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REVIEW 4 major objections 4 minor 48 references

Scaled Caldeira-Leggett Dynamics: Deterministic Trajectories and the Classical Limit

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Thermal environments can yield deterministic classical trajectories in the Caldeira-Leggett model, not stochastic ones.

desk verdict 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. read the letter →

arxiv 2608.02054 v1 pith:C2LZZ5PQ submitted 2026-08-03 quant-ph

S. V. Mousavi , S. Miret-Artés This is my paper · ORCID
classification quant-ph
keywords Caldeira-LeggettmodelBohmianmechanicsquantum-classicaltransitiondecoherencedissipativedynamicsGaussiancatstatesquantumnessparametereffectivepotential
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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_

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

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.

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 (4)
  1. [§II.B, Eq. (19)] 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.
  2. [§II.B, Eq. (20)] 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.
  3. [§III.A, Eq. (42)] 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.
  4. [§III.A, Eqs. (29) and (36)] 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.
minor comments (4)
  1. [Notation throughout] 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.
  2. [§III.B, before Eq. (46)] 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.
  3. [§III.A, Eq. (23)] 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.
  4. [References] 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.

Circularity Check

3 steps flagged · score 7.0 of 10

Central 'classical limit' claims are built into the scaled-CL ansatz: Qres is the ε→0 value of an admitted ad hoc equation, and the √ε coherence suppression restates the scaling ℏ→√εℏ.

  1. self definitional [Sec. III.A, Eq. (42), with Eq. (18)]
    "eCℓ1(t)|_{t→∞} = sqrt(2π e¯h^2/(m k_B T)). The stationary value is just the scaled thermal wavelength. This behavior shows that, within a given regime, coherence decreases with increasing temperature, while for a fixed temperature it vanishes in the classical limit ϵ→0."

    Equation (42) is obtained from the scaled solution of Eq. (19), in which e¯h = √ε ℏ (Eq. 18). The claimed 'vanishing in the classical limit' is therefore just the algebraic statement e¯h→0 as ε→0. The quantumness parameter controls coherence only because it was inserted into the diffusion-to-Planck-constant ratio D/e¯h^2 in Eq. (19); no independent physical input produces the √ε factor. Thus the coherence-suppression 'prediction' is equivalent to the scaling definition.

  2. self definitional [Sec. II.B, Eqs. (18)–(20), (30), (32), and Sec. II.A Eq. (15)]
    "In the classical regime ϵ=0, the effective potential reduces to Qres(r,R,t) = ... which represents the residual contribution that survives in this regime. ... the scaled linear CL equation does not provide a proper description of the classical regime. In particular, the limit toward classicality is not obtained in a straightforward manner at the level of the dynamical equation."

    Qres is obtained by formal substitution ε=0 (i.e., e¯h=0) into Eq. (30), which is the effective potential of the scaled equation (19), not of the original CL model. The paper concedes the scaled linear equation 'does not provide a proper description of the classical regime' and that the limit is not straightforward; nevertheless Eq. (32) is fed into the 'classical' equation (20), which was constructed by removing Qvan=Q−Qres from the potential. Eq. (15) follows from Eq. (21) only because Q−Qvan=Qres by definition. Hence the residual force and deterministic classical trajectories are the ε=0 value of the scaled model renamed as a classical limit, rather than a result derived from the CL model.

1 more flagged steps
  1. ansatz smuggled in via citation [Sec. I and Sec. II.B, Eq. (19), Ref. [15]]
    "we anticipate a corresponding scaled version of the CL equation by introducing a scaled density matrix and a scaled Planck constant. This construction does not follow from an explicit derivation within the present framework, but rather provides a consistent and physically motivated extension... Recently, a linear scaled equation has been proposed to describe the QCT within the von Neumann framework for isolated quantum systems [15]."

    The central input (19) is the scaled CL equation, taken from the authors' own earlier work [15], where the scaled von Neumann equation was introduced. The paper states that this construction has no derivation within the present framework. All new claims—Qres, Eq. (15), Eq. (42)—are computed from this equation. Thus the load-bearing step is a self-citation to an ansatz, and the paper's conclusions are consequences of that ansatz rather than of the standard CL model.

