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Microcausality without Lorentz invariance

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Pith's one-line read Microcausality forces the commutator's spatial Fourier transform to be analytic in $k^2$ and bounded by $e^{|\mathrm{Im} k||t|}$; these two conditions are necessary and sufficient, and they are checkable inside Lorentz-breaking effective…

desk verdict A genuinely useful microcausality criterion in (t,k) space, with an explicit stability assumption that should be proved or sharply stated; the examples and the inflation bound make it worth refereeing. read the letter →

arxiv 2502.04215 v2 pith:5RU5ZHX3 submitted 2025-02-06 hep-th

Lam Hui , Alberto Nicolis , Alessandro Podo , Shengjia Zhou This is my paper · ORCID
classification hep-th
keywords microcausalityPaley-Wienertheoremcommutatortwo-pointfunctionLorentz-breakingstateseffectivefieldtheorygroupvelocitypositivityboundsinflationaryEFT
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

This paper derives the full imprint of microcausality on two-point functions evaluated on Lorentz-breaking states. For any spatially homogeneous and isotropic state, the commutator Green's function $G_c(t,\vec x)$ vanishes outside a ball of radius $|t|$, so by the Paley-Wiener theorem its spatial Fourier transform $\tilde G_c(t,\vec k)$ must be analytic in the complex momentum and bounded by a polynomial times $e^{|\mathrm{Im} k||t|}$. The paper shows these conditions are both necessary and sufficient, and that they can be tested within the regime of validity of a low-energy effective field theory by taking $k$ below the cutoff and $kt\gg 1$. This converts microcausality into a practical consistency check for theories with spontaneously broken Lorentz invariance, and it yields concrete results: subluminal group velocities for stable excitations, a qualified allowance of superluminal group velocities for broad resonances, and a positivity condition on the EFT-of-inflation coefficient $M_2^4$.

What carries the argument

The load-bearing object is the Paley-Wiener theorem applied to $G_c(t,\vec x)$ as a distribution in $\vec x$ for fixed $t$. Microcausality says the support of $G_c$ is contained in the ball $|\vec x|\le |t|$; Paley-Wiener converts this into the statement that the spatial Fourier transform is entire in complex $\vec k$ and exponentially bounded with rate $|\mathrm{Im}\,\vec k|\,|t|$. Isotropy then reduces the complex-vector condition to analyticity in the single complex variable $k^2$, which is what the paper checks in each example. The theorem is also what makes the criterion necessary and sufficient, and it supplies the precise limit---$k$ below the EFT cutoff, $t$ large, $kt\gg 1$---in which exponential boundedness can be probed without knowing the UV completion.

What would settle it

Compute the one-loop corrected commutator $\tilde G_c(t,\vec k)$ for superfluid phonons from the self-energy in Eq. (6.19), taking complex $k$ with $|k|$ below the EFT cutoff and $kt\gg 1$. If the result has a branch point in $k^2$ away from $k^2=0$, or if $|\tilde G_c|$ exceeds $C(D+|k|)^N e^{|\mathrm{Im}\,k||t|}$ at any such $k$, the paper's necessary-and-sufficient criterion would be falsified in a theory that still obeys microcausality.

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Extended reading notes

Core claim

The central discovery is that microcausality alone---with no input from Lorentz invariance---fixes the analytic structure of the mixed-representation correlator $\tilde G_c(t,\vec k)=\int d^3x\, e^{-i\vec k\cdot\vec x}\langle[\phi(t,\vec x),\phi(0)]\rangle$: it must be an entire function of $k^2\equiv\vec k\cdot\vec k$ and must obey $|\tilde G_c(t,\vec k)|<C(D+|k|)^N e^{|\mathrm{Im} k||t|}$ for all complex $k$. These are the Paley-Wiener conditions for a distribution supported inside the lightcone at fixed $t$, and the converse direction of the theorem makes them sufficient. The paper verifies them in free and interacting examples---relativistic scalars, non-relativistic fields, superfluids with double-square-root dispersion relations, solids and framids, relativistic hydrodynamics, ocean gravity waves, de Sitter scalars, and slow-roll inflation---finding that apparent non-analyticities from individual excitations cancel in the full commutator exactly when microcausality holds. Within an EFT, exponential boundedness reduces in the large-$kt$ limit to a constraint on the low-energy dispersion relation, which for stable excitations is subluminal group velocity and for inflationary scalars is $M_2^4\ge 0$.

