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arxiv: 2604.27459 · v1 · submitted 2026-04-30 · 🪐 quant-ph

Galilean boost invariance does not survive the trace: symmetry breaking in open quantum systems

Leonardo F. Calder\'on , Esteban Marulanda , Santiago Morales , Leonardo A. Pach\'on This is my paper

Pith reviewed 2026-05-07 10:05 UTC · model grok-4.3

classification 🪐 quant-ph
keywords open quantum systemsGalilean invariancesymmetry breakingCaldeira-Leggett modelHu-Paz-Zhang master equationdissipationfluctuation-dissipation theoremboost covariance
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The pith

Tracing out a Galilean-invariant environment breaks boost covariance of the reduced quantum dynamics

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

The paper shows that Galilean boost invariance of the full closed system does not carry over once the environment is traced out, leaving the reduced dynamics without boost covariance. Spatial translations and rotations continue to hold as symmetries of the open system. The breaking originates solely in the dissipative anticommutator term of the master equation, which is fixed by the fluctuation-dissipation relation to the bath's absorptive spectrum. This incompatibility means that bilinear coupling to any non-trivial bath spectrum prevents Galilean invariance, the fluctuation-dissipation theorem, and reduced boost covariance from holding together. The result applies directly to systems such as cold atoms in optical lattices and ultracold molecules, which sit near the crossover set by the ratio of damping strength to temperature.

Core claim

Tracing out a Galilean-invariant Caldeira-Leggett environment breaks Galilean boost covariance of the reduced dynamics, while spatial translations and rotations survive intact. An operator-level analysis of the exact Hu-Paz-Zhang master equation localizes the violation entirely in the dissipative anticommutator term, scaling with the damping coefficient Γ(t)f(t). The fluctuation-dissipation theorem ties this coefficient to the absorptive bath response that drives equilibrium momentum diffusion, so for any non-trivial bath spectral density bilinear-coupled Galilean invariance, the fluctuation-dissipation theorem, and reduced boost covariance cannot hold simultaneously. The stochastic decompo

What carries the argument

The dissipative anticommutator term in the Hu-Paz-Zhang master equation that scales with Γ(t)f(t) and is fixed by the fluctuation-dissipation theorem to the bath absorptive response

Load-bearing premise

The environment is bilinearly coupled to the system and obeys the fluctuation-dissipation theorem through its absorptive response

What would settle it

An explicit check of whether the reduced master equation remains form-invariant when the system operators are transformed under a Galilean boost, specifically testing if the dissipative term acquires an extra contribution

Figures

Figures reproduced from arXiv: 2604.27459 by Esteban Marulanda, Leonardo A. Pach\'on, Leonardo F. Calder\'on, Santiago Morales.

Figure 1
Figure 1. Figure 1: FIG. 1. Dimensionless boost-violation parameter view at source ↗
read the original abstract

Tracing out a Galilean-invariant Caldeira-Leggett environment breaks Galilean boost covariance of the reduced dynamics, while spatial translations and rotations survive intact. An operator-level analysis of the exact Hu-Paz-Zhang master equation localizes the violation entirely in the dissipative anticommutator term, scaling with the damping coefficient $\Gamma(t)f(t)$. The fluctuation-dissipation theorem ties this coefficient to the absorptive bath response that drives equilibrium momentum diffusion, so for any non-trivial bath spectral density bilinear-coupled Galilean invariance, the fluctuation-dissipation theorem, and reduced boost covariance cannot hold simultaneously. The stochastic decomposition of the influence functional extends the mechanism beyond the quadratic regime. The dimensionless ratio $\hbar\gamma/k_\mathrm{B} T$ delineates the crossover: cold atoms in dissipative optical lattices and ultracold molecules sit at its edge. Parametric driving offers a one-directional escape: the squeezing rate that protects nonequilibrium entanglement above the standard quantum limit also suppresses boost-breaking over a driving cycle.

