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arxiv: 2606.16037 · v1 · pith:MTZBZ5U5new · submitted 2026-06-14 · 🪐 quant-ph · cond-mat.mtrl-sci· hep-th

Adiabatically-induced Kawaguchi geometry and jerk in quantum-classical systems

Athanasios Chatzistavrakidis , Larisa Jonke , Ryan Requist This is my paper

Pith reviewed 2026-06-27 03:15 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mtrl-scihep-th
keywords adiabatic eliminationKawaguchi geometryjerkquantum-classical systemseffective actionnon-Newtonian dynamicsnonadiabatic effects
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The pith

Adiabatic elimination of quantum degrees of freedom produces jerk-dependent forces in classical equations of motion at third order.

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

The paper derives a hierarchy of effective actions for classical variables in mixed quantum-classical systems by applying successive near-identity unitary transformations to the quantum state. This process eliminates the quantum degrees of freedom adiabatically to any order in the small parameter. At third order, the resulting Euler-Lagrange equation becomes non-Newtonian, with the force depending on the jerk. The third-order terms induce Kawaguchi geometry on the classical configuration space, featuring an almost symplectic structure and a line element that incorporates acceleration alongside velocity. These results enable more accurate modeling of nonadiabatic effects in simulations such as molecular dynamics.

Core claim

By applying a sequence of near-identity unitary transformations, a hierarchy of effective actions is obtained for the classical variables. The third order Euler-Lagrange equation is non-Newtonian as the force depends on the jerk. The third order terms induce a special kind of Kawaguchi geometry on the space of classical variables characterized by an almost symplectic structure and a differential line element that depends on the acceleration in addition to the velocity.

What carries the argument

The Kawaguchi geometry induced by the third-order terms, characterized by an almost symplectic structure and an acceleration-dependent differential line element.

If this is right

  • The effective dynamics captures higher-order nonadiabatic corrections in quantum-classical systems.
  • Classical equations of motion gain dependence on the third time derivative of position.
  • Molecular dynamics simulations can incorporate these effects efficiently.
  • The geometry on classical variables is almost symplectic rather than strictly symplectic.

Where Pith is reading between the lines

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

  • Similar higher-order geometries might appear in other adiabatic approximation schemes beyond quantum-classical systems.
  • Testing the almost symplectic property could involve checking conservation laws or Poisson bracket structures in numerical simulations.
  • Extensions to fourth order or higher might reveal further geometric structures on the classical space.

Load-bearing premise

A sequence of near-identity unitary transformations can generate a hierarchy of increasingly accurate effective actions for the classical variables to arbitrary order without the approximation breaking down.

What would settle it

A direct numerical comparison in a specific quantum-classical model, such as a two-level system coupled to a classical oscillator, showing whether the predicted jerk-dependent force matches the exact dynamics up to third order in the adiabatic parameter.

read the original abstract

Adiabatically eliminating the quantum degrees of freedom in a mixed quantum-classical system produces an effective force in the classical equation of motion. The elimination can be made to any order in the adiabatic parameter, generating a series of higher order forces. By applying a sequence of near-identity unitary transformations to the quantum state, we derive a hierarchy of increasingly accurate effective actions for the classical variables. The third order Euler-Lagrange equation is non-Newtonian as the force depends on the jerk, the third order time derivative of position. We find that the third order terms induce a special kind of Kawaguchi geometry on the space of classical variables. This geometry is characterized by an almost symplectic structure and a differential line element that depends on the acceleration in addition to the velocity. Our results can be used to efficiently capture higher order nonadiabatic effects in molecular dynamics simulations.

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 adiabatically eliminating quantum degrees of freedom in mixed quantum-classical systems via a sequence of near-identity unitary transformations produces a hierarchy of effective classical actions to arbitrary order in the adiabatic parameter. At third order this yields a non-Newtonian Euler-Lagrange equation whose force depends on jerk, inducing a Kawaguchi geometry on the classical variables characterized by an almost symplectic structure and a line element that depends on acceleration as well as velocity. The results are positioned as a tool for capturing higher-order nonadiabatic effects in molecular dynamics simulations.