Load-bearing premise

The load-bearing premise is that the commutator at a fixed time can be treated as a distribution supported inside a ball of radius $|t|$, with all non-exponential prefactors growing only polynomially (or at least slower than exponentially) in $|t|$; if the quantum state is unstable enough that these prefactors grow exponentially in time, the exponential-bound test would stop being equivalent to microcausality, and the paper assumes this stability rather than proving it for the interacting, finite-density, and cosmological states it applies the test to.

Editorial extensions

If this is right

  • A Lorentz-breaking EFT whose two-point function is not analytic in $k^2$---such as the 'khronon' action of Sec. 7---is incompatible with microcausality in every regime, so microcausality can rule out candidate effective theories without any reference to their UV completion.
  • For a stable low-energy excitation, exponential boundedness requires subluminal group velocity, $|v_g|\le 1$; an excitation with width $\Gamma_k$ can have $|v_g|\le 1+\Gamma_k/(2|\mathrm{Im}\,k|)$, so superluminal group velocity is possible only when the resonance is broad enough.
  • In the superfluid EFT beyond leading order, the correlator-level bound is stronger than group-velocity subluminality: it forbids a positive $\alpha$ of order $(1-c_s)$ as $c_s\to 1$ even though a negative $\alpha$ would escape the group-velocity test.
  • For slow-roll inflation, the exponential bound is satisfied only for subluminal sound speed, which yields the positivity condition $M_2^4\ge 0$ in the EFT of inflation and $\partial^2 P/\partial X^2\ge 0$ in $P(X,\phi)$ theories, assuming the Null Energy Condition holds.
  • Nested commutators extend the same analyticity-and-boundedness logic to higher-point functions: the $n$-point correlator vanishes if any spatial separation exceeds $|t|$, so its Fourier transform obeys analogous Paley-Wiener bounds.

Reading between the lines

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

  • One direction the paper leaves open is to turn the analyticity check into a systematic diagnostic: for any proposed Lorentz-breaking EFT, compute the full one-loop (and, where necessary, resummed) commutator and test for branch points in $k^2$; theories that fail at any order would be excluded even if their tree-level dispersion relations look causal.
  • The stability assumption behind the Paley-Wiener step---that non-exponential prefactors do not grow exponentially in $t$---deserves testing in out-of-equilibrium or unstable states; if it can be violated, the exponential-bound criterion would need to be weakened, and causality would not automatically fail.
  • Applied to the nested-commutator generalization, microcausality could constrain inflationary bispectra and trispectra in a way that is independent of model-building details, since the bound is purely kinematic.
  • The positivity bound $M_2^4\ge 0$ is stated only under slow roll and a fixed gravitational background; extending it beyond slow roll, or to theories with dynamical gravity where the lightcone is not sharp, is an open direction the paper flags.
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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 / 5 minor

Summary. The paper derives necessary-and-sufficient-style conditions for microcausality in spatially homogeneous and isotropic states of a QFT, using the Paley-Wiener theorem in a mixed (t, k) representation. The central claim is that the commutator two-point function G_c(t, x) has compact spatial support of radius |t|, so its spatial Fourier transform is analytic in complex k (or in k^2 under isotropy) and exponentially bounded by e^{|Im k| |t|} up to subexponential prefactors. The authors verify these conditions in a wide set of examples: free and non-relativistic scalars, the EFT of superfluids including a one-loop self-energy computation, solids and framids, relativistic hydrodynamics, ocean gravity waves, de Sitter scalars, and the EFT of inflation. They use the conditions to derive constraints on sound speeds and group velocities, a positivity condition on an inflationary EFT coefficient, and a brief extension to higher-point nested commutators. The paper is well organized and states its assumptions explicitly, including the key stability assumption on the t-dependence of the Paley-Wiener constants.