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

2 major / 2 minor

Summary. The manuscript claims that tracing out a Galilean-invariant Caldeira-Leggett environment from the exact Hu-Paz-Zhang master equation breaks Galilean boost covariance of the reduced dynamics while preserving spatial translations and rotations. The violation is localized entirely to the dissipative anticommutator term scaling with the damping coefficient Γ(t)f(t), which is tied via the fluctuation-dissipation theorem to the bath absorptive response. For any non-trivial bilinearly coupled bath spectral density, Galilean invariance, the FDT, and reduced boost covariance cannot hold simultaneously. A stochastic decomposition of the influence functional extends the result beyond the quadratic regime, with the dimensionless ratio ħγ/k_B T marking a crossover relevant to cold atoms and ultracold molecules; parametric driving is proposed as a partial mitigation.

Significance. If the central claim is substantiated, the result identifies a fundamental tension between total-system Galilean invariance, the fluctuation-dissipation theorem, and covariance of reduced open-system dynamics under bilinear coupling. This has direct implications for symmetry considerations in dissipative quantum systems, particularly in ultracold atomic and molecular experiments where the ħγ/k_B T ratio is experimentally accessible. The operator-level analysis of the HPZ equation and the stochastic extension of the influence functional are positive features that strengthen the mechanistic insight.

major comments (2)
  1. [operator-level analysis of the Hu-Paz-Zhang master equation] The operator-level analysis asserts that the boost violation is localized exclusively to the dissipative anticommutator term Γ(t)f(t) after applying the Galilean boost operator U(v) = exp(-i v (X_total t - t P_total)/ℏ) to the total system+bath Hilbert space and performing the partial trace. However, the transformed coefficients for the unitary (Lamb-shift) term, the diffusion kernel, and the time-dependent functions are not explicitly derived or displayed, leaving open the possibility that non-covariant contributions appear elsewhere or that the boost does not commute with the trace in the expected manner. This step is load-bearing for the symmetry-breaking conclusion.
  2. [discussion of the fluctuation-dissipation theorem and parametric driving] The claim that Galilean invariance, the fluctuation-dissipation theorem, and reduced boost covariance are mutually incompatible for any non-trivial bath spectral density relies on linking Γ(t) to the absorptive bath response. An explicit check of how the boosted bath spectral density transforms under the total boost and subsequent tracing would strengthen this incompatibility statement, particularly for the crossover behavior governed by the dimensionless ratio ħγ/k_B T.
minor comments (2)
  1. The time-dependent functions Γ(t) and f(t) appearing in the dissipative term should be defined explicitly with their relation to the bath spectral density at the first occurrence in the main text.
  2. The stochastic decomposition of the influence functional is invoked to extend the result beyond the quadratic regime, but a brief outline of the key steps in that decomposition would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. These have prompted us to strengthen the explicit derivations and discussion of the fluctuation-dissipation relation. We address each point below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: The operator-level analysis asserts that the boost violation is localized exclusively to the dissipative anticommutator term Γ(t)f(t) after applying the Galilean boost operator U(v) = exp(-i v (X_total t - t P_total)/ℏ) to the total system+bath Hilbert space and performing the partial trace. However, the transformed coefficients for the unitary (Lamb-shift) term, the diffusion kernel, and the time-dependent functions are not explicitly derived or displayed, leaving open the possibility that non-covariant contributions appear elsewhere or that the boost does not commute with the trace in the expected manner. This step is load-bearing for the symmetry-breaking conclusion.

    Authors: We agree that displaying the full set of transformed coefficients improves transparency. In the revised manuscript we have added an appendix that explicitly computes the action of the total boost operator U(v) on each term of the Hu-Paz-Zhang master equation before the partial trace. The resulting expressions for the Lamb-shift (unitary) coefficients, the diffusion kernel, and all time-dependent functions are shown; only the anticommutator term proportional to Γ(t)f(t) acquires a non-covariant piece linear in the boost velocity v. All other terms remain form-invariant under the subsequent trace, confirming that the boost commutes with the partial trace in the manner required by the total-system Galilean invariance. This explicit calculation removes the ambiguity noted by the referee. revision: yes

  2. Referee: The claim that Galilean invariance, the fluctuation-dissipation theorem, and reduced boost covariance are mutually incompatible for any non-trivial bath spectral density relies on linking Γ(t) to the absorptive bath response. An explicit check of how the boosted bath spectral density transforms under the total boost and subsequent tracing would strengthen this incompatibility statement, particularly for the crossover behavior governed by the dimensionless ratio ħγ/k_B T.

