  Light-matter interfaces have now entered a new stage marked by the ability to
engineer quantum correlated states under driven-dissipative conditions. To
propel this new generation of experiments, we are confronted with the need to
model non-unitary many-body dynamics in strongly coupled regimes, by
transcending traditional approaches in quantum optics. In this work, we
contribute to this program by adapting a functional integral technique,
conventionally employed in high-energy physics, in order to obtain
non-equilibrium dynamics for interacting light-matter systems. Our approach is
grounded in constructing 'two-particle irreducible' (2PI) effective actions,
which provide a non-perturbative and conserving framework for describing
quantum evolution at a polynomial cost in time. We apply our method to
complement the analysis of spin glass formation in the context of frustrated
multi-mode cavity quantum electrodynamics, initiated in our accompanying work
[H. Hosseinabadi, D. Chang, J. Marino, arXiv:2311.05682]. Finally, we outline
the capability of the technique to describe other near-term platforms in
many-body quantum optics, and its potential to make predictions for this new
class of experiments.
