extraction of Qres as load-bearing and underived. My stress-test sharpens this into a specific technical failure: the limit used to define Qres is singular, because the scaled density matrix collapses in r as epsilon->0 while Qres is read off from an expression that assumes a regular amplitude. This makes the central claim of deterministic temperature-dependent classical trajectories unsupported by the presented derivation. I do not see a way to salvage Eq. (15) from the material in the paper without an independent derivation of a classical limit from the standard CL model. The verdict REJECT remains appropriate, so no adjustment is needed.","tokens_in":16785,"tokens_out":4599,"duration_ms":46692,"concrete_test":"Take the standard (unscaled) CL equation (1) for the same Gaussian initial state at finite hbar, compute Q from Eq. (7b) and the Bohmian acceleration Eq. (14), then examine the limit hbar->0 with D=2m gamma k_B T held fixed. If the limiting force does not match Eq. (36) (or if the polar decomposition itself becomes ill-defined because the density matrix collapses to the diagonal), then Eq. (15) is an artifact of the epsilon-scaling rather than the classical limit of CL dynamics. Independently, one can also test the epsilon->0 limit by first rescaling the relative coordinate r = sqrt(epsilon) u in Eq. (19) before taking epsilon->0; if Qres changes or diverges under this rescaling, the naive epsilon=0 substitution in Eq. (30) is not a valid limit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Eq. (15): deterministic, temperature-dependent classical trajectories with residual force from Qres. This requires a well-defined classical limit of the CL dynamics. The paper's route is the scaled CL equation (19) with hbar_tilde = sqrt(epsilon) hbar. In Sec. II.B the authors concede this linear scaled equation 'does not provide a proper description of the classical regime.' The problem is worse: at epsilon->0, the diffusion term D/hbar_tilde^2 r^2 in (19) diverges, so the Gaussian solution (23)-(27) becomes singular in r, and the amplitude A whose second derivative defines Q via (7b) is not defined. Qres in Eq. (32) is obtained by formally setting epsilon->0 in Eq. (30), but this does not commute with taking the limit of the dynamics. A finite Qres can survive even when the underlying density matrix collapses, so the residual force may be an artifact of the scaling rather than a genuine classical-limit force. Moreover, Eq. (20) is proposed rather than derived, and it requires Qvan, which is defined only after one already knows Qres from the quantum solution; this makes the classical equation circular. The standard hbar->0 limit of the CL master equation (in Wigner form) is a Fokker-Planck/Kramers equation with stochastic forces, not a deterministic trajectory equation with a residual potential. Thus the paper does not establish that the standard CL model yields deterministic temperature-dependent classical trajectories.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the high-temperature Caldeira-Leggett (CL) master equation in a Bohmian hydrodynamic formulation. Writing the reduced density matrix in polar form leads to generalized Hamilton-Jacobi and continuity equations, with an effective potential Q that depends on both the quantum state and the thermal environment. The authors then introduce a 'scaled' CL equation in which the Planck constant is replaced by sqrt(epsilon) hbar and the density matrix is correspondingly rescaled. For Gaussian initial states they obtain closed-form expressions for the density matrix, coherence, and effective potential. Setting epsilon = 0 in the effective potential yields a residual potential Q_res, which is used to write a deterministic, temperature-dependent 'classical' equation of motion, Eq. (15). The paper also analyzes single-Gaussian and cat-state coherence, showing suppression by temperature and by decreasing epsilon. The central claim is that deterministic classical trajectories with residual thermal forces emerge from the CL model, in contrast to stochastic Langevin dynamics.","tokens_in":17228,"tokens_out":4949,"duration_ms":49922,"significance":"If the central claim were established, it would be a notable conceptual result: temperature would enter deterministic trajectory equations in the classical limit through a residual effective potential, rather than through noise. The paper also provides explicit analytic Gaussian solutions and coherence formulas, which are useful reference calculations within the CL framework. However, the load-bearing steps of the argument are not derived: the scaled CL equation is postulated, and the classical limit is