What does it mean to have a QFT that can not be encoded by an action. What is by far the most powerful approach of study in such a case. What is the best studied physical theory that falls into this category. What is QFT? I think it is a set of rules that facilitate descriptions of things we can observe, but what is the most mathematically accurate way of capturing what it is? I have read of cases where the path integral approach fails what does this mean? Is there any geometric structure whose properties describe the space of all posible QFT's? What happens when you can't use an action?
Answer
It is possible to abstract the notion of a QFT away from the notion of Lagrangians/Hamiltonians, one axiomatic way are the Wightman axioms. As one can see, they reduce the quantum theory to its very heart: A Hilbert space where the states live and a field operator that acts upon it, generating "particles", all of this happening in a Lorentz covariant way.
Concrete example of QFTs without an action are CFTs in 2 dimensions, they are almost completely fixed by demanding that they are a QFT in the axiomatic sense which has a Virasoro symmetry.
The "space of all possible QFTs" is quite a complicated (and open, I think) question, since it is quite difficult to prove that a given QFT is really a QFT in the sense of the Wightman (or Osterwalder-Schrader) axioms.
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