From what I hear, some modern mathematical approach quantum field theory uses the following definition
"A $d$-dimensional $S$-structured quantum field theory $Q$ is a mathematical object, consisting of its partition function $Z_Q$, its space of states $H_Q$, and its submanifold operators $V_Q$, satisfying various axioms." (taken from some lecture notes)
In physics textbooks, there is a definition of the partition function $Z$ in terms of an classical field action (much like the wikipedia entry).
I would like to know if there is a way to recover the partition function of a Wightman QFT (theory given in terms of fields operator valued distributions) or a Haag-Kastler QFT (given in terms of a local net of operator algebras). I ask this because, from what I understand, those need not to come from a classical field theory derived from an action principle.