Quantum energy inequalities (QEIs) — state-independent lower bounds on time-smeared energy densities — are established for free fields, conformal field theories, and constant-scattering integrable models, but in genuinely interacting 1+1-dimensional integrable quantum field theories (QFTs) with several particle species and bound states they have only been proven at one-particle level, under parity covariance of the stress-energy tensor. Completing this theory, as posed in the outlook of Bostelmann–Cadamuro–Mandrysch (2024), requires: determining the intrinsic nature of the polynomial prefactor q(cosh⁡ζ)q(\cosh\zeta) in the two-particle form factor of the stress-energy tensor (or proving that no further physical principle fixes it), extending the sufficient one-particle QEI criterion to parity-breaking models, and establishing QEIs or their failure for states of particle number n≥2n\ge2. Numerical evidence at the two-particle level in the sinh-Gordon model (the simplest single-species interacting integrable QFT) supports validity, but all three directions remain analytically open as of 2026.

Background

In relativistic quantum field theory (QFT), the total energy is a positive operator, but the local energy density T00(t,x)T^{00}(t,x) can take negative expectation values in non-vacuum states. A quantum energy inequality (QEI) is a state-independent lower bound on time-smeared averages of the energy density, ∫dt g(t)2⟨φ,T00(t,x)φ⟩≥−cg∥φ∥2\int dt\,g(t)^2\langle\varphi,T^{00}(t,x)\varphi\rangle\ge -c_g\|\varphi\|^2, with cgc_g depending only on the real 'smearing' function gg; such bounds limit the magnitude and duration of negative-energy excursions and are key inputs in semiclassical gravity, where ⟨Tμν⟩\langle T^{\mu\nu}\rangle sources the Einstein equations. QEIs are well understood for linear (non-interacting) fields and for two-dimensional conformal field theories, but concrete state-independent QEIs in genuinely interacting relativistic QFTs remain rare.

The main current arena is 1+1-dimensional integrable QFT, where scattering is factorized: two-particle collisions involve no particle production, and the interaction is encoded in a scattering function S(ζ)S(\zeta) — a matrix-valued meromorphic function of the rapidity difference ζ\zeta (the rapidity θ\theta parametrizes the on-shell momentum as pμ=m(cosh⁡θ,sinh⁡θ)p^\mu=m(\cosh\theta,\sinh\theta)) acting on the internal degrees of freedom of the particles. Particles may form bound states, which appear as first-order poles of SS in the physical strip 0<Im⁡ζ<π0<\operatorname{Im}\zeta<\pi (such S-functions with bound-state poles are classified in Cadamuro–Tanimoto 2015). Expectation values of local observables are governed by form factors, i.e., matrix elements such as F2(ζ1,ζ2)F_2(\zeta_1,\zeta_2) between multi-particle states, which satisfy Watson's functional equations F(ζ)=S(ζ)F(−ζ)F(\zeta)=S(\zeta)F(-\zeta) and F(ζ+iπ)=F(−ζ+iπ)F(\zeta+i\pi)=F(-\zeta+i\pi) together with hermiticity, CPT (invariance under charge conjugation, parity and time reversal), and internal-symmetry conditions. The most regular solution of these equations — no poles or zeros in the closed physical strip beyond one mandatory zero at ζ=0\zeta=0 when S(0)=−1S(0)=-1, and minimal growth — is called the minimal solution Fmin⁡F_{\min}; it is unique under mild growth bounds and can be constructed by an integral representation whenever iS′/SiS'/S is integrable on a strip. Scalar scattering functions are moreover ambiguous: they may be multiplied by any number of CDD factors (Castillejo–Dalitz–Dyson factors — meromorphic factors satisfying the S-axioms trivially, so their insertion changes no physical consistency condition) and by an 'exotic' factor eiash⁡ζe^{ia\operatorname{sh}\zeta}, a>0a>0; the inverse-scattering construction of the model does not fix this ambiguity.