Significance. If the central theorem is valid, the paper provides a genuinely useful framework: it converts microcausality into concrete analyticity and boundedness tests that can be applied within low-energy Lorentz-breaking EFTs, and it derives several new constraints, including a positivity condition on M_2^4 in the EFT of inflation. The worked examples are nontrivial and often require careful cancellation of non-analytic structures, e.g., the superfluid double square-root contributions and the longitudinal/transverse cancellation in solids and framids. The paper is also honest in flagging its main limitation, namely that the Paley-Wiener prefactors are assumed to grow slower than exponentially in t. That limitation is load-bearing, so the contribution is significant conditional on this stability assumption being either proved for the relevant states or incorporated as an explicit hypothesis in all statements.

major comments (4)
  1. [Sec. 2, after Eq. (2.5)] The Paley-Wiener theorem states the bound (2.5) for each fixed t, with constants C, D, N that may depend on t. The paper's third bullet explicitly assumes that these coefficients grow slower than an exponential in t, but this is neither derived from microcausality nor from the stated hypotheses of spatial homogeneity and isotropy. This assumption is load-bearing: it is used to derive the retarded-correlator falloff (3.1), the group-velocity bound (5.22), and the inflationary bound (9.30)-(9.33). If C(t) were to grow like e^{gamma |t|}, the effective exponent would become |Im k| + gamma and the subsequent subluminality tests would need modification. Please either prove this stability condition for the classes of states considered or restate the main theorem and all downstream claims as conditional on it, and verify it explicitly in each worked example.
  2. [Sec. 6.1] The UV-complete superfluid example is one of the main interacting finite-density checks of the boundedness criterion, but the exponential-boundedness verification is omitted: after Eq. (6.6) the text says 'for the sake of brevity here we omit the proof.' Since this is a central example and the omitted proof concerns the second half of the paper's main criterion, the manuscript should either provide the proof or state precisely which assumptions on the mode functions and spectral overlaps are needed for the bound to hold.
  3. [Secs. 9.2 and 9.3] The positivity bound M_2^4 >= 0 in Eq. (9.32) is derived under several non-trivial assumptions: a fixed FRW background, the slow-roll approximations leading to Eq. (9.16), and the neglect of M_Pl-suppressed effects. The text acknowledges the fixed-background caveat in Sec. 9.3, but the abstract and Sec. 9.2 present the bound as a consequence of microcausality. The statement of the result should explicitly include all these conditions, since without them the bound is not established. The same applies to the P(X,phi) version in Eq. (9.33).
  4. [Sec. 3, Eq. (3.1)] The claim that analyticity plus the asymptotic properties (3.1) are sufficient for microcausality is attributed in footnote 7 to an unpublished result of Salehian and Creminelli. Since this is used to justify an equivalence statement in the frequency-domain discussion, the paper should either provide a self-contained proof or cite a publicly available reference; otherwise the sufficiency claim should be labeled as conjectural.
minor comments (5)
  1. [Eq. (6.4)] Equation (6.4) contains typesetting artifacts ('radicaltp/radicalvertex') that make the formula unreadable; it must be typeset properly.
  2. [Eqs. (2.5) and (2.7)] The constants C, D, N in (2.5) and (2.7) are not necessarily the same; use different symbols or a subscript to avoid implying identical constants.
  3. [Sec. 2, footnote 3] The text says G_c is a distribution in x for fixed t, but the examples include step functions and oscillatory terms; clarify the sense in which these are distributions and how the t-dependence is treated in the distributional statement.
  4. [Sec. 8.2] The analytic continuation of the retarded Green's function (8.16) is argued from the deformation of the contour in Fig. 2, but the actual integral is not computed; a short numerical check or an explicit bound on the contour deformation would make the analyticity claim easier to verify.
  5. [Sec. 7, Eq. (7.5)] The factor t^3 f(k^2 t^2) in the cancellation (7.5) is stated without derivation; a few lines showing the Taylor expansion and the leading cancellation would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central Paley-Wiener criterion is an external mathematical equivalence, and the examples are independent consistency checks, not fitted inputs renamed as predictions.

