It's well-known that there are no rigorously constructed and physically relevant QFTs. There is, however, a lot of mathematical work on effective field theories and renormalization, such as the books by Costello and by Salmhofer. My question is: does this mathematical work allow one to give mathematically rigorous (albeit effective, possibly depending upon empirical parameters) derivations of the physical computations one does with QFT (such as the electron magnetic dipole moment, say, or Compton scattering, this sort of thing)? If not, how far are we from being able to do so?


I ask because as somewhat of an outsider it seems hard for me to tell. Books that intend to give an account of these computations, like Folland's Quantum Field Theory, are very far from rigorous. The mathematical books on renormalization and effective field theories such as the ones I mentioned *seem* to be rigorous (although I have not read them in any amount of detail), but they also don't seem to discuss fundamental physics (or maybe I just missed it).



State of rigorous effective quantum field theories
I will leave aside what is meant by "effective field theory" in a purely mathematical context and just presume that the question asks whether renormalized interactive perturbative QFT (using formal power series in $\hbar$ and the coupling constants) can be mathematically well-defined. The answer is Yes (in multiple different ways), which has been repeated on this website several times (it is sufficient to search with the corresponding keywords). But it seems that there was not much interest in references before. So here are some references for what I consider to be a fairly clean approach of *perturbative algebraic QFT*. The list mixes older and newer references, as well as more and less readable references.



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> * *Scharf, G.*, Finite quantum electrodynamics. The causal approach., Texts and Monographs in Physics. Berlin: Springer-Verlag. x, 409 p. (1995). [ZBL0844.53052](https://zbmath.org/?q=an:0844.53052).
> * *Steinmann, Othmar*, Perturbative quantum electrodynamics and axiomatic field theory, Texts and Monographs in Physics. Berlin: Springer. ix, 355 p. (2000). [ZBL0946.81079](https://zbmath.org/?q=an:0946.81079).
> * *Hollands, Stefan*, [**Renormalized quantum Yang-Mills fields in curved spacetime**](http://dx.doi.org/10.1142/S0129055X08003420), Rev. Math. Phys. 20, No. 9, 1033-1172 (2008). [ZBL1161.81022](https://zbmath.org/?q=an:1161.81022).
> * *Brunetti, Romeo (ed.); Dappiaggi, Claudio (ed.); Fredenhagen, Klaus (ed.); Yngvason, Jakob (ed.)*, [**Advances in algebraic quantum field theory**](http://dx.doi.org/10.1007/978-3-319-21353-8), Mathematical Physics Studies. Cham: Springer (ISBN 978-3-319-21352-1/hbk; 978-3-319-21353-8/ebook). xii, 453 p. (2015). [ZBL1329.81022](https://zbmath.org/?q=an:1329.81022).
> * *Rejzner, Kasia*, [**Perturbative algebraic quantum field theory. An introduction for mathematicians**](http://dx.doi.org/10.1007/978-3-319-25901-7), Mathematical Physics Studies. Cham: Springer (ISBN 978-3-319-25899-7/hbk; 978-3-319-25901-7/ebook). xi, 180 p. (2016). [ZBL1347.81011](https://zbmath.org/?q=an:1347.81011).
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The work of Costello and Gwilliam is a different formalism, but could also be cited as an example. Perhaps others can add answers with references to their own favorite approach.