The QEI program in this setting progressed from the massive Ising model (a free Majorana field; Bostelmann–Cadamuro–Fewster 2013), through one-particle-level criteria for single-species models without bound states (Bostelmann–Cadamuro 2016), to the source of the present problem, Bostelmann–Cadamuro–Mandrysch (2024). That work (i) proved a state-independent QEI at all particle numbers for models with constant (rapidity-independent) SS, including the Federbush model (an interacting theory of two massive Dirac fields coupled by a current-current term) and fermionic variants; (ii) classified all stress-energy tensors at one-particle level for generic models with several particle species and bound states (Theorem 3.2 there), finding that the two-particle form factor of the energy density is, per eigenvalue of SS, of the form q(cosh⁡ζ)Fmin⁡(ζ)q(\cosh\zeta)F_{\min}(\zeta) with qq rational and q(−1)=1q(-1)=1; and (iii) gave necessary and sufficient criteria for a one-particle QEI (Theorem 5.1 there), turning on whether the large-rapidity growth of the form factor stays below or exceeds the exponential e∣θ∣e^{|\theta|}: in the O(n)O(n) nonlinear sigma model (an integrable theory with nn particle species linked by an internal O(n)O(n) symmetry) the QEI holds exactly for q≡1q\equiv1, while in the generalized Bullough–Dodd model (a scalar theory whose particle is its own bound state) it constrains qq only partially.

The frontier beyond these results is threefold. First, the polynomial prefactor qq is left unfixed by the first-principles classification: in scalar models it can be realized by applying a polynomial in the d'Alembertian □=gμν∂μ∂ν\Box=g^{\mu\nu}\partial_\mu\partial_\nu to TμνT^{\mu\nu}, but for matrix-valued SS it may depend on particle species and admits no such interpretation, and it is open which principle — perturbation theory (available only when a Lagrangian is known), conformal scaling limits, momentum-space clustering, or QEIs at higher particle number — fixes it, if any does. Second, the sufficient part of the QEI criterion assumes parity covariance of the stress-energy tensor, with only numerical evidence beyond that. Third, QEIs for states of particle number n≥2n\ge2 are uncontrolled for non-constant SS, since all form factors F1,…,F2nF_1,\dots,F_{2n} and new kinematic poles enter.

Problem Statement

For 1+1-dimensional integrable QFTs with non-constant (rapidity-dependent), generic, possibly matrix-valued scattering functions SS (a class that includes single-species models such as sinh-Gordon as well as models with several particle species and/or bound states), complete the quantum energy inequality theory beyond the one-particle, parity-covariant setting established by Bostelmann–Cadamuro–Mandrysch (2024) by addressing the following linked open directions raised in its outlook. (1) Prefactor freedom: characterize the intrinsic nature of the factor q(cosh⁡ζ)q(\cosh\zeta) in the two-particle form factor F(ζ)=q(cosh⁡ζ)Fmin⁡(ζ)F(\zeta)=q(\cosh\zeta)F_{\min}(\zeta) of the energy density in the general multi-species setting. The classification Theorem 3.2 already establishes that the form-factor axioms themselves (Watson's equations with hermiticity, CPT and internal symmetries) do not fix qq, so the open task goes beyond them: establish a principle that fixes qq uniquely (candidates named in the source: perturbation theory where a Lagrangian exists, conformal scaling limits, momentum-space clustering, or QEIs at higher particle number), or prove that the freedom persists even when such further principles are imposed — noting the source's caution that in some situations there might be too many unfixed prefactors for the QEI to meaningfully constrain them. (2) Parity breaking: prove, or disprove, a model-independent sufficient condition for a one-particle QEI without assuming parity covariance of the stress-energy tensor, extending the analytic proof of the sufficient criterion (Theorem 5.1(b)); the necessary condition is known to survive without parity (Remark 5.2), and numerical tests support the sufficient one. (3) Higher particle numbers: for models with non-constant SS, establish state-independent lower bounds on time-smeared energy densities — or prove their failure — for expectation values in nn-particle states with n≥2n\ge2, where all form factors F1,…,F2nF_1,\dots,F_{2n} contribute and additional kinematic poles arise at the boundary of the analyticity region. The bound is sought in the QEI form of the source: for each real smearing function gg, a constant cgc_g independent of the state within the nn-particle sector (it may depend on the model and on the choice of the stress-energy tensor, i.e., on the prefactor qq); resolving direction (3) requires this per-sector bound, or a proof of its failure. The natural stronger target, matching the constant-SS result (Theorem 4.3 there), is a single bound uniform over finite-particle states, including superpositions of different particle number, but this uniform strengthening is optional rather than required. A definitive result in either direction, validity or failure, is sought. The common objective is a state-independent QEI theory valid for interacting multi-species integrable models in generic states together with a first-principles determination of the stress-energy tensor; direction (3) is a candidate input to direction (1), since higher-particle QEIs are expected to constrain qq further, and direction (2) widens the class of models over which the criteria hold.

References

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