    Authors: We have incorporated the requested explicit transformation. Under the total-system Galilean boost the bath spectral density J(ω) is unchanged because the bilinear coupling and the bath Hamiltonian are both Galilean invariant; consequently the absorptive response χ''(ω) that enters the fluctuation-dissipation theorem remains identical. After the partial trace, however, the reduced damping coefficient Γ(t) inherits a velocity-dependent correction that violates boost covariance of the master equation. We now display this step-by-step transformation in the main text and have expanded the discussion of the dimensionless ratio ħγ/k_B T, including its numerical value for typical cold-atom and ultracold-molecule parameters and the manner in which parametric driving suppresses the breaking term over a drive cycle. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard HPZ equation and FDT independently

full rationale

The paper's central claim follows from applying the established Hu-Paz-Zhang master equation (derived in prior independent literature) to a bilinearly coupled Galilean-invariant bath, then using the fluctuation-dissipation theorem (a standard physical relation, not fitted here) to identify the dissipative anticommutator term as the sole source of boost non-covariance. No step reduces a prediction to a fitted input by construction, renames a known result, or relies on a load-bearing self-citation whose content is unverified or tautological. The operator-level localization and stochastic extension are presented as direct consequences of the master-equation structure and FDT without self-referential closure. The derivation remains self-contained against external benchmarks such as the standard open-systems literature.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The claim rests on the Caldeira-Leggett bilinear-coupling model and the validity of the Hu-Paz-Zhang master equation plus the fluctuation-dissipation theorem for the bath response.

free parameters (2)
  • damping coefficient Γ(t)
    Scales the dissipative term that produces the boost breaking; taken from the master equation.
  • dimensionless ratio ħγ/k_B T
    Delineates the temperature-damping crossover; introduced to characterize regimes.
axioms (2)
  • domain assumption The environment is a Caldeira-Leggett bath with bilinear system-bath coupling.
    Required to obtain the Hu-Paz-Zhang master equation whose dissipative term is analyzed.
  • domain assumption The fluctuation-dissipation theorem holds and relates the dissipative coefficient to the bath's absorptive response.
    Used to connect Γ(t)f(t) to the bath spectrum and establish the incompatibility.

pith-pipeline@v0.9.0 · 5483 in / 1460 out tokens · 66825 ms · 2026-05-07T10:05:23.230154+00:00 · methodology

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev. Mod. Phys.79, 555 (2007)

    work page 2007

  2. [2]

    Zanardi, Phys

    P. Zanardi, Phys. Rev. Lett.87, 077901 (2001)

    work page 2001

  3. [3]

    Schlosshauer,Decoherence and the Quantum-to- Classical Transition(Springer, Berlin, 2007)

    M. Schlosshauer,Decoherence and the Quantum-to- Classical Transition(Springer, Berlin, 2007)

    work page 2007

  4. [4]

    Gasbarri, M

    G. Gasbarri, M. Toroš, and A. Bassi, Phys. Rev. Lett.119, 100403 (2017)

    work page 2017

  5. [5]

    A. S. Holevo, J. Math. Phys.37, 1812 (1996)

    work page 1996

  6. [6]

    A. S. Holevo, Rep. Math. Phys.32, 211 (1993)

    work page 1993

  7. [7]

    A. S. Holevo, Rep. Math. Phys.33, 95 (1993)

    work page 1993

  8. [8]

    A. S. Holevo, Izv. Math.59, 427 (1995)

    work page 1995

  9. [9]

    Bassi, K

    A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Rev. Mod. Phys.85, 471 (2013)

    work page 2013

  10. [10]

    Lombardi, M

    O. Lombardi, M. Castagnino, and J. S. Ardenghi, Stud. Hist. Philos. Mod. Phys.41, 93 (2010)

    work page 2010

  11. [11]

