Most people prefer to do research in the Path Integral Formalism (PIF), instead of the Operator Formalism (OF), because it is "easier." Easier means that whatever property you have in the OF, if you have a PIF for that theory, you can have more properties in the PIF. For example, Noether's theorem is only valid, as far as I know, in a Lagrangian formulation.
The fact you find more (almost everything) of the work in Euclidean space-time using the PIF is due to the above explanation.
I only know about two cases where people work on the (Euclidean) OF without referring to the PIF:
1.- The Conformal Bootstrap
2.- When there's no Lagrangian.
For the second point I am not expert, and I cannot guide you into the whole literature. If you are interested only in N=2 4D SCFTs you might want to take a look at the Tachikawa's books
For the Conformal Bootstrap, there are two good reviews out there:
which was pointed out by Peter Kravchuk pinted, and
I personally love the Rychkov's review.
This post imported from StackExchange Physics at 2016-05-31 07:23 (UTC), posted by SE-user CGH