# Statistical Mechanics as a Field Theory

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Statistical field theory is concerned with studying statistical mechanics as a field theory. I think this approach has its roots in Landau's phenomenological theory of phase transitions and critical phenomena, where the so-called Landau free energy is assumed to be analytic on its order parameter (at least for small values of it) and given by its Taylor expansion:

$$L = \int d^{d}{\bf{r}} \bigg{[} \alpha (\nabla m)^{2}+\beta m^{2}+\gamma m^{4}-H({\bf{r}})m({\bf{r}})\bigg{]}$$

The above expression is already the most general form of Landau's free energy. It is built under some really phenomenological considerations and it turns out to show resemblance with the Lagrangian of the $\varphi^{4}$ Euclidean Field Theory.

Now, I'd like to better understand the motivations which lead to consider statistical mechanics as a field theory. Most books on statistical field theory just introduce Landau's theory by studying the 2D Ising model and then they generalize it to the above free energy, which is basically the transition to field theory, almost as an inductive step which is (in my opinion) poorly justified. Basically all the justifications I encountered are based on the fact that our systems may be inhomogeneous and we'd like to generalize it. However, Mehran Kardar states in his book that a coarse-graining of some of these statistical mechanics systems may be appropriate because observations indicate that fluctuations have long wavelenghts in the vicinity of the critical point, so that the interactions in the system involve many particles and this justifies the process of coarse-graining.

So, I'd like to ask if this is all we have to justify the transition from statistical mechanics to a field theory or if there are more (maybe more deep) justifications for it. Also, for the mathematical perspective, does treating the constituents of the system as a field instead of regular spins turns things easier in some sense? It sounds to be much more difficult to treat evaluate functional integrals than to adopt other analytical procedure to study, say, the Ising model.

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