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My question is what can Hairer's theory of regularity structures contribute to the existence problem in relativistic QFT. I want to know whether his existence result would help in proving existence of the corresponding Minkowski QFT. For this one needs to be able to prove both reflection positivity and O(4) invariance.
See also the discussion at http://www.physicsoverflow.org/18021/submission-creation-requests?show=22205#c22205
The O(4) invariance is automatic, because the SPDE solutions are (statistically) O(4) invariant if you use a O(4) invariant regulator. His regulator is a smooth function convolved with the noise, and you can make the smooth function rotationally invariant. The reflection positivity is more involved, but it shouldn't be hard. The main issue is overcoming the superrenormalizability requirement, as you said in other comments, but I believe it is not insuperable (although Hairer doesn't even try to do this, as he is not particularly interested in unitary QFTs, but in SPDEs).
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