Causality in relativistic quantum interactions without mediators

Eduardo Mart\'in-Mart\'inez; Eirini C. Telali; T. Rick Perche

REVIEW 2 major objections 4 minor 2 cited by

Causality in relativistic quantum interactions without mediators

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Field-free quantum interactions that mimic QFT always carry retrocausal signals, yet today's gravity-entanglement experiments cannot resolve them.

desk verdict Quantitative retrocausality bounds for the qc-model, internally consistent, with a minor slip that doesn't change the GME conclusion; deserves serious refereeing. read the letter →

arxiv 2412.16288 v1 pith:M7CYH3ZZ submitted 2024-12-20 quant-ph gr-qchep-th

Eirini C. Telali , T. Rick Perche , Eduardo Martín-Martínez This is my paper · ORCID

classification quant-phgr-qchep-th

keywords quantum-controlledmodelretrocausalitysignallingestimatorgravity-mediatedentanglementrelativisticquantuminformationHuygensprinciplefieldtheoryUnruh-DeWittdetector

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 quantum-controlled (qc) model, an effective description in which two quantum systems interact through a retarded relativistic potential but the mediating field has no quantum degrees of freedom of its own. Comparing this model to a full quantum field theory (QFT) description, the authors show that the qc-model can approximate QFT only in regimes where it also predicts retrocausal signalling: a system can be affected by another system that lies in its causal future. They quantify this effect with a signalling estimator and find that in 3+1 dimensions with a massless field the retrocausal part is a fraction L/T of the total signal, vanishing for long interactions, while in 1+1 dimensions it can dominate. Applied to gravity-mediated entanglement proposals, the numbers yield retrocausal phase shifts far below experimental resolution, so those experiments cannot distinguish the qc-model from QFT and cannot, on these grounds, claim to witness the quantum degrees of freedom of gravity.

What carries the argument

The machinery is the decomposition of the qc-model's evolution in terms of the symmetric propagator $\$\Delta$(x, x') = G_R(x, x') + G_A(x, x')$, the sum of retarded and advanced Green's functions, which replaces the field's quantum degrees of freedom. The signalling estimator $C_a = \$\Delta$(\Lambda_a, \Lambda_b)$ splits into causal and retrocausal parts, and the strong Huygens principle in 3+1 dimensions — the retarded propagator is a delta-function on the light cone — is what keeps the retrocausal part bounded by $L/T$. The trade-off is encoded in the identity relating the symmetric and causal propagators: the qc-model discards the antisymmetric combination $E = G_R - G_A$, and the paper proves that discarding $E$ while retaining $\$\Delta$$ necessarily permits advanced propagation.

What would settle it

Measure the phase of the signalling estimator in a gravity-mediated entanglement experiment with time resolution better than the light-crossing time L between the masses: the qc-model predicts a constant retrocausal phase shift of order $\$lambda^{2}$/(4\pi)$, which a local QFT description does not produce.

Watch

Extended reading notes

Core claim

The central claim is that the qc-model inevitably contains retrocausal contributions whenever one system has support in the causal future of the other, and these contributions are inseparable from the model's ability to match QFT: approximating QFT forces the evolution to depend on the symmetric propagator $\$\Delta$ = G_R + G_A$, which carries advanced (future-to-past) information. The paper shows quantitatively that for massless fields in 3+1 dimensions, where the retarded propagator is delta-supported on the future light cone, the retrocausal part of the signalling estimator is $L/T$ of the total for interaction times $T > 2L$, and the absolute retrocausal contribution is bounded by the squared coupling. For the proposed gravity-mediated entanglement parameters ($L \sim $10^{{-6}}$$ m, $T \sim 1$ s, $\$lambda^{2}$ \sim $10^{{-14}}$$), this is $$10^{{-14}}$$ of the signal, requiring a time resolution of order the light-crossing time to observe. The conclusion is that current GME proposals are experimentally indistinguishable from a field-free qc-description, so they do not yet access quantum aspects of the gravitational interaction through retrocausality.

Load-bearing premise

The bound on retrocausality in 3+1 dimensions assumes the mediating field obeys the strong Huygens principle, so the retarded propagator is concentrated exactly on the future light cone; if the field has tails, retrocausal effects are no longer bounded, as the 1+1-dimensional case shows.

