One way to approach QFT in mathematical terms is provided by the so-called Gårging-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 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?

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 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?