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 ${\tilde{\Gamma }}_{\mathrm{gap}}\ge {\Gamma }_{\mathrm{gap}}/[\gamma (1+v{v}_{\max })]$, 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.