Editorial extensions

If this is right

Any qc-model that faithfully approximates QFT during a finite interaction is necessarily retrocausal, so fully causal and fully field-free relativistic models cannot both match QFT.
In 3+1 dimensions, retrocausal effects in qc-interactions are suppressed by the factor $L/T$, so long interaction times make the model effectively causal even though it is not exactly causal.
In settings where the strong Huygens principle fails, such as 1+1-dimensional massless fields or massive mediators, the retrocausal signal is not bounded and can dominate the causal signal.
With the parameters of current gravity-mediated entanglement proposals, the retrocausal contribution is $\lambda^2 \sim 10^{-14}$ of the signal, so the qc-model and the full QFT description are experimentally indistinguishable in those experiments.
Detecting the retrocausal signature would require resolving relative state variations of order $10^{-14}$ or time intervals of order the light-crossing time ($10^{-14}$ s for the proposed setups), far beyond current experimental capabilities.

Reading between the lines

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

A future GME experiment with time resolution near the light-crossing time could convert the retrocausal phase shift into an unambiguous discriminator between a field-free qc-description and a local QFT description, a test the paper's bounds make concrete.
The same trade-off between matching QFT and introducing advanced-propagation artefacts likely applies to any effective model that truncates mediator degrees of freedom, not just the specific qc-construction studied here.
Since the qc-model cannot reproduce the Hadamard (noise) contribution of the quantum field, measuring local decoherence of the sources — rather than their mutual entanglement — may be a cleaner experimental route to detect the field's quantum degrees of freedom in regimes where retrocausality is negligible.
The conclusions about GME experiments rest on gravity behaving like a massless higher-spin field with no tails; computing the same estimators for massive-graviton or non-linear corrections would test whether the retrocausal bound survives more realistic gravitational dynamics.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read it. This paper makes the retrocausality worry in quantum-controlled models quantitative, and it does so cleanly. The new pieces are the explicit definitions of retrocausality (III.1, III.2), the split of the signalling estimator into causal and retrocausal parts, and the L/T scaling conditions that follow from strong Huygens propagation in 3+1 dimensions. The nonperturbative phase-shift analysis for degenerate detectors is also new and consistent with the perturbative result. It builds on your group's earlier qc-model work, so the novelty is incremental, but it is real: the earlier papers didn't derive a bound on retrocausal effects.

The derivations hold up. I re-derived the integral estimates in Section IV.A and the phase-shift argument in IV.B; the algebra is consistent. The conclusion that the qc-model always has some retrocausal content when it approximates QFT is derived from Eq. (8) and the symmetric propagator, not assumed. The GME application is the payoff: retrocausal effects in the proposed parameter range are below experimental resolution by a large margin, so the paper reinforces the earlier claim that these experiments won't witness field degrees of freedom.

Soft spots are minor. The stress test flagged an offset in Eq. (91): the causal split should be lambda^2 (T-L)/(2 pi L) rather than lambda^2 T/(2 pi L) if you use the same windows as Eq. (65). This is a genuine slip, but it changes an L/T-relative offset of order 10^-14 and leaves the retrocausal phase lambda^2/4 pi, the time shift delta T = L, and the GME conclusion untouched. Worth a footnote, not a rewrite.

The load-bearing assumption is strong Huygens propagation for linearized gravity in Minkowski space. That's standard for massless higher-spin fields in flat space, so I don't see it as a flaw, but the paper should say more explicitly that the gravity conclusion lives or dies with that assumption. The 1+1 example shows the bound disappears with tails, so a sentence acknowledging the tail caveat would be honest.

Self-citation is present but not a problem here; the qc-model is the authors' construction and the GME claim inherits from Ref. [5]. The argument doesn't loop.

Who is this for: people working on GME experiments, effective models of relativistic interactions, and causality in nonlocal models. It deserves a serious referee. I'd accept after minor revision, with the Eq. (91) fix and a more explicit caveat on Huygens.

Recommendation: send to peer review.

Referee Report

2 major / 4 minor

Summary. The paper analyzes the quantum-controlled (qc) model, in which two quantum systems interact through a direct, retarded two-body coupling with no local quantum field degrees of freedom, and compares it with a full QFT description. It provides operational definitions of non-retrocausal interactions (Definitions III.1 and III.2) and argues from the symmetric-propagator structure of the qc evolution (Eqs. (8), (16)-(18), (39)) that the qc-model necessarily contains retrocausal contributions whenever one system lies in the causal future of the other, so that any regime in which the qc-model approximates QFT carries some retrocausal signalling. The paper quantifies this in explicit qubit setups: for a massless scalar in (3+1) dimensions with window switching and pointlike smearing, the retrocausal part of the signalling estimator is C_r/C_a = L/T for T > 2L (Eqs. (66)-(70)), while in (1+1) dimensions the retrocausal contribution can dominate, with C_r/C_a tending to 1 as S → ∞ (Eqs. (75)-(80)). A nonperturbative (gapless) analysis gives a retrocausal phase shift of order λ^2/(4π) and a time shift δT = L (Eqs. (91)-(97)). These results are applied to gravity-mediated entanglement (GME) experiments with L ~ 10^-6 m, T ~ 1 s, λ^2 ~ 10^-14, concluding that the retrocausal predictions of the qc-model are below the resolution of current proposals, so those experiments do not yet discriminate between a qc-description and a full quantum-gravitational one.

