Path integrals for classical-quantum dynamics

The paper provides a strong, well-structured bridge from CPTP CQ master equations to path integrals, elegantly highlighting a fundamental decoherence–diffusion trade-off and giving workable formulas in the continuous/quadratic sector. This is the first CPTP-consistent CQ path-integral that cleanly unifies Fokker-Planck, FV, and CQ coupling with an explicit trade-off.

Brief overview:

1- In this paper, the authors derive a general path‐integral formulation for classical–quantum (CQ) dynamics that is linear, CPTP (completely positive, trace preserving), and keeps a clean split between classical and quantum dofs. The path integral decomposes into: a purely classical (stochastic) part, a quantum part (with a Feynman–Vernon term), and a CQ coupling that penalizes deviations from the \pm-averaged drift/Euler–Lagrange equations.

2- For the continuous (Kramers–Moyal up to second order) class, they integrate out response variables and obtain a compact phase–space path integral with a decoherence–diffusion trade-off: D_0\succeq0, D_2\succeq0, and 4D_2 \succeq D_0^{-1}. When the bound is saturated, the \pm paths decouple and purity is preserved conditional on the classical trajectory.

3- Under standard quadratic-in-momenta assumptions they map master equations to configuration-space path integrals (proto-action W_{\rm CQ}); deviations from \pm-averaged Hamilton/Euler–Lagrange flow are exponentially suppressed. 

The paper deserves publication for the development of the formal framework. Before that, however, I recommend that the authors address the following.

Foundational issues:

1- The D matrices encode diffusion and decoherence phenomenologically. Outside of specific measurement models, it’s unclear how to derive D_{0,1,2} from a microscopic CQ theory (especially for fundamental classical fields like gravity). Without such microfoundations, experimental constraints (e.g., on the trade-off) risk being model-dependent.

2- The “saturated” regime is mathematically beautiful (factorization of \pm branches, conditional purity), yet it corresponds to D_0 = D_1 D_2^{-1} D^\dagger_1. Small perturbations generically break this equality, reintroducing \pm cross-terms and mixedness. The paper doesn’t address stability under renormalization or coarse-graining; a practical scheme would need to show that near-saturation still yields controlled errors. Saturation is a fine-tuned surface in parameter space, so it is unclear how natural or stable it is under perturbations and coarse-graining. 

3- In a similar vein, the inequality 4D_2 \succeq D_0^{-1} (which is the specialisation of the general conditions (27)) is central to the decoherence-diffusion trade-off. It ensures CPTP but implies you cannot simultaneously have small classical diffusion and small quantum decoherence when there is back‐reaction. Doesn't this clash with semiclassical regimes often desired in gravity and measurement theory (low classical noise yet negligible decoherence)? 4. In the limits D_2 \to 0 and small coupling, does the CQ path integral reduce to (i) standard classical Liouville dynamics or (ii) standard Feynman–Vernon influence functionals? An explicit recovery of these limiting cases would anchor the formalism physically.

Technical issues:

1- After diagonalizing D_2(z,t), integrating out response variables produces a factor (42) which the authors absorb into Dz (45), likening it to known curved‐space/Langevin measure corrections. Why does this not deserve a careful derivation to address global issues? Such measure renormalisation can carry physical content (anomalies, curvature terms). Is this an assumption that this redefinition is harmless?

2- When D_2 has zero modes, the corresponding response integrals produce delta function constraints forcing the classical drift component to match D_1 in those directions, and, by CP constraints, eliminate back-reaction there. While internally consistent, this seems non-robust: arbitrarily small perturbations that lift the zero eigenvalues could qualitatively change the dynamics (re-enabling back-reaction). Physical models that nearly (but not exactly) have deterministic classical coordinates may be singular in this formalism.

3- In the Hamiltonian-drift example, back-reaction is encoded with a CQ bracket and supplemented by diffusion/decoherence terms (Eq. 50–52). While CPTP resolves positivity issues, questions remain about energy conservation, symplectic structure, and Jacobi identity violations at the hybrid level. The path integral suppresses deviations from \pm-averaged Hamilton flow (not exact Hamilton flow), which may be physically nonunique: different choices of averaging (or different operator orderings/Lindblad bases) could give inequivalent drifts yet all be CPTP. Can something be said about uniqueness and physical selection of the bracket in interacting field settings?

