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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.

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The partition function is a special case of the matrix element

$\langle \psi_f| e^{i\sum_\alpha c_\alpha \mathcal{O}_\alpha} | \psi_i\rangle$,

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 Haag-Kastler setup, I'm not sure how you can go farther than this, since they don't make use of local fields.

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.

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