Probability Density Functionals in Perturbation Theory within the QFT in-in Formalism

We delve into the in-in formalism and derive general expressions for the computation of n-point functions for a self-interacting scalar field by means of a probability density functional (PDF). Even though in typical situations these formulae are not simpler than the standard approaches to Quantum Field Theory (QFT), we find that in specific situations, such as a first-order computation with a non-trivial potential, they can significantly reduce the effort needed to evaluate quantities of interest. The presented formulae may be readily generalized to other types of fields (e.g., fermionic Dirac fields), and to interacting theories that contain various types of fields. An analogous result for the in-out formalism is presented using the previously employed strategies. Finally, we apply the obtained results to first-order computations and work out specific examples where we explore their limitations and possible prospects for future computations.