Issues that deserve emphasis in the introduction or summary so that they don't get lost in detail: 

1- The formalism admits a Pawula-type dichotomy: either infinitely many moments (jump processes) or at most second order (continuous). The full “general” path integral (Eq. 20) becomes unwieldy with infinite moments; all tractable results appear to come only after assuming the continuous subclass and integrating out response fields. That leaves the important jump CQ dynamics essentially without a practical path‐integral calculus (beyond a formal expression). This limits applicability to realistic hybrid scenarios with discrete events/feedback kicks. The reader would greatly benefit from an overview (even if very brief) of how strong this limitation is, possibly with some examples.

2- The Feynman–Vernon part depends on the chosen L_\alpha. Because different Lindblad unravelings/bases can represent the same master equation, path integrals built at the unraveling level may inherit gauge-like redundancies. The paper does not articulate equivalence classes of actions under Lindblad basis changes, nor how to identify basis‐invariant observables directly at the path‐integral level. The Hermitian-L case is showcased, where the physics is clearer (decoherence in the L-eigenbasis) but the general non-Hermitian case is opaque. Since the path integral doubles the configuration variables, what is the interpretation of complex drifts generated by non-Hermitian L_\alpha? Does this compromise positivity of the classical marginal? The paper could clarify whether all admissible non-Hermitian choices correspond to physically realisable stochastic unravelings.

3- To reach configuration space, they assume quadratic momentum dependence, minimal coupling (no momentum dependence in V_I), and diffusion only in momenta. These are strong modeling choices; many realistic hybrid couplings (e.g., velocity-dependent forces, gauge couplings, or position diffusion) fall outside this scope, at which point the advertised clean configuration‐space form and proto-action W_{CQ} no longer hold. Is this fair?

4- The central penalty term is \frac12\,D_1^{\rm diff}!\cdot D_2^{-1}!\cdot D_1^{\rm diff}, where D_1^{\rm diff} encodes the difference between the actual classical velocity and its expected drift (including back-reaction). This is elegant, but it builds in that the most probable path is the averaged one, not necessarily the solution of a microscopic CQ equation of motion. In systems where rare events or non-Gaussian statistics dominate, this Gaussian large-deviation structure may mislead. 

5- It is stated that covariant and even diffeomorphism-invariant CQ path integrals (e.g., gravity) are obtainable and Lorentz-invariant examples and a companion paper are mentioned. But the present derivation leans on time-slicing and autonomous Markovian dynamics; it’s not shown in detail how microcausality, constraint algebras, and gauge fixings are handled in field theory beyond toy forms. Until those ingredients are fully spelled out, shouldn't claims of “general covariant CQ path integrals” be read as promissory notes? 

6- All results are for autonomous Markovian semigroups. Many realistic hybrid scenarios (especially measurement-feedback with delays, or effective gravity with retardation) are non-Markovian; extending these constructions to CP-divisible but time-nonlocal kernels is nontrivial and not covered here. (The authors note autonomy and use Trotterization; nothing guarantees an equally neat path integral with memory kernels.)

Ask for major revision

This paper works out a path integral formalism for what the authors call "classical-quantum dynamics". I have a question for the authors, and then a few comments:

1) Isn't this theory really just a special case of quantum mechanics, where there are some degrees of freedom which do not have entangling terms in the Hamiltonian? For example we could have some qubits where the Hamiltonian involved only the Z operator and not the X operator. Here I also mean that we should also only consider states which are in the image of a quantum-to-classical channel, e.g. in the qubit case the channel which decoheres the quantum state in the Z basis. If so, then shouldn't the paper acknowledge this up front? If not, then why not?

2) As a terminology quibble, by "classical" the authors really mean "stochastic" - true classical mechanics as taught in university is deterministic. I understand that it doesn't really make sense to couple a deterministic classical system to a quantum system, and I also understand that the authors know this, but some readers may not so I think this should be clarified the first time the term "classical" is used.

3) I found the introduction to be rather lacking in motivation. Why should we go through all of this? There are some sketchy applications at the end, but for the most part the paper doesn't have much physics in it which makes it hard to judge its potential importance.

4) The authors mention that they have not worked out the renormalization of a classical field coupled to a quantum field. I suspect that this will lead to problems, as entanglement is an essential feature of quantum field theory and states without an infinite amount of entanglement have infinite energy. This is related to the old Banks-Peskin-Susskind criticism of information loss as rapidly heating the universe. The authors mention a "diffusion/decoherence" tradeoff" in the paper, which sounds like it has the flavor of what worries me, and it would be interesting to understand the consequences of this tradeoff for the renormalization problem. This point is not an actionable item for this paper, but I encourage the authors to pursue it in later work.

Ask for minor revision