Significance. If the results hold, the paper delivers a quantitative, checkable bound on the regime of validity of relativistic direct-coupling models and sharpens the GME debate: for the currently proposed parameters, a qc-description without gravitational field degrees of freedom remains experimentally indistinguishable, so those experiments would not by themselves reveal the quantum nature of gravity. The paper's strengths are its explicit operational definitions of retrocausality, the fully analytic perturbative computations of the signalling estimators (Eqs. (66)-(68) and (75)-(79), which I verified), the nonperturbative gapless check, and the honest identification of the strong-Huygens assumption through the (1+1)-dimensional counterexample in which retrocausal signalling is not bounded. The central conclusions are conditional on the absence of tails in the mediating propagator, which is standard for massless fields in (3+1)-dimensional Minkowski spacetime and for linearized gravity in harmonic gauge; the paper itself demonstrates where the assumption fails.

major comments (2)

[§IV.B, Eqs. (91)-(93)] The claimed additive split ∆ab = ∆(c)ab + ∆(r)ab is inconsistent as written. With the same window functions as in Eq. (65), the causal part of the perturbative signalling estimator for T > 2L is C(c)a = (T−L)/(2πL) (from Eq. (67)), so the nonperturbative split should read ∆(c)ab = λ²(T−L)/(2πL) and ∆(r)ab = λ²/(2π); as stated, Eq. (91) gives ∆(c)ab = λ²T/(2πL), and then ∆(c)ab + ∆(r)ab = λ²(T+L)/(2πL) ≠ ∆ab = λ²T/(2πL). Correspondingly, Eq. (93) should use Na = |sin(λ²T/(4πL))| and N(c)a = |sin(λ²(T−L)/(4πL))|. I emphasize that the downstream conclusions survive: the relative phase shift between Na and N(c)a is still λ²/(4π), the time shift is still δT = L, and the tolerance conditions (95)-(97) and the GME estimates are unchanged. The error is local, but Eqs. (91)-(93) should be corrected.
[§III.B, Eq. (46)] The argument for the claim that approximating QFT inescapably implies retrocausal signalling is stated in a way that can mislead. Since the qc-model's reduced state depends only on ∆ab and not on Eab, the non-retrocausality condition ρa(∆ab, Eab) = ρa(∆ab − G̃ba, Eab + G̃ba) fails whenever G̃ba ≠ 0 regardless of whether Eab vanishes; the sentence 'imposing Eab = 0 prevents Eq. (46) from being satisfied' suggests that a nonzero Eab could restore non-retrocausality, which is not the case for the unitary direct-coupling form. The intended point—that a non-retrocausal model would need to depend on the antisymmetric combination in a way the symmetric-propagator structure cannot provide—is correct, but the paragraph should be reformulated for precision.

minor comments (4)

[§IV.C, paragraph on GME parameters] The sentence in the Conclusions stating that retrocausal effects are 'upper bounded by the square of the interaction strength' should be made more precise: the signalling estimator C(r)a = 1/(2π) is O(1) and λ-independent, while it is the phase shift λ²/(4π) that is O(λ²); the bound C(r)a/Ca = L/T is geometric. Please clarify which quantity is being bounded.
[§IV.C and §V] The extension of the (3+1) bound to linearized gravity rests on the strong Huygens property (delta-supported retarded propagator, Eq. (62)). The paper makes this assumption at the outset but should state explicitly in Section IV.C that a massive graviton, background curvature, or nonlinearities would reintroduce tails and invalidate the L/T bound, as the (1+1) computation in Eqs. (75)-(80) demonstrates; the current text leaves this as an implicit inference.
[§II.B.2, Eq. (37)] The index structure of the fourth operator term in Eq. (37) appears unbalanced: the term written as ˆjb(a)(x)ˆρab,0ˆja(a)(x′) should presumably involve a b-index on the last current to match the Hermitian structure of the adjacent terms. Please check all index pairings in Eqs. (37)-(38).
[Throughout] There are several typos and redundancies that should be cleaned up: 'the the setups' (Section I), 'stystem' (Section II.B.3), 'von Newman algebras' (Section II.B.1), 'quantum quantum degrees of freedom' (Section V), 'are are of the order' (Section IV.C), and 'buf for completeness' (footnote 1).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the retrocausality result is derived from the model's own definitions and propagator structure, and the GME application cites prior published work rather than repackaging this paper's inputs.