full rationale

The paper's central derivation is a direct application of the Paley-Wiener theorem (Sec. 2): microcausality, i.e. compact support of G_c(t,x) in a ball of radius |t|, is equivalent to analyticity plus exponential boundedness of its spatial Fourier transform. This equivalence is an external mathematical fact, not an input redefined as an output. The subsequent examples are independent checks: free scalars, non-relativistic EFT, the derivatively coupled scalar, superfluids, solids/framids, hydrodynamics, gravity waves, and inflationary correlators are all computed from their own Lagrangians or dispersion relations and then tested against the conditions. No parameter is fitted to the criterion and then called a prediction. The superfluid UV example uses dispersion relations and overlaps from the authors' earlier paper [18], but those are independent derivations in a different context, not a restatement of the present microcausality conditions, so the self-citation is not load-bearing for the central claim. The only explicit qualification is the assumption in Sec. 2, third bullet after Eq. (2.5), that the non-exponential Paley-Wiener prefactors grow slower than exponentially in t; this is a genuine extra stability/spectral hypothesis, and without it the uniform-in-time exponential bound used later is not guaranteed. That is a limitation and a correctness risk, but it is not circularity: the paper does not define microcausality as this bound, nor does it derive the bound from itself. Omitted demonstrations, such as the unproved exponential-boundedness check for the superfluid UV completion in Sec. 6.1, are gaps rather than circular reductions. Overall, no step in the derivation chain reduces to its own input by construction.

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

The paper introduces no new entities and fits no free parameters to data. Its results rest on the standard Paley-Wiener theorem plus several state- and background assumptions, all explicitly stated. The most fragile of these are the stability of the state and the fixed-background light cone for cosmology.

assumptions (5)
  • domain assumption The state is such that the non-exponential prefactors C(D+|k|)^N in the Paley-Wiener bound grow slower than exponentially in time.
    Assumed in Section 2, third bullet, and needed to exclude pathological states with exponential-in-time growth of the polynomial prefactors; if violated, the exponential boundedness criterion in (2.5) would not follow.
  • domain assumption The equal-time commutator vanishes, or time smearing is used so that the Paley-Wiener argument applies.
    Assumed in Section 3, footnote 5; for Lorentzian correlators, smearing in time is generally required as in [10]. The authors state their mixed-representation approach is valid in either case, but the precise distributional setup is not fully proven.
  • domain assumption For the cosmological bound, the gravitational background is treated as fixed and its light cone defines microcausality.
    Explicitly discussed in Sections 9.3 and 10: with dynamical gravity, the light cone is not sharply defined; without this assumption the positivity bound on M2^4 and the combined f1+f9 positivity does not follow.
  • domain assumption At late times and for the pole approximation, the one-loop corrected Green's function can be trusted for the low-energy poles, while high-energy spurious poles from the denominator polynomial are discarded.
    Section 6.2: the authors restrict to perturbative corrections of the two known poles and disregard new high-energy poles. This is standard EFT logic, but it is an assumption that the discarded poles cannot affect the causality test with exponential boundedness.
  • standard math The Paley-Wiener theorem applies to tempered distributions of compact support.
    Standard result, cited in Section 2; the paper relies on it to go from vanishing of the commutator outside the light cone to analyticity and exponential boundedness.

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Pith. "Pith review of Microcausality without Lorentz invariance." pith.science (2026). https://pith.science/paper/5RU5ZHX3

@misc{pith2026250204215,
  author       = {Pith},
  title        = {Pith review of: Microcausality without Lorentz invariance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5RU5ZHX3}},
  note         = {Machine review of arXiv:2502.04215}
}
abstract

Microcausality -- the vanishing of commutators outside the lightcone -- is a fundamental property of relativistic quantum field theories. We derive its implications for two-point functions of scalar operators on {\it Lorentz-breaking} states. We restrict to spatially homogeneous and isotropic states, at zero and finite temperature, such as finite-density states of matter and primordial inflationary states. In a mixed $(t, \vec k \, )$ representation, we find certain analyticity and exponential boundedness conditions, which we verify in a variety of examples. Crucially, we discuss how our conditions can be tested within the regime of validity of Lorentz-breaking low-energy effective field theories, clarifying the role of the group velocity of low-energy excitations. In the cosmological case, we derive a positivity condition on an EFT coefficient in an inflationary background. Lastly, we comment on how microcausality can be used to constrain higher-point correlation functions, via suitable nested commutators.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

41 extracted references · 10 canonical work pages · cited by 7 Pith papers

  1. [6]

    Gavassino, M

    L. Gavassino, M. M. Disconzi and J. Noronha, Dispersion Relations Alone Cannot Guarantee Causality, Phys. Rev. Lett. 132 (2024) 162301 [ 2307.05987]

    arXiv 2024

  2. [1]