extracted by formally setting epsilon = 0 in quantities obtained from the scaled quantum solution. These gaps directly undermine the claimed emergence of deterministic temperature-dependent classical trajectories.","major_comments":[{"comment":"The scaled CL equation is not derived. It is introduced by replacing hbar with sqrt(epsilon) hbar in the standard CL equation, and the authors explicitly concede that the construction 'does not follow from an explicit derivation within the present framework.' Moreover, Eq. (19) is singular at epsilon = 0 because the diffusion term D/\\tilde{hbar}^2 r^2 diverges, so epsilon = 0 is not a regular limit of the dynamics. Despite this, Eq. (32) defines Q_res by formally setting epsilon = 0 in Eq. (30), which is a solution obtained for epsilon > 0. The limit of the solution and the limit of the equation do not commute, so Q_res is not shown to be the classical limit of the CL model.","section":"§II.B, Eq. (19)"},{"comment":"Equation (20) is proposed as the 'classical' CL equation, but it contains Q_van, defined in Eq. (9) as Q - Q_res, where both Q and Q_res are taken from the quantum solution of the scaled equation. Thus Eq. (20) is not a self-contained classical equation: one must already know the quantum solution to write it. The resulting Eq. (15) for deterministic classical trajectories inherits this circularity. No independent derivation of Eq. (20) is given, and no proof is provided that a solution rho_cl exists with the imported Q_van.","section":"§II.B, Eq. (20)"},{"comment":"The claimed suppression of coherence in the classical limit is imposed by the scaling rather than derived. Eq. (42) gives the stationary l1-norm of coherence as sqrt(2 pi \\tilde{hbar}^2/(m k_B T)) = sqrt(2 pi epsilon) hbar / sqrt(m k_B T). As the paper itself notes, this is just the scaled thermal wavelength. Therefore the statement that coherence vanishes as epsilon -> 0 is a direct consequence of the ansatz (18), not of the CL dynamics. The same applies to the short-time terms in Eq. (41) that scale as 1/epsilon and 1/epsilon^2.","section":"§III.A, Eq. (42)"},{"comment":"The temperature dependence of the claimed classical trajectories is weaker than advertised. The centroid x_t in Eq. (29) is independent of temperature, and the residual force F_res in Eq. (36) vanishes exactly at x = x_t. Thus for the Gaussian example the central trajectory is not temperature-dependent; only trajectories starting away from the center feel temperature through F_res. Moreover, F_res is computed from the epsilon -> 0 limit of the scaled quantum potential, so it carries the same circularity as Eq. (20). The paper should state this limitation explicitly and should not claim that the CL model generically yields temperature-dependent deterministic classical trajectories without a derivation independent of the scaled ansatz.","section":"§III.A, Eqs. (29) and (36)"}],"minor_comments":[{"comment":"The typesetting of tilde quantities is inconsistent: Eqs. (23)-(27) use symbols like e\\rho, e\\hbar, and e\\Gamma that are hard to distinguish from the constant e. Please use a consistent tilde notation for scaled variables.","section":"Notation throughout"},{"comment":"The phrase 'Due to the linearity of the master equation (19) with r = 0' is unclear. Linearity of Eq. (19) does not depend on setting r = 0; the subsequent superposition argument should be phrased in terms of linearity of the full equation.","section":"§III.B, before Eq. (46)"},{"comment":"The expression for \\tilde{\\rho}(r,R,t) contains \\tilde{w}_t in the prefactor; it would help to state explicitly that the inverse Fourier transform is carried out with respect to the center-of-mass coordinate, since the derivation is only summarized in the text.","section":"§III.A, Eq. (23)"},{"comment":"The paper relies heavily on the scaled von Neumann framework of Ref. [15], but the equivalence between the nonlinear QCT equation and the linear scaled equation is only cited, not reproduced. A brief self-contained statement of that equivalence would make the paper more readable.","section":"References"}],"recommendation":"reject","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the polar-form decomposition of the standard Caldeira-Leggett equation (Eqs. 5–6) is clean, and the Gaussian wave-packet calculations are carried out carefully. Second, the central claim—that the effective potential survives in the classical limit and yields deterministic temperature-dependent classical trajectories—does not survive contact with the paper's own scaling assumptions. The scaled CL equation (19) is explicitly admitted not to be derived, and its epsilon→0 limit is singular: the diffusion term D/(ħ̃²)r² diverges, so the Gaussian solution becomes singular in r and the amplitude whose second derivative defines Q is not well defined. Taking Eq. (30) and formally setting epsilon=0 gives Q_res (32), but that limit does not commute with taking the limit of the dynamics. The paper itself says the linear scaled equation “does not provide a proper description of the classical regime.” On top of that, the proposed classical equation (20) is simply asserted, and it requires Q_van, which is only defined after one has already solved the quantum problem for Q. So Eq. (15) is not established as a consequence of the CL model.