full rationale

The paper's central claim—that the qc-model always contains retrocausal contributions when one system has support in the causal future of the other—is derived, not assumed. The qc interaction Hamiltonian is defined in Eq. (8) from the retarded propagator; Eq. (16) and Eq. (17) then show that the time-integrated evolution depends on the symmetric propagator Delta = G_R + G_A. Equation (39) gives the leading-order qc update, and Eq. (43) isolates the advanced-propagator term that makes the reduced state of A depend on G_ba, in contrast to the retarded-only QFT expression in Eq. (42). The retrocausality definitions (Defs. III.1 and III.2) are introduced independently, and the quantitative estimator Ca = Delta(Lambda_a, Lambda_b) of Eq. (59) is then evaluated with explicit propagators and switching functions. No parameter is fitted to data and repackaged as a prediction. The GME application imports the qc-description of gravitation from the authors' prior Ref. [5] and the QFT-approximation regime from [4]; these are published external results with their own derivations, and even if one set them aside, the retrocausality bound remains as a conditional statement linking the qc model's symmetric-propagator structure to the Huygens support of the retarded propagator. The strong Huygens assumption in Eq. (62) is stated explicitly and is standard for massless fields in 3+1 dimensions; the paper also includes the 1+1 tail case where the bound fails, so the assumption is not smuggled in. The minor split inconsistency around Eq. (91) affects an offset of order L/T but does not change the leading retrocausal phase or the GME time-resolution estimate. I find no step that reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data; lambda, L, T, and masses are physical inputs. The paper introduces no new particles, forces, or conserved quantities. Its 'entities' are models (the qc-field) and definitions (causal/retrocausal regions), not new degrees of freedom.

assumptions (6)

domain assumption The qc Hamiltonian (Eq. 8) is obtained from the classical retarded interaction under the adiabatic approximation and neglect of self-interactions (Appendix A).
The retrocausality analysis is for this qc-model. If the adiabatic approximation fails, the direct-coupling model is different and the estimates do not apply.
domain assumption The initial field state is quasi-free (zero-mean Gaussian) and uncorrelated with the sources (Section II.B.3).
This ensures the first-order QFT correction vanishes and makes the second-order comparison well defined; it is an assumed preparation.
domain assumption The massless scalar retarded propagator in 3+1 Minkowski spacetime obeys the strong Huygens principle (Eq. 62), and linearized gravity is treated as a massless higher-spin field with the same causal propagation (Section IV.C).
The bounded retrocausal contribution C_a^(r) = 1/(2pi) for T > 2L depends on delta-function light-cone support. The GME application inherits this via a footnote and Ref. [5], but the paper does not re-derive it for linearized gravity.
ad hoc to paper Retrocausality is defined operationally as any dependence of A's reduced state on the advanced-propagation kernel G_ba when G_ab = 0 (Definitions III.1 and III.2).
This is a proposed definition, not a theorem; all quantitative conclusions are relative to accepting it.
standard math Standard algebraic QFT background: the canonical commutation relations [Phi(f), Phi(g)] = -i E(f,g) 1 and the Wightman/Feynman/Hadamard decompositions (Eqs. 23, 28-29).
Used without proof to separate QFT corrections into Hadamard, retarded/advanced, and symmetric-propagator pieces.
domain assumption Perturbation theory truncated at second order for gapped detectors, with a nonperturbative check only for gapless systems.
Higher-order corrections could in principle alter the estimates; the nonperturbative gapless case is consistent with the leading-order scaling.

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Cite this review

Pith. "Pith review of Causality in relativistic quantum interactions without mediators." pith.science (2026). https://pith.science/paper/M7CYH3ZZ

@misc{pith2026241216288,
  author       = {Pith},
  title        = {Pith review of: Causality in relativistic quantum interactions without mediators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7CYH3ZZ}},
  note         = {Machine review of arXiv:2412.16288}
}

read the original abstract

We analyse the interaction between two quantum systems in spacetime and we compare two possible models to describe it: 1) a fully quantum field theoretical (QFT) description of the coupling of two quantum systems mediated by a quantum field and 2) a quantum-controlled model (qc-model), which is an effectively relativistic direct-coupling in which the interaction of two quantum systems is not mediated by a field with local quantum degrees of freedom. We show that while there are regimes where the qc-model can approximate QFT arbitrarily well, it can suffer from retrocausal effects. We discuss in what regimes those retrocausal predictions of the qc-model are non-negligible and whether they can be used to argue that gravity induced entanglement experiments can reveal genuinely quantum aspects of the gravitational interaction or not.