    Strocchi, Selected Topics on the General Properties of Quantum Field Theory

    F. Strocchi, Selected Topics on the General Properties of Quantum Field Theory . World Scientific, 1993, 10.1142/1807

    work page doi:10.1142/1807 1993

  3. [2]

    Dubovsky, A

    S. Dubovsky, A. Nicolis, E. Trincherini and G. Villadoro, Microcausality in curved space-time, Phys. Rev. D 77 (2008) 084016 [ 0709.1483]

    arXiv 2008

  4. [3]

    Creminelli, O

    P. Creminelli, O. Janssen and L. Senatore, Positivity bounds on effective field theories with spontaneously broken Lorentz invariance, JHEP 09 (2022) 201 [ 2207.14224]

    arXiv 2022

  5. [4]

    M. P. Heller, A. Serantes, M. Spali´ nski and B. Withers, Rigorous Bounds on Transport from Causality, Phys. Rev. Lett. 130 (2023) 261601 [ 2212.07434]

    arXiv 2023

  6. [5]

    Creminelli, O

    P. Creminelli, O. Janssen, B. Salehian and L. Senatore, Positivity bounds on electromagnetic properties of media, JHEP 08 (2024) 066 [ 2405.09614]

    arXiv 2024

  7. [7]

    E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis. Princeton University Press, 2003

    work page 2003

  8. [8]

    N. N. Bogolubov, A. A. Logunov, A. I. Oksak and I. T. Todorov, eds., General Principles of Quantum Field Theory , vol. 10 of Mathematical Physics and Applied Mathematics. Springer, 1990, 10.1007/978-94-009-0491-0

    work page doi:10.1007/978-94-009-0491-0 1990

Show all 41 references
  1. [9]

    Kravchuk, J

    P. Kravchuk, J. Qiao and S. Rychkov, Distributions in CFT. Part II. Minkowski space , JHEP 08 (2021) 094 [ 2104.02090]

    2021 arXiv

  2. [10]

    Witten, Algebras, regions, and observers., Proc

    E. Witten, Algebras, regions, and observers., Proc. Symp. Pure Math. 107 (2024) 247 [2303.02837]

    2024 arXiv

  3. [11]

    T. J. Hollowood and G. M. Shore, Causality and Micro-Causality in Curved Spacetime , Phys. Lett. B 655 (2007) 67 [ 0707.2302]

    2007 arXiv

  4. [12]

    T. J. Hollowood and G. M. Shore, The Refractive index of curved spacetime: The Fate of causality in QED , Nucl. Phys. B 795 (2008) 138 [ 0707.2303]. – 42 –

    2008 arXiv

  5. [13]

    T. J. Hollowood and G. M. Shore, Causality Violation, Gravitational Shockwaves and UV Completion, JHEP 03 (2016) 129 [ 1512.04952]

    2016 arXiv

  6. [14]

    A. V. Manohar, Introduction to Effective Field Theories , 1804.05863

    arXiv

  7. [15]

    Adams, N

    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion , JHEP 10 (2006) 014 [hep-th/0602178]

    2006 arXiv

  8. [16]

    J. D. Jackson, Classical Electrodynamics. Wiley, 1998

    1998

  9. [17]

    E. M. Lifshitz and L. P. Pitaevskii, Statistical physics: theory of the condensed state , vol. 9. Elsevier, 2013

    2013

  10. [18]

    L. Hui, I. Kourkoulou, A. Nicolis, A. Podo and S. Zhou, S-matrix positivity without Lorentz invariance: a case study , JHEP 04 (2024) 145 [ 2312.08440]

    2024 arXiv

  11. [19]

    Creminelli, M

    P. Creminelli, M. Delladio, O. Janssen, A. Longo and L. Senatore, Non-analyticity of the S-matrix with spontaneously broken Lorentz invariance , JHEP 06 (2024) 201 [2312.08441]

    2024 arXiv

  12. [20]

    Esposito, R

    A. Esposito, R. Krichevsky and A. Nicolis, Solidity without inhomogeneity: Perfectly homogeneous, weakly coupled, UV-complete solids, JHEP 11 (2020) 021 [ 2004.11386]

    2020 arXiv

  13. [21]

    Nicolis, R

    A. Nicolis, R. Penco, F. Piazza and R. Rattazzi, Zoology of condensed matter: Framids, ordinary stuff, extra-ordinary stuff , JHEP 06 (2015) 155 [ 1501.03845]