Significance. If the derivation is placed on a rigorous footing, the work supplies a systematic route to higher-order effective dynamics without fitted parameters, which would be useful for molecular-dynamics applications. The explicit construction of the third-order jerk term and the resulting Kawaguchi geometry constitute a concrete, falsifiable prediction that can be checked against known adiabatic expansions.

major comments (2)
  1. [Section deriving the unitary transformations and effective Lagrangian] The central construction (sequence of near-identity unitaries generating the effective action hierarchy) provides no spectral-gap assumptions, convergence bounds, or explicit remainder estimates; this directly undermines the claim that the third-order jerk term and associated Kawaguchi geometry are reliable to arbitrary adiabatic order.
  2. [Section presenting the third-order Euler-Lagrange equation] No comparison of the derived third-order force against exact diagonalization, known perturbative limits, or numerical integration of the full quantum-classical equations is supplied; without such validation the non-Newtonian character cannot be confirmed as physical rather than an artifact of truncation.
minor comments (2)
  1. [Geometry section] The definition of the almost symplectic form and the differential line element should be written explicitly (e.g., as a two-form or metric expression) rather than described only in words.
  2. [Notation and preliminaries] Notation for the adiabatic parameter and the successive unitary generators is introduced without a consolidated table or list of symbols, making cross-referencing between orders cumbersome.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Section deriving the unitary transformations and effective Lagrangian] The central construction (sequence of near-identity unitaries generating the effective action hierarchy) provides no spectral-gap assumptions, convergence bounds, or explicit remainder estimates; this directly undermines the claim that the third-order jerk term and associated Kawaguchi geometry are reliable to arbitrary adiabatic order.

    Authors: The derivation proceeds perturbatively by constructing a sequence of near-identity unitaries order by order in the adiabatic parameter, yielding an explicit hierarchy of effective actions. This approach is formal and does not include spectral-gap assumptions or remainder estimates in the present manuscript, consistent with many derivations in adiabatic perturbation theory that prioritize the explicit form of the corrections. We agree that a clarifying discussion of the underlying assumptions would strengthen the work. In revision we will add a paragraph in the section on the unitary transformations noting that a spectral gap in the quantum Hamiltonian is implicitly required for the adiabatic elimination to hold and that the expansion is assumed valid for sufficiently small adiabatic parameter. revision: yes

  2. Referee: [Section presenting the third-order Euler-Lagrange equation] No comparison of the derived third-order force against exact diagonalization, known perturbative limits, or numerical integration of the full quantum-classical equations is supplied; without such validation the non-Newtonian character cannot be confirmed as physical rather than an artifact of truncation.

    Authors: The manuscript is a theoretical derivation; the jerk-dependent term arises directly as the Euler-Lagrange equation of the third-order effective Lagrangian obtained from the unitary transformations. Within this perturbative framework the non-Newtonian structure is a mathematical consequence rather than a truncation artifact. No numerical comparisons are included because the primary contribution is the analytical construction and the induced Kawaguchi geometry. We will add a sentence in the conclusions noting that direct numerical validation against exact quantum-classical dynamics constitutes a natural direction for subsequent work. revision: partial

standing simulated objections not resolved
  • Providing explicit convergence bounds or remainder estimates for the infinite hierarchy of unitary transformations would require a separate rigorous analysis that is outside the scope of the current perturbative derivation.

Circularity Check

0 steps flagged

No circularity: derivation proceeds from unitary transformations without reduction to inputs

full rationale

The paper derives the hierarchy of effective actions and the third-order non-Newtonian force (jerk dependence) by applying a sequence of near-identity unitary transformations to the quantum state, followed by adiabatic elimination. This is a standard constructive procedure in quantum-classical systems and does not reduce any claimed result to a fitted parameter, self-definition, or self-citation chain. No equations in the provided abstract or claims equate a prediction to its own input by construction, and the Kawaguchi geometry emerges as a consequence of the third-order terms rather than being presupposed. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the derivation is described at the level of unitary transformations and adiabatic ordering without numerical fitting or new postulated objects.