    L. E. Ballentine,Quantum Mechanics: A Modern Develop- ment, 2nd ed. (World Scientific, Singapore, 1998)

    work page 1998

  12. [12]

    A. O. Caldeira and A. J. Leggett, Physica A121, 587 (1983)

    work page 1983

  13. [13]

    Hakim and V

    V . Hakim and V . Ambegaokar, Phys. Rev. A32, 423 (1985)

    work page 1985

  14. [14]

    B. L. Hu, J. P. Paz, and Y . Zhang, Phys. Rev. D45, 2843 (1992)

    work page 1992

  15. [15]

    Smirne and B

    A. Smirne and B. Vacchini, Phys. Rev. A82, 022110 (2010)

    work page 2010

  16. [16]

    L. A. Pachón, J. F. Triana, D. Zueco, and P. Brumer, J. Chem. Phys.150, 034105 (2019)

    work page 2019

  17. [17]

    Grynberg and C

    G. Grynberg and C. Robilliard, Phys. Rep.355, 335 (2001)

    work page 2001

  18. [18]

    Nimmrichter and K

    S. Nimmrichter and K. Hornberger, Phys. Rev. Lett.110, 160403 (2013)

    work page 2013

  19. [19]

    J. T. Stockburger and H. Grabert, Phys. Rev. Lett.88, 170407 (2002)

    work page 2002

  20. [20]

    W. Koch, F. Großmann, J. T. Stockburger, and J. Ankerhold, Phys. Rev. Lett.100, 230402 (2008)

    work page 2008

  21. [21]

    Diósi, N

    L. Diósi, N. Gisin, and W. T. Strunz, Phys. Rev. A58, 1699 (1998)

    work page 1998

  22. [22]

    Grabert, P

    H. Grabert, P. Schramm, and G.-L. Ingold, Phys. Rep.168, 115 (1988)

    work page 1988

  23. [23]

    Štelmachovi ˇc and V

    P. Štelmachovi ˇc and V . Bužek, Phys. Rev. A64, 062106 (2001)

    work page 2001

  24. [24]

    Talkner, E

    P. Talkner, E. Lutz, and P. Hänggi, Phys. Rev. E73, 046131 (2006)

    work page 2006

  25. [25]

    Galve, L

    F. Galve, L. A. Pachón, and D. Zueco, Phys. Rev. Lett.105, 180501 (2010)

    work page 2010

  26. [26]

    A. F. Estrada and L. A. Pachón, New J. Phys.17, 033038 (2015)

    work page 2015

  27. [27]

    J. J. Halliwell and T. Yu, Phys. Rev. D53, 2012 (1996)

    work page 2012

  28. [28]

    Anastopoulos and B

    C. Anastopoulos and B. L. Hu, Phys. Rev. A62, 033821 (2000)

    work page 2000

  29. [29]

    C. H. Fleming and B. L. Hu, Ann. Phys. (NY)327, 1238 (2012)

    work page 2012

  30. [30]

    J. R. Anglin, J. P. Paz, and W. H. Zurek, Phys. Rev. A55, 4041 (1997)

    work page 1997

  31. [31]

    Arndt and K

    M. Arndt and K. Hornberger, Nat. Phys.10, 271 (2014)

    work page 2014

  32. [32]

    Gonzalez-Ballestero, M

    C. Gonzalez-Ballestero, M. Aspelmeyer, L. Novotny, R. Quidant, and O. Romero-Isart, Science374, eabg3027 (2021)

    work page 2021

  33. [33]

    Aspelmeyer, T

    M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys.86, 1391 (2014)

    work page 2014

  34. [34]

    Leibfried, R

    D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys.75, 281 (2003)

    work page 2003

  35. [35]

    J. L. Bohn, A. M. Rey, and J. Ye, Science357, 1002 (2017)

    work page 2017

  36. [36]

    See Supplemental Material at [URL] for the strictly invari- ant Lagrangian variant and the free-particle limit, the ex- plicit cancellation pattern of the master equation under boost transformations, and the Matsubara analysis of the high- temperature expansion off(t)

    work page