Relation between statistical mechanics and quantum field theory

Question

I was talking with a friend of mine, he is a student of theoretical particle physics, and he told me that lots of his topics have their foundations in statistical mechanics. However I thought that the modern methods of statistical mechanics, for example the renormalization group or the Parisi-Sourlas dimensional reduction, come from the methods of quantum field theory or many-body techniques (Feynman diagrams and so on). I notice that books also regarding modern concepts, such as spin glasses, don't require any other knowledge than basic calculus.

Can someone explain which is the relation between these subjects?

What topics should I study of field theory or similar to have a deep understanding in statistical mechanics?

Answer 1

Quantum statistical mechanics is usually worked out within the framework of second quantization, in which a system with a variable number of particles is described as a field theory. Much of statistical mechanics deals with the nonrelativistic case, which is far simpler than relativistic QFT as all renormalizations are finite. Therefore one can see QFT working there without having to understand the cancellation of infinities.

The intuition gained from statistical mechanics is then very useful for treating problems in relativistic QFT. This is also the historical way things were worked out.

Answer 2

Statistical field theory is equivalent to quantum field theory if you perform a Wick rotation in time. Inverse temperature $1/T$ is identified as time.

Of course, the metrics are different. In QFT, it is Minkowski while in SFT, it is Euclidean.

Answer 3

I think it works better the other way around (understand Statistical Mechanics to get a feel for QFTs). This is not an answer "per se", since one take too much space, but you can find good lectures online:

Perimeter Scholars - Quantum Field Theory 2 - Francois David

The first two lectures should be enough for you to get all the parallels.