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discussion (0)

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Reference graph

Works this paper leans on

60 extracted references

  1. [1]

    N. G. Van Kampen. Elimination of fast variables.Physics Reports, 124:69, 1985

    1985

  2. [2]

    C. A. Mead and D. G. Truhlar. On the determination of Born DOppenheimer nu- clear motion wave functions including complications due to conical intersections and identical nuclei.J. Phys. Chem., 70:2284 D2296, 1979

    1979

  3. [3]

    H. C. Longuet-Higgins, U. ¨Opik, M. H. L. Pryce, and R. A. Sack. Studies of the Jahn- Teller effect II. The dynamical problem.Proc. R. Soc. London, Ser. A, 244:1–16, 1958

    1958

  4. [4]

    F. S. Ham. Berry’s geometric phase and the sequence of states in the Jahn-Teller effect.Phys. Rev. Lett., 58:725, 1987

    1987

  5. [5]

    Alden Mead

    C. Alden Mead. The geometric phase in molecular systems.Rev. Mod. Phys., 64:51–85, 1992

    1992

  6. [6]

    J. P. Provost and G. Vallee. Riemannian structure on manifolds of quantum states. Commun. Math. Phys., 76:289, 1980

    1980

  7. [7]

    R. Jackiw. Berry’s phase: Topological ideas from atomic, molecular and optical physics.Commun. At. Molec. Phys., 21:71, 1988

    1988

  8. [8]

    M. V. Berry.The quantum phase, five years after, pages 7–28. World Scientific, Singapore, 1989. in Geometric Phases in Physics, Eds. A. Shapere and F. Wilczek

    1989

  9. [9]

    M. V. Berry and R. Lim. The Born-Oppenheimer electric gauge force is repulsive near degeneracies.J. Phys. A: Math. Gen, 23:L655–L657, 1990

    1990

  10. [10]

    M. V. Berry and J. M. Robbins. Classical geometric forces of reaction: an exactly solvable model.Proc. R. Soc. Lond. A, 442:641–658, 1993

    1993

  11. [11]

    M. V. Berry. Quantal phase factors accompanying adiabatic changes.Proc. Roy. Soc. Lond. A, 392:45, 1984. 28

    1984

  12. [12]

    F. A. Bornemann, P. Nettesheim, and C. Sch¨ utte. Quantum-classical molecular dy- namics as an approximation to full quantum dynamics.J. Chem. Phys., 105:1074, 1996

    1996

  13. [13]

    O. V. Prezhdo and V. V. Kisil. Mixing quantum and classical mechanics.Phys. Rev. A, 56:162, 1997

    1997

  14. [14]

    J. C. Tully. Mixed quantum-classical dynamics.Faraday Discuss., 110:407, 1998

    1998

  15. [15]

    T. N. Todorov. Time-dependent tight binding.J. Phys.: Condens. Matter, 13:10125, 2001

    2001

  16. [16]

    J. L. Alonso, A. Castro, J. Clemente-Gallardo, J. C. Cuch´ ı, P. Echenique, and F. Fal- ceto. Statistics and Nos´ e formalism for Ehrenfest dynamics.J. Phys. A: Math. Theor., 44:395004, 2011

    2011

  17. [17]

    Born and V

    M. Born and V. Fock. Beweis des Adiabatensatzes.Z. Physik, 51:165 D180, 1928

    1928

  18. [18]

    T. Kato. On the adiabatic theorem of quantum mechanics.J. Phys. Soc. Jpn., 5:435– 439, 1950

    1950

  19. [19]

    M. V. Berry and P. Shukla. High-order classical adiabatic reaction forces: slow mani- fold for a spin model.J. Phys. A: Math. Theor., 43:045102, 2010

    2010

  20. [20]

    R. G. Littlejohn and S. Weigert. Adiabatic motion of a neutral spinning particle in an inhomogeneous magnetic field.Phys. Rev. A, 48:924, 1993

    1993

  21. [21]

    Weigert and R

    S. Weigert and R. G. Littlejohn. Diagonalization of multicomponent wave equations with a Born-Oppenheimer example.Phys. Rev. A, 47:3506, 1993

    1993

  22. [22]

    Ceresoli, R

    D. Ceresoli, R. Marchetti, and E. Tosatti. Electron-corrected Lorentz forces in solids and molecules in a magnetic field.Phys. Rev. B, 75:161101(R), 2007

    2007

  23. [23]

    Culpitt, L

    T. Culpitt, L. D. M. Peters, E. I. Tellgren, and T. Helgaker. Ab initio molecular dynamics with screened Lorentz forces. I. calculation and atomic charge interpretation of Berry curvature.J. Chem. Phys., 155:024104, 2021

