Quantum energy inequalities in interacting integrable quantum field theories beyond one-particle level: stress-tensor prefactor freedom, parity breaking, and higher particle numbers

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 $q(\cosh\zeta)$ 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 $n\ge2$. 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 $T^{00}(t,x)$ 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, $\int dt\,g(t)^2\langle\varphi,T^{00}(t,x)\varphi\rangle\ge -c_g\|\varphi\|^2$, with $c_g$ depending only on the real 'smearing' function $g$; such bounds limit the magnitude and duration of negative-energy excursions and are key inputs in semiclassical gravity, where $\langle T^{\mu\nu}\rangle$ 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 $S(\zeta)$ — a matrix-valued meromorphic function of the rapidity difference $\zeta$ (the rapidity $\theta$ parametrizes the on-shell momentum as $p^\mu=m(\cosh\theta,\sinh\theta)$) acting on the internal degrees of freedom of the particles. Particles may form bound states, which appear as first-order poles of $S$ in the physical strip $0<\operatorname{Im}\zeta<\pi$ (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 $F_2(\zeta_1,\zeta_2)$ between multi-particle states, which satisfy Watson's functional equations $F(\zeta)=S(\zeta)F(-\zeta)$ and $F(\zeta+i\pi)=F(-\zeta+i\pi)$ 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 $\zeta=0$ when $S(0)=-1$, and minimal growth — is called the minimal solution $F_{\min}$; it is unique under mild growth bounds and can be constructed by an integral representation whenever $iS'/S$ 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 $e^{ia\operatorname{sh}\zeta}$, $a>0$; 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) $S$, 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 $S$, of the form $q(\cosh\zeta)F_{\min}(\zeta)$ with $q$ rational and $q(-1)=1$; 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 $e^{|\theta|}$: in the $O(n)$ nonlinear sigma model (an integrable theory with $n$ particle species linked by an internal $O(n)$ symmetry) the QEI holds exactly for $q\equiv1$, while in the generalized Bullough–Dodd model (a scalar theory whose particle is its own bound state) it constrains $q$ only partially.

The frontier beyond these results is threefold. First, the polynomial prefactor $q$ is left unfixed by the first-principles classification: in scalar models it can be realized by applying a polynomial in the d'Alembertian $\Box=g^{\mu\nu}\partial_\mu\partial_\nu$ to $T^{\mu\nu}$, but for matrix-valued $S$ 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 $n\ge2$ are uncontrolled for non-constant $S$, since all form factors $F_1,\dots,F_{2n}$ and new kinematic poles enter.

Problem Statement

For 1+1-dimensional integrable QFTs with non-constant (rapidity-dependent), generic, possibly matrix-valued scattering functions $S$ (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 $q(\cosh\zeta)$ in the two-particle form factor $F(\zeta)=q(\cosh\zeta)F_{\min}(\zeta)$ 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 $q$, so the open task goes beyond them: establish a principle that fixes $q$ 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 $S$, establish state-independent lower bounds on time-smeared energy densities — or prove their failure — for expectation values in $n$-particle states with $n\ge2$, where all form factors $F_1,\dots,F_{2n}$ 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 $g$, a constant $c_g$ independent of the state within the $n$-particle sector (it may depend on the model and on the choice of the stress-energy tensor, i.e., on the prefactor $q$); resolving direction (3) requires this per-sector bound, or a proof of its failure. The natural stronger target, matching the constant-$S$ 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 $q$ further, and direction (2) widens the class of models over which the criteria hold.

References

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