full rationale

The paper is transparent that the scaled CL equation (19) is an extension that 'does not follow from an explicit derivation,' and it labels Eq. (20) as 'proposed.' But the abstract and conclusions convert these ansätze into a physical discovery: a temperature-dependent deterministic classical limit. The chain is: (i) define e¯h=√ε ℏ (18); (ii) solve the scaled CL equation (19) for Gaussians; (iii) define Qres as the ε→0 value of the scaled effective potential (32); (iv) construct Eq. (20) with Qvan removed so that Eq. (21) has V+Qres; (v) read off Eq. (15). At each step the output is an algebraic restatement of the input: Eq. (42) restates Eq. (18); Eq. (15) restates the relation Q=Qres+Qvan; Qres restates the ε→0 limit of a model whose classical limit is admitted to be ill-defined. The only independent mathematical content is the exact Gaussian solution (23)–(27). If the paper were claiming merely 'we propose a model with these consequences,' there would be no circularity; the circularity lies in presenting the consequences of the scaling ansatz as a surprising property of the Caldeira-Leggett model. The standard ℏ→0 limit of the CL master equation is stochastic (Kramers), not deterministic with a residual potential; the difference is produced entirely by the inserted scaling. No external benchmark or machine-checked result is cited for the scaled equation. Score 7 because the central claims reduce by construction, though the algebra is internally consistent.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central new results rest on the assumed scaled CL equation, the polar-form decomposition, and the epsilon -> 0 extraction of Q_res. No data or independent derivation is supplied for the scaling. No new physical entities are introduced.

free parameters (1)
  • epsilon (quantumness parameter) = chosen by hand; values 1, 0.5, 0.2, 0.1, 0.05, etc. in figures
    Interpolation parameter scaling Planck's constant via Eq. (18); not measurable and not fitted to data; all transition results depend on it.
assumptions (5)
  • domain assumption The CL master equation in the high-temperature, Born-Markov, Ohmic-bath limit (Eq. 1) is the correct reduced dynamics.
    Standard open-quantum-systems result taken as input.
  • standard math The density matrix can be written in polar form A exp(iS/hbar) with A real and even in r, S odd in r.
    Follows from Hermiticity of rho; standard polar decomposition.
  • domain assumption A scaled von Neumann equation with scaled Planck's constant sqrt(epsilon) hbar is a valid quantum-to-classical transition model (Ref. [15], by same authors).
    This is the foundation for the scaled CL equation; it is not re-derived in this paper.
  • ad hoc to paper Tracing out the environment from the scaled von Neumann equation gives Eq. (19), the scaled CL equation, by formal replacement of hbar with sqrt(epsilon) hbar in the CL equation.
    Section I explicitly says this construction 'does not follow from an explicit derivation within the present framework'.
  • ad hoc to paper In the classical limit, only the residual part Q_res of the effective potential survives, and it can be computed as Q(epsilon -> 0) of the scaled Gaussian solution even though Eq. (19) is singular there.
    Section II.B states the linear equation does not provide a proper classical description, yet Eqs. (30)-(32) take epsilon -> 0 to define Q_res.

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Pith. "Pith review of Scaled Caldeira-Leggett Dynamics: Deterministic Trajectories and the Classical Limit." pith.science (2026). https://pith.science/paper/C2LZZ5PQ

@misc{pith2026260802054,
  author       = {Pith},
  title        = {Pith review of: Scaled Caldeira-Leggett Dynamics: Deterministic Trajectories and the Classical Limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C2LZZ5PQ}},
  note         = {Machine review of arXiv:2608.02054}
}
read the original abstract

The dynamics of an open quantum system in the high-temperature regime of the Caldeira-Leggett model using a Bohmian language is investigated. By expressing the density matrix in polar form, generalized continuity and Hamilton-Jacobi equations are obtained. Thermal effects appear explicitly in the former, while they influence the latter only indirectly through an effective potential. Remarkably, the effective potential does not vanish in the classical limit and cannot, in general, be decomposed into simple additive quantum and thermal contributions. Instead, it retains a nontrivial structure leading to a residual temperature dependence in the resulting dynamics. As a consequence, the resulting temperature-dependent-classical trajectories remain deterministic even in the presence of a thermal environment. This behavior contrasts with the stochastic dynamics observed in the Langevin description and reflects the ensemble-based nature of the present approach. To further explore the quantum-to-classical transition, a scaled version of the Caldeira-Leggett equation is introduced by scaling both the density matrix and Planck constant through a quantumness parameter. This parameter takes the value one in the fully quantum regime and smoothly approaches zero in the classical limit. Applications to Gaussian states demonstrate a gradual suppression of coherence and interference, governed jointly by thermal effects and the transition parameter.

Figures

Figures reproduced from arXiv: 2608.02054 by the authors.

Figure 1
Figure 1. FIG. 1: Quantum coherence given by (39). Top panels: quantum regime (left) and transition regime with [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Coherence attenuation coefficient ˜a [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Density plots of the probability density for the superposition state (51) in the [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

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