    2021

  24. [24]

    A. S. Goldhaber. Newtonian adiabatics unified.Phys. Rev. A, 71:062102, 2005

    2005

  25. [25]

    Geometry in an n-Dimensional space with the Arc Lengths=R {ai(x, x′)x′′i +b(x, x ′)}1/pdt.Trans

    Akitsugu Kawaguchi. Geometry in an n-Dimensional space with the Arc Lengths=R {ai(x, x′)x′′i +b(x, x ′)}1/pdt.Trans. Amer. Math. Soc., 44(2):153–167, 1938

    1938

  26. [26]

    M. V. Losik. On a certain class of Kawaguchi spaces.Dokl. Akad. Nauk SSSR, 134(6):1299–1302, 1960

    1960

  27. [27]

    Panzhenskii

    V.I. Panzhenskii. Motions in Kawaguchi spaces with special metric.Russ Math., 51:78–83, 2007

    2007

  28. [28]

    Bauer, P

    W. Bauer, P. Bergold, F. Gay-Balmaz, and C. Tronci. Koopman trajectories in nona- diabatic quantum-classical dynamics.Multiscale Model. Simul., 22:1365, 2024. 29

    2024

  29. [29]

    R. Requist. Quantum covariant derivative: a tool for deriving adiabatic perturbation theory to all orders.J. Phys. A, 56(46):465301, 2023

    2023

  30. [30]

    Teufel G

    S. Teufel G. Panati, H. Spohn. Space-adiabatic perturbation theory.Adv. Theor. Math. Phys., 7:145, 2003

    2003

  31. [31]

    Requist and O

    R. Requist and O. Pankratov. Adiabatic approximation in time-dependent reduced- density-matrix functional theory.Phys. Rev. A, 81:042519, 2010

    2010

  32. [32]

    Scherrer, F

    A. Scherrer, F. Agostini, D. Sebastiani, E. K. U. Gross, and R. Vuilleumier. On the mass of atoms in molecules: beyond the Born-Oppenheimer approximation.Phys. Rev. X, 7:031035, 2017

    2017

  33. [33]

    M´ atyus and S

    E. M´ atyus and S. Teufel. Effective non-adiabatic Hamiltonians for the quantum nuclear motion over coupled electronic states.J. Chem. Phys., 151:014113, 2019

    2019

  34. [34]

    Littlejohn, J

    R. Littlejohn, J. Rawlinson, and J. Subotnik. Diagonalizing the Born-Oppenheimer Hamiltonian via Moyal perturbation theory, nonadiabatic corrections, and transla- tional degrees of freedom.J. Chem. Phys., 160:114103, 2024

    2024

  35. [35]

    Y. Ren. Nonadiabatic origin of quantum-metric effects via momentum-space metric tensor.Phys. Rev. B, 113:134309, 2026

    2026

  36. [36]

    M. V. Berry. Quantum phase corrections from adiabatic iteration.Proc. R. Soc. Lond. A, 414:31, 1987

    1987

  37. [37]

    G. Hori. Theory of general perturbations with unspecified canonical variables.Pub. Astron. Soc. Jap., 18:287, 1966

    1966

  38. [38]

    A. Deprit. Canonical transformations depending on a small parameter.Cel. Mech., 1:12, 1969

    1969

  39. [39]

    A. J. Dragt and J. M. Finn. Lie series and invariant functions for analytic symplectic maps.J. Math. Phys., 17:2215, 1976

    1976

  40. [40]

    J. R. Cary. Lie transform perturbation theory for Hamiltonian systems.Phys. Rep., 79:129, 1981

    1981

  41. [41]

    R. Requist. Adiabatic perturbation theory for two-component systems with one heavy component.Phys. Rev. B, 111:024311, 2025

    2025

  42. [42]

    J.-T. L¨ u, M. Brandbyge, and P. Hedegard. Blowing the fuse: Berry’s phase and runaway vibrations in molecular conductors.Nano Lett., 10:1657, 2010

    2010

  43. [43]

    T. N. Todorov, D. Dundas, J.-T. L¨ u, M. Brandbyge, and P. Hedegard. Current-induced forces: a simple derivation.Eur. J. Phys., 35:065004, 2014

    2014

  44. [44]