Figures

Figures reproduced from arXiv: 2412.16288 by the authors.

**Figure 2.** FIG. 2. A spacetime diagram of the interaction regions of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3. The dependence of the different estimators [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Spacetime diagram of the interaction regions of A and [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

Discussion (0). Continue with ORCID to comment.

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Reference graph

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Comparison with the quantum-controlled model Up to this point we have presented the interaction be- tween two quantum systems A and B through quantum- controlled fields in II A and quantum fields in II B 2. Now we compare the evolution of the state of the system AB up to second order in λ and discuss the requisites for qc-fields to be a good approximation...

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Classical sources interacting via classical fields Assume two classical systems A and B in Minkowski spacetime, M, that interact via a classical field ϕ(a), where ( a) ≡ (µ, ν, . . .) is an arbitrary set of Lorentz indices. The two systems couple to the field through currents ja (a), jb (a). The total system is described by the action: S = Z dV (Lϕ + La +...

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Quantum sources interacting via classical propagation The model above can be extended to the case where systems A and B are quantum. This gives rise to the quantum-controlled model ( qc-model) studied in [4, 5]. The goal of the qc-model is to prescribe an interaction be- tween two quantum sources that preserves the relativis- tic nature of the classical m...

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Local interactions with a quantum field Now let us discuss the role of the propagators when two quantum systems A and B locally interact with a quan- tum field ˆϕ(a). In analogy to the classical interaction (5), we prescribe the interaction Hamiltonian as ˆHint(t) = λ Z d3x ˆja (a)(x) ˆϕ(a)(x) + ˆjb (a)(x) ˆϕ(a)(x) . (30) We will refer to this model as QF...

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [5]

Comparison with the quantum-controlled model Up to this point we have presented the interaction be- tween two quantum systems A and B through quantum- controlled fields in II A and quantum fields in II B 2. Now we compare the evolution of the state of the system AB up to second order in λ and discuss the requisites for qc-fields to be a good approximation...

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[2] [1]

) is an arbitrary set of Lorentz indices

Classical sources interacting via classical fields Assume two classical systems A and B in Minkowski spacetime, M, that interact via a classical field ϕ(a), where ( a) ≡ (µ, ν, . . .) is an arbitrary set of Lorentz indices. The two systems couple to the field through currents ja (a), jb (a). The total system is described by the action: S = Z dV (Lϕ + La +...

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[3] [2]

putting hats

Quantum sources interacting via classical propagation The model above can be extended to the case where systems A and B are quantum. This gives rise to the quantum-controlled model ( qc-model) studied in [4, 5]. The goal of the qc-model is to prescribe an interaction be- tween two quantum sources that preserves the relativis- tic nature of the classical m...

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[4] [3]

In the context of QFT, it will be convenient to define the retarded and advanced propagators as bi-distributions with kernels given by G(ab) R (x, x′), G(ab) A (x, x′)

Quantum field propagators In quantum field theory propagation is encoded in n-point functions. In the context of QFT, it will be convenient to define the retarded and advanced propagators as bi-distributions with kernels given by G(ab) R (x, x′), G(ab) A (x, x′). Given a set F (M) of smooth 3 Symmetry of ∆ (ab)(x, x′) follows from G(ab) A (x, x′) = G(ba) ...

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[5] [4]

In analogy to the classical interaction (5), we prescribe the interaction Hamiltonian as ˆHint(t) = λ Z d3x ˆja (a)(x) ˆϕ(a)(x) + ˆjb (a)(x) ˆϕ(a)(x)

Local interactions with a quantum field Now let us discuss the role of the propagators when two quantum systems A and B locally interact with a quan- tum field ˆϕ(a). In analogy to the classical interaction (5), we prescribe the interaction Hamiltonian as ˆHint(t) = λ Z d3x ˆja (a)(x) ˆϕ(a)(x) + ˆjb (a)(x) ˆϕ(a)(x) . (30) We will refer to this model as QF...

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[6] [6]

The Hadamard contribution in Eq. (37),

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[7] [7]

The asymmetric self-interaction terms in (38),

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[8] [8]

Indeed, in [4] it was shown that there are regimes where the conditions above are fulfilled

The antisymmetric causal propagator terms in (38). Indeed, in [4] it was shown that there are regimes where the conditions above are fulfilled. Concretely, it was found that when the interaction between the sys- tems lasts for sufficiently long times, the systems are in causal contact with each other, and the coupling with the field is weak enough, the tw...

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