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We delve into the in-in formalism and derive general expressions for thecomputation of n-point functions for a self-interacting scalar field by meansof a probability density functional (PDF). Even though in typical situationsthese formulae are not simpler than the standard approaches to Quantum FieldTheory (QFT), we find that in specific situations, such as a first-ordercomputation with a non-trivial potential, they can significantly reduce theeffort needed to evaluate quantities of interest. The presented formulae may bereadily generalized to other types of fields (e.g., fermionic Dirac fields),and to interacting theories that contain various types of fields. An analogousresult for the in-out formalism is presented using the previously employedstrategies. Finally, we apply the obtained results to first-order computationsand work out specific examples where we explore their limitations and possibleprospects for future computations.

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