Numerical results on quantum energy inequalities in integrable models at the two-particle level

In this article, we study the impact of self-interaction and multiparticle states on sustaining negative energies in relativistic quantum systems. For physically reasonable models, one usually requires bounds on both magnitude and duration of the accumulation of negative energy, typically given in form of a quantum energy inequality (QEI). Such bounds have applications in semiclassical gravity where they exclude exotic spacetime geometries and imply the formation of singularities. The essence of this article is a novel numerical method for determining optimal QEI bounds at the one- or two-particle level, extending previous work focused on the one-particle case and overcoming a new type of technical challenge associated with the two-particle scenario. Our method is tailored for integrable models in quantum field theory constructed via the S-matrix boostrap. Applying the method to a representative model, the sinh-Gordon model, we confirm self-interaction as the source of negative energy, with stronger interactions leading to more pronounced negativities. Moreover, we establish the validity of QEIs and the averaged weak energy condition (AWEC) at the one- and two-particle level. Lastly, we identify a constrained one-parameter class of nonminimal stress tensor expressions satisfying QEIs at both levels, with more stringent constraints emerging from the QEI bounds at the two-particle level.

Figure 1

Absolute value of lowest eigenvector $|\phi ({\theta }_1,{\theta }_2)|$ of $T_2^{00}(g^2)$ for the maximally interacting sinh-Gordon model ($B=1$). Parameters: $n=80$ and, from upper left to lower right, $(R,\sigma )=(8.0,0.008{\mathrm{m}}^{-1})$, $(7.0,0.021{\mathrm{m}}^{-1})$, $(5.0,0.086{\mathrm{m}}^{-1})$, $(4.0,0.220{\mathrm{m}}^{-1})$.

Figure 2

Lowest eigenvalue of $T_k^{00}(g^2)$ for varying coupling constant $B$ in the sinh-Gordon model. We compare the lowest eigenvalues in one- and two-particle states and indicate its variance for increasing $n$ for the latter. Parameters: $R=4.5$ or 5.0, $\sigma =0.1{\mathrm{m}}^{-1}$.

Figure 3

Lowest eigenvalue of $T_k^{00}(g^2)$ for varying averaging scale $\sigma$ in the maximally interacting sinh-Gordon model ($B=1$) at the one- and two-particle level for Gaussian and Lorentzian averaging. Parameters: $n=60$, and adaptively chosen $R$. Dotted lines indicate the asymptotic scaling ($\propto {\sigma }^{-2}$).

Figure 4

Lowest eigenvalue of $T_{k,\text{nonmin},\nu }^{00}(g^2)$ for nonminimal choice of the stress tensor as in (37) in the maximally interacting sinh-Gordon model ($B=1$). We compare the lowest eigenvalues in one- and two-particle states for varying $\nu$ and indicate convergence with increasing $n$ for the latter. The asymptotes of the curves are indicated by dotted lines. Parameters: $R=4.5$ or 5.0, $\sigma =0.1{\mathrm{m}}^{-1}$.