# partition function for Wightman and Haag-Kastler QFT

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.

$\langle \psi_f| e^{i\sum_\alpha c_\alpha \mathcal{O}_\alpha} | \psi_i\rangle$,
where $\psi_f$ and $\psi_i$ are final and initial states, the $\mathcal{O}_\alpha$ are operators that appear in your theory. It's a function of the parameters $c_\alpha$; it's meant to be a generating function for the matrix elements of observables and amplitudes.
In the Wightman scheme you can go farther, since you have operator-valued distributions to play with. The QFT partition function is supposed to be a generating function for vacuum expectation values of products of local observables, aka Wightman functions. To do this, you assume $\psi_f$ and $\psi_i$ are both the vacuum vector, and that the $\mathcal{O}_\alpha$ have the form $\langle j_\alpha,\phi_\alpha \rangle \simeq \int j_\alpha(x) \mathcal{\phi}_\alpha (x) dx$, where the $\phi_\alpha$ are the Wightman fields, and the $j_\alpha$ are nice test functions.
In more general frameworks, you let the $\phi_\alpha$ be generators of an OPE-algebra, and test them only on test functions which are appropriately conserved currents.