Mathematicians absolutely have been trying to formalise QFT for decades, with excellent success in many areas! Have a quick look at the nLab page for quantum fi

Mathematicians absolutely have been trying to formalise QFT for decades, with excellent success in many areas! Have a quick look at the nLab page for quantum field theory: https://ncatlab.org/nlab/show/quantum+field+theory https://ncatlab.org/nlab/show/quantum+field+theory . It is a complicated and hairy problem, and one that's frankly not even well-defined, because QFT can be approached from different perspectives. Often, this is done by using analogies with simpler quantum mechanical theories or classical field theories, but these end up not holding up to mathematical scrutiny when you try to implement them in a naive way. But this is really quite normal for mathematics. Nobody argues that quantum mechanics is poorly-defined, even though the quantum mechanics that physics undergraduates learn is certainly poorly-defined, and most physicists will never learn about rigged Hilbert spaces or know enough functional analysis to define it properly. This never results in any actual disagreements. All trained physicists will come to the same predictions if they start from the same assumptions, even if they don't know the strict, mathematical formalities. Definitional issues almost always arises because of infinities, and true infinities are rare in physics -- it's almost always just an approximation for "very big" or "very many". QFT has a lot of infinities. Quantum mechanics promotes classical theories with finite numbers of degrees of freedom into infinite-dimensional wave functions. QFT attempts to do the same, but with classical theories having uncountably infinite degrees of freedom. It's no surprise that there are definitional issues. And even if you have an axiomatic mathematical model, you'd be surprised how little added value that is to physics as a whole. It's certainly an interesting and important area of research, but the fact that it's ongoing doesn't somehow invalidate the predictions made using formal methods, because those methods are reproducible. Renormalisation isn't magic, it's just a set of rules that you apply, and in fact, those rules have a quite natural physical interpretation (in terms of bare vs effective theories).