Mourre theory and spectral analysis of energy-momentum operators in relativistic quantum field theory

Abstract: A central task of theoretical physics is to analyse spectral properties of quantum mechanical observables. In this endeavour, Mourre's conjugate operator method emerged as an effective tool in the spectral theory of Schr\"odinger operators. This paper introduces a novel class of examples from relativistic quantum field theory that are amenable to Mourre's method. By assuming Lorentz covariance and the spectrum condition, we derive a limiting absorption principle for the energy-momentum operators and provide new proofs of the absolute continuity of the energy-momentum spectra. Moreover, under the assumption of dilation covariance, we show that the spectrum of the relativistic mass operator is purely absolutely continuous in $(0,\infty)$.