    M. Borneas. On a Generalization of the Lagrange function.Am. J. Phys., 27(4):265D267, 1959. 30

    1959

  45. [45]

    S.-S. Chern. Finsler geometry is just Riemannian geometry without the quadratic restriction.Notices Amer. Math. Soc., 43:959 D963, 1996

    1996

  46. [46]

    Bao, S.-S

    D. Bao, S.-S. Chern, and Z. Shen.An Introduction to Riemannian-Finsler Geometry. Graduate Texts in Mathematics 200. Springer DVerlag, 2000

    2000

  47. [47]

    PhD thesis, Palacky University, Olomouc, 2013

    Erico Tanaka.Parameter invariant Lagrangian formulation of Kawaguchi geometry. PhD thesis, Palacky University, Olomouc, 2013

    2013

  48. [48]

    Kirill C. H. Mackenzie.General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series. Cambridge University Press, 2005

    2005

  49. [49]

    Zermelo.Untersuchungen zur Variationsrechnung

    E. Zermelo.Untersuchungen zur Variationsrechnung. PhD thesis, Friedrich-Wilhelms- Universitaet, Berlin, 1894

  50. [50]

    Urban and D

    Z. Urban and D. Krupka. The Zermelo conditions and higher order homogeneous functions.Publ. Math. Debrecen, 82:59 D76, 2013

    2013

  51. [51]

    V. G. Lemleyn. Spaces of symmetric almost symplectic connectivity.Dokl. Akad. Nauk SSSR (N.S.), 115:655–658, 1957

    1957

  52. [52]

    Symplectic connections.Int

    Pierre Bieliavsky, Michel Cahen, Simone Gutt, John Rawnsley, and Lorenz Schwachh¨ ofer. Symplectic connections.Int. J. Geom. Methods Mod. Phys., 03(03):375– 420, 2006

    2006

  53. [53]

    Theoretical and Mathematical Physics

    Ralph Blumenhagen, Dieter L¨ ust, and Stefan Theisen.Basic concepts of string theory. Theoretical and Mathematical Physics. Springer, Heidelberg, Germany, 2013

    2013

  54. [54]

    Princeton University Press, 8 1994

    Marc Henneaux and Claudio Teitelboim.Quantization of Gauge Systems. Princeton University Press, 8 1994

    1994

  55. [55]

    Variational point- obstacle avoidance on Riemannian manifolds.Math

    Anthony Bloch, Margarida Camarinha, and Leonardo Colombo. Variational point- obstacle avoidance on Riemannian manifolds.Math. Control Signals Systems, 33:109D121, 2021

    2021

  56. [56]

    Dynamic interpola- tion for obstacle avoidance on Riemannian manifolds.Int

    Anthony Bloch, Margarida Camarinha, and Leonardo Colombo. Dynamic interpola- tion for obstacle avoidance on Riemannian manifolds.Int. J. Control., 94:588–600, 2021

    2021

  57. [57]

    On the analysis of movement smoothness.J

    Sivakumar Balasubramanian, Alejandro Melendez-Calderon, Agnes Roby-Brami, and Etienne Burdet. On the analysis of movement smoothness.J. Neuroeng. Rehabil., 12(1):112, 2015

    2015

  58. [58]

    Emanuel Todorov and Michael I. Jordan. Smoothness maximization along a predefined path accurately predicts the speed profiles of complex arm movements.J. Neurophys- iol., 80(2):696–714, 1998. PMID: 9705462

    1998

  59. [59]

    The Two- Thirds Power Law Derived from a Higher-Derivative Action.Physics, 6(4):1251D1263, 2024

    Nicolas Boulanger, Fabien Buisseret, Fr´ e d´ e ric Dierick, and Olivier White. The Two- Thirds Power Law Derived from a Higher-Derivative Action.Physics, 6(4):1251D1263, 2024. 31

    2024

  60. [60]

    From brain to motion: harnessing higher-derivative mechanics for neural control.Frontiers in Applied Mathematics and Statistics, Volume 11 - 2025, 2025

    Olivier White, Fabien Buisseret, Fr´ e d´ e ric Dierick, and Nicolas Boulanger. From brain to motion: harnessing higher-derivative mechanics for neural control.Frontiers in Applied Mathematics and Statistics, Volume 11 - 2025, 2025. 32

    2025