Dynamical observables from spectral quantum field theory

We present a framework for the direct, non-perturbative computation of real-time observ- ables in quantum field theories using spectral functional methods. Building on spectral representations of correlators, particularly the K¨all´en-Lehmann representation of the prop- agator, this approach enables the evaluation of functional diagrammatic relations, such as the Dyson-Schwinger equations and the functional renormalisation group, across the entire complex frequency domain. The primary focus of this work is on quantum chro- modynamics (QCD), where we develop efficient methods for computing the propagator spectral functions of various elementary fields. These include the spectral function of a scalar field in three dimensions, the graviton in asymptotically safe quantum gravity, and, in particular, the light quark in QCD. To study the real-time properties of dynamical chi- ral symmetry breaking, we construct a causal quark-gluon interaction model. Building on the non-perturbative spectral functions of the propagators, we investigate the properties of bound states and show how the spectral approach can be applied to compute bound- state masses directly on the real frequency axis. Finally, we use spectral reconstruction techniques to compute the pion distribution amplitude directly on the light front.