\n\nWhat is genuinely new? The decomposition of the effective potential into residual and vanishing parts, and the explicit formulas for Q and Q_res for a Gaussian state, are not in the earlier literature as far as I can tell. The Bohmian treatment of CL is standard in outline, but the algebra here is thorough. The coherence-decay results are also explicit and internally consistent—though the stationary coherence scaling as sqrt(epsilon) is just the scaled thermal wavelength, so calling it a “suppression of coherence” is partly tautological.\n\nThe soft spots are load-bearing, not cosmetic. A referee would need the authors to either derive the scaled equation from a microscopic model or show that the epsilon→0 limit makes sense as a limit of the CL dynamics. The residual force also vanishes at the trajectory center, so the advertised “temperature-dependent classical trajectories” only affect off-center trajectories—a weaker statement than the abstract suggests.\n\nWho is this for? People working on Bohmian mechanics and the quantum-classical transition may find the polar decomposition useful, and the Gaussian solution is a nice reference calculation. But I would not cite the main claim. It deserves a serious referee, though my own verdict is skeptical: the classical-limit trajectory picture is unsupported as it stands.","headline":"Polar decomposition of the Caldeira-Leggett equation is clean, but the central claim of deterministic temperature-dependent classical trajectories rests on an ad hoc scaled equation with a singular epsilon→0 limit.","tokens_in":17583,"tokens_out":3071,"would_cite":false,"duration_ms":28841,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Thermal environments can yield deterministic classical trajectories in the Caldeira-Leggett model, not stochastic ones.","keywords":["Caldeira-Leggett model","Bohmian mechanics","quantum-classical transition","decoherence","dissipative dynamics","Gaussian cat states","quantumness parameter","effective potential"],"falsifier":"Solve the scaled Caldeira-Leggett equation for a non-Gaussian initial state and take the ε -> 0 limit of the effective potential: if the residual potential Q_res diverges or acquires an explicit hbar dependence, the deterministic classical-trajectory claim collapses. Alternatively, measure individual trajectories of particles in a thermal bath: deterministic temperature-dependent forces would produce reproducible, non-random off-center deviations, whereas Langevin noise would produce shot-to-shot stochastic spreading.","tokens_in":16687,"feed_emoji":"🌡️","tokens_out":4569,"duration_ms":41123,"temperature":0.7,"pith_summary":"The paper analyzes the high-temperature Caldeira-Leggett model in a Bohmian polar-form language and claims that the effective potential governing the phase does not disappear in the classical limit. Instead of vanishing, it leaves a residual, temperature-dependent term that modifies the classical equations of motion. As a result, the ensemble of classical trajectories remains deterministic even under thermal noise, in sharp contrast to the Langevin picture where thermal effects appear as random forces. The authors also introduce a scaled version of the Caldeira-Leggett equation using a quantumness parameter that interpolates between quantum and classical regimes, and show for Gaussian states that decoherence is suppressed as the parameter decreases. A sympathetic reader would care because this offers an alternative, ensemble-based route to classicality that avoids stochastic trajectories.","feed_headline":"Thermal baths need not make classical trajectories random","feed_subtitle":"In a Caldeira-Leggett Bohmian treatment, thermal effects survive as a deterministic residual potential, unlike the stochastic Langevin pictu","key_machinery":"The key machinery is the polar decomposition of the reduced density matrix, ρ = A e^{iS/hbar}, which yields a generalized Hamilton-Jacobi equation (5) and continuity equation (6). From these, the effective