Lorentz-Covariant Spectral Bounds from Thermal Quantum Field Theory

Abstract

We derive rigorous Lorentz-covariant bounds on relaxation spectra directly from the analytic structure of retarded Green's functions in thermal quantum field theory, using only causality, unitarity, the Kubo-Martin-Schwinger condition, and Lorentz covariance, without reference to any specific dynamical model. A single rest-frame quasinormal pole is generically smeared into a continuum of excitations in boosted frames, with width set by the maximal signal velocity. We prove that the non-hydrodynamic gap $\Gamma_{\mathrm{gap}}$ transforms as $\tilde{\Gamma}_{\mathrm{gap}} \geq \Gamma_{\mathrm{gap}}/[\gamma(1 + v v_{\mathrm{max}})]$, and that the convergence radius of the hydrodynamic gradient expansion satisfies $\tilde{k}_c \in [k_c/\gamma(1+v v_s), k_c/\gamma(1-v v_s)]$ under a boost of velocity $v$. We verify the bounds by a numerical quasinormal-mode computation in the ${N}=4$ super-Yang-Mills plasma: the leading boosted pole moves deeper into the complex plane -- the observed relaxation rate increases with boost velocity, in sharp contrast to naive time dilation -- while respecting the bound throughout. The results apply non-perturbatively to the quark-gluon plasma, neutron star merger dynamics, and quantum critical systems.