Operator formalism in QFT in Euclidean space-time

Question

In QFT there are two very useful general approaches to study quantum fields (on the Minkowski space-time): path integrals and operator formalism. Sometimes they give the same results, sometimes one formalism is better than another.

My question is what happens in the Euclidean space-time (e.g. statistical physics)? As far as I know the formalism of path integrals is widely used in the field, but what about the operator formalism? In particular I would be happy to have a reference to a detailed discussion of the free scalar field in Euclidean space time in dimension greater than 2 (the 2d massless case is discussed in several books on CFT).

Answer 1

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 http://arxiv.org/abs/1312.2684

For the Conformal Bootstrap, there are two good reviews out there:

https://arxiv.org/abs/1602.07982

which was pointed out by Peter Kravchuk pinted, and

https://arxiv.org/abs/1601.05000

I personally love the Rychkov's review.