potential Q = -hbar^2/(m A) A_{rR} is defined; its classical-limit residual Q_res is obtained by taking the ε -> 0 limit of the scaled potential. The scaled Caldeira-Leggett equation (19) is introduced by scaling both hbar and ρ via the quantumness parameter, and the classical equation (20) subtracts Q_van from the external potential to maintain the classical Hamilton-Jacobi form.","core_discovery":"Using the polar form of the reduced density matrix in the high-temperature Caldeira-Leggett equation, the authors derive a generalized Hamilton-Jacobi equation in which the effective potential Q does not vanish in the classical limit. They split Q into a term Q_van that vanishes in that limit and a residual term Q_res that survives. The residual term is generally non-additive in quantum and thermal parts and depends on temperature. As a result, the classical Newtonian-like equation retains a deterministic force from the environment: x¨ = -2γ x˙ - (1/m) V' - (1/m) Q_res'. The authors verify this structure explicitly for Gaussian wave packets, computing Q_res (Eq. 32) and the residual force F_","pith_inferences":["The paper's deterministic-trajectory picture suggests that Bohmian-type trajectory ensembles for open systems could be reconciled with thermal environments without invoking hidden noise; whether this extends to non-Markovian baths is untested.","The residual force F_res might be observable as a temperature-dependent spread in the velocity field of a prepared ensemble, distinguishable from stochastic Brownian motion by its deterministic, reproducible form.","The scaling construction implies a smooth interpolation of decoherence rates with the quantumness parameter; this could serve as a benchmark for numerical simulations of the quantum-classical transition in more complex potentials."],"forward_implications":["If correct, thermal decoherence does not necessarily require stochastic forces; the environment acts through a deterministic effective force field.","The stationary coherence of a Gaussian state is the scaled thermal wavelength sqrt(2π hbar^2/(m k_B T)), decreasing with temperature and with the quantumness parameter.","The interference pattern of a Schrödinger cat state is suppressed monotonically, with the attenuation coefficient approaching zero as ε -> 0.","The classical limit is not obtained by a straightforward ε -> 0 of the linear scaled equation; a nonlinear classical equation (20) is needed, highlighting a structural discontinuity in the transition."],"fun_headline_variants":["Thermal baths don't randomize classical paths in scaled Caldeira-Leggett","Residual potential keeps classical trajectories deterministic despite heat","Bohmian open system: deterministic classical dynamics from thermal bath","Scaled Caldeira-Leggett exposes residual force in classical limit","Quantum-to-classical transition retains deterministic thermal effect"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The central claim rests on the scaled Caldeira-Leggett equation being a valid model of the quantum-to-classical transition even though it is constructed by analogy rather than derived from a microscopic system-plus-bath Hamiltonian, and on the identification of Q_res as the ε -> 0 limit of Q despite the linear scaled equation not providing a proper classical description.","fun_headline_variants_meta":{"raw":{"variants":["Thermal baths don't randomize classical paths in scaled Caldeira-Leggett","Residual potential keeps classical trajectories deterministic despite heat","Bohmian open system: deterministic classical dynamics from thermal bath","Scaled Caldeira-Leggett exposes residual force in classical limit","Quantum-to-classical transition retains deterministic thermal effect"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1226,"prompt_tokens":766,"completion_tokens":460,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":373}},"tokens_in":510,"tokens_out":460,"duration_ms":4628,"temperature":1.0,"reasoning_tokens":373,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T16:01:58.094852+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the scaled Caldeira-Leggett equation for a non-Gaussian initial state and take the ε -> 0 limit of the effective potential: if the residual potential Q_res diverges or acquires an explicit hbar dependence, the deterministic classical-trajectory claim collapses. Alternatively, measure individual trajectories of particles in a thermal bath: deterministic temperature-dependent forces would produce reproducible, non-random off-center deviations, whereas Langevin noise would produce shot-to-shot stochastic spreading.","supporting_citations":[],"review_version":1}