# Relation between statistical mechanics and quantum field theory

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 theorem, 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 then 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?

• This is nicely discussed in "An introduction to lattice gauge theory and spin systems - B. Kogut". You can find it for free on some site if you google search. Aug 29, 2012 at 10:02
• Related: physics.stackexchange.com/q/27416/2451 and links therein. Aug 29, 2012 at 11:37
• The article @dushya points out is in Reviews of Modern Physics, by the way. I second the recommendation. The state-of-the-art has evolved only slightly in the 30+ years since it was written, and if you work through that you'll be on very firm ground.
– wsc
Aug 29, 2012 at 13:17
• [arXiv:hep-th/9403084v2] (arXiv.org/abs/hep-th/9403084) Sep 4, 2012 at 18:46

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.

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 mechnaics deals with the nonrelativistic case, which is far simpler than realtivistic QFT as all rnormalizations are finite. Therefore one can see QFT working there without having to understand the cancellation of infinities.

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

• So in order to understand statistical mechanics I don't need any deep understanding in QFT (a basic course of many body should be fine). What about conformal field theory? Is it necessary? en.wikipedia.org/wiki/Conformal_field_theory Sep 2, 2012 at 8:32
• To understand statistical mechanics at the level of the book by Reichl, say you don't need any QFT. Some conformal field theory is needed only if you want to rigorously study critical scaling in 2D theories. Of course, for a deep understanding, one needs both. Sep 2, 2012 at 16:24

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.