One way to approach QFT in mathematical terms is provided by the so-called Gårding-Wightman axioms, which defines in rigorous mathematical terms what a quantum field theory is supposed to be. If I'm not mistaken, this is what is called *axiomatic QFT*. Although it is a precise mathematical theory, it is widely known that to construct examples of quantum field theories which are proved to satisfy these axioms is no easy task and, as far as I know, only a few such theories were constructed by now.


Now, to better pose my question, let me use a specific theory in QFT, which is the Klein-Gordon theory, as an example. We can, for instance, follow the exposition of [Folland's book](https://www.ams.org/publications/authors/books/postpub/surv-149) and give mathematical meaning to such theory by defining the correct (single particle) Hilbert space $\mathscr{H}$, giving meaning to creation and annihilation operators as operator-valued distributions in the associated Fock space and obtaining the expressions for the associated quantum fields.


Although this construction is, in practice, using some ingredients of what's expected in the axiomatic construction, it *seems* to me that these two approaches have different phylosophies: the axiomatic approach tries to construct a theory which fulfills every one of its axioms while the second one seems more pragmatic and tries to take what is written in the physics literature and convert it to a mathematical precise language.


The non-axiomatic approach does not intend to check if its objects satisfy any kind of axioms; it is almost like it was a 'dictionary' that translates the physical theory to a mathematical audience. As a consequence, it seems less technical and more compatible with what is written in the physics literature.


**Question:** So, my question is: why do we need an axiomatic approach to formalize QFT? Is there any limitation with the non-axiomaic approach? Are both approaches research material, i.e. different practices of the current attempts to give mathematical meaning to QFT?



QFT and mathematical rigor
As Abdelmalek Abdesselam pointed in his comment to the OP, the axiomatic approach to QFT is rather concerned with answering the question "what is a quantum field?". This is stated right at the Preface of the book of Streater and Wightman, *PCT, Spin and Statistics, and All That*. More precisely, it lists a minimal set of desiderata for a reasonable concept of a "quantum field", and deals with the consequences of such requirements, which can be thought of as "structural" or "model-independent". It leaves open the question of which fields used in physics actually satisfy these requirements, apart from checking that free fields do. This is important as a first relevance check, since free fields are the easiest to mathematically control - if not even a free field complies with a list of axioms for QFT, then these axioms are useless as they stand and should be modified. More generally, any list of mathematical axioms applied to a class of physical theories is bound not to be set in stone, since our knowledge of Nature is always approximate due to our limited experimental precision. One should rather see such axioms (and their limitations in the given context) as general physical principles whose robustness has to be constantly tested, in addition to forming a well-defined mathematical framework by themselves. Such is the way of natural sciences.


In that respect, it must be remarked that indeed *not all* physically relevant fields satisfy the Garding-Wightman axioms - most notably, fields acting in Krein spaces (i.e. "Hilbert" spaces with a possibly indefinite scalar product), such as the electromagnetic potential in a covariant gauge, do not. The corresponding axiom for vacuum expectation values that fails is that of *positivity*. There are ways to extend the Wightman formalism to such fields, but the results are nowhere near as mighty of even as rigorous, since positive definiteness of the scalar product is a powerful constraint. Another tricky example are *perturbative* (renormalized) quantum fields, since these are *formal* power series in the coupling constant (convergence of the renormalized perturbative series is usually expected to fail). One must in this case keep track of the order-by-order structure of all series involved, and to define certain concepts invoked by the Garding-Wightman axioms such as positivity is far from trivial. Regarding renormalization, the main conceptual challenge is not so much the ultraviolet problem (which is rigorously well understood on a formal perturbative level), but the infrared problem, which plagues all interacting QFT models with massless fields and is not completely understood on a rigorous level, even in formal perturbation theory. And all that before even considering how to extract some non-perturbative definition of the model from perturbative data by employing e.g. some generalized summability concept for the perturbative series.


This order of doing things - namely, stating a list of general desiderata and then checking if models comply with it or not - is what is often called a "top-down" approach to QFT. What people such as Folland try to do is rather a "bottom-up" approach: trying to make sense of the formal procedures physicists actually use in (perturbative) QFT. A more successful way of doing what Folland tries to do is achieved by the so-called *perturbative algebraic QFT*, which draws ideas from the Haag-Kastler algebraic approach to QFT (which is a different axiomatic scheme from that of Garding and Wightman, aiming at aspects which do not depend on the particular Hilbert space the fields live in) in order to mathematically understand formal renormalized perturbative QFT (summability of the perturbative series has not been addressed yet in this approach). The so-called *constructive* approach to QFT also fits in such a "bottom-up" philosophy: it tries to build QFT models by means other than perturbation theory, in view of its expected divergence in relevant cases (Abdelmalek can tell you far more than me on that). Once one has obtained the model, one may try to e.g. check the Wightman axioms (or some suitable modification thereof) in order to see if the model obtained complies with that particular notion of QFT, which is undoubtedly relevant.


To sum up, I would say that both approaches are important and complement each other.