    2015 arXiv

  14. [22]

    Blas and S

    D. Blas and S. Sibiryakov, Horava gravity versus thermodynamics: The Black hole case, Phys. Rev. D 84 (2011) 124043 [ 1110.2195]

    2011 arXiv

  15. [23]

    Creminelli, J

    P. Creminelli, J. Norena, M. Pena and M. Simonovic, Khronon inflation, JCAP 11 (2012) 032 [ 1206.1083]

    2012 arXiv

  16. [24]

    Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories , J

    P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories , J. Phys. A 45 (2012) 473001 [ 1205.5040]

    2012 arXiv

  17. [25]

    Joyce, A

    A. Joyce, A. Nicolis, A. Podo and L. Santoni, Integrating out beyond tree level and relativistic superfluids, JHEP 09 (2022) 066 [ 2204.03678]

    2022 arXiv

  18. [26]

    Nicolis, A

    A. Nicolis, A. Podo and L. Santoni, The connection between nonzero density and spontaneous symmetry breaking for interacting scalars , JHEP 09 (2023) 200 [2305.08896]

    2023 arXiv

  19. [27]

    Baier, P

    R. Baier, P. Romatschke, D. T. Son, A. O. Starinets and M. A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance, and holography , Journal of High Energy Physics 2008 (2008) 100

    2008

  20. [28]

    L. D. Landau and E. M. Lifshitz, Fluid Mechanics: Volume 6 , vol. 6. Elsevier, 1987

    1987

  21. [29]

    Baumann, D

    D. Baumann, D. Green and T. Hartman, Dynamical Constraints on RG Flows and Cosmology, JHEP 12 (2019) 134 [ 1906.10226]. – 43 –

    2019 arXiv

  22. [30]

    de Rham, A

    C. de Rham, A. J. Tolley and J. Zhang, Causality Constraints on Gravitational Effective Field Theories, Phys. Rev. Lett. 128 (2022) 131102 [ 2112.05054]

    2022 arXiv

  23. [31]

    Green, Y

    D. Green, Y. Huang, C.-H. Shen and D. Baumann, Positivity from Cosmological Correlators, JHEP 04 (2024) 034 [ 2310.02490]

    2024 arXiv

  24. [32]

    Carrillo Gonz´ alez,Bounds on EFT’s in an expanding universe , Phys

    M. Carrillo Gonz´ alez,Bounds on EFT’s in an expanding universe , Phys. Rev. D 109 (2024) 085008 [ 2312.07651]

    2024 arXiv

  25. [33]

    Donath and E

    Y. Donath and E. Pajer, The in-out formalism for in-in correlators , JHEP 07 (2024) 064 [2402.05999]

    2024 arXiv

  26. [34]

    Cheung, P

    C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, The Effective Field Theory of Inflation , JHEP 03 (2008) 014 [ 0709.0293]

    2008 arXiv

  27. [35]

    Weinberg, Effective Field Theory for Inflation , Phys

    S. Weinberg, Effective Field Theory for Inflation , Phys. Rev. D 77 (2008) 123541 [0804.4291]

    2008 arXiv

  28. [36]

    Bordin, G

    L. Bordin, G. Cabass, P. Creminelli and F. Vernizzi, Simplifying the EFT of Inflation: generalized disformal transformations and redundant couplings , JCAP 09 (2017) 043 [1706.03758]

    2017 arXiv

  29. [37]

    Visser, B

    M. Visser, B. Bassett and S. Liberati, Superluminal censorship, Nucl. Phys. B Proc. Suppl. 88 (2000) 267 [ gr-qc/9810026]

    2000 arXiv

  30. [38]

    Gao and R

    S. Gao and R. M. Wald, Theorems on gravitational time delay and related issues , Class. Quant. Grav. 17 (2000) 4999 [ gr-qc/0007021]

    2000 arXiv

  31. [39]

    Caron-Huot, D

    S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Sharp boundaries for the swampland, JHEP 07 (2021) 110 [ 2102.08951]

    2021 arXiv

  32. [40]

    N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space , Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, UK, 1982, 10.1017/CBO9780511622632

    1982 doi

  33. [41]

    Weinberg, The Quantum theory of fields

    S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations . Cambridge University Press, 6, 2005, 10.1017/CBO9781139644167. – 44 –

    2005 doi

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