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In (rigorous) statistical mechanics and statistical field theory one is usually concerned in giving meaning to integrals of the form: \begin{eqnarray} \langle \mathcal{O}\rangle = \frac{1}{Z}\int D\phi e^{-S(\phi)} \mathcal{O}(\phi) \tag{1}\label{1} \end{eqnarray} where $D\phi$ is some measure on the space of fields $\phi$. One way to approach the problem is to study the discretized version of the theory and look for the existence of some limits (thermodynamic and continuous). Well, I'm not interested in the technical aspects of the theory here. The point is: one of the most important actions in the literature is given by: \begin{eqnarray} S(\phi) = \int_{\mathbb{R}^{d}}d^{d}x\bigg{(}\frac{1}{2}\nabla\phi(x)^{2}+\frac{1}{2}m^{2}\phi(x)^{2}+\lambda\phi(x)^{4}\bigg{)}. \tag{2}\label{2} \end{eqnarray} This is called the $\phi^{4}$-model. If the field $\phi$ and its derivatives have enough decay, the integrand in (\ref{2}) can be rewritten in terms of the massive Laplacian $-\Delta+m^{2}$.

I'm starting to write some personal notes about my studies on this topic and I plan to write an introductory section where I give the motivations to study the $\phi^{4}$ model. But I'm having a hard time trying to find a nice way to do it. Let me elaborate a little more. I know that many interesting models can be realized as $\phi^{4}$-models. For instance, I know that the Ising model is some sort of limit of the above scenario and also that $\phi^{4}$ models are fundamental to study random walks and white noise. But I'd prefer not to attain myself to explicit models but rather to give a more general motivation. I've been thinking about it and I think the most natural motivation to this model is to consider the Landau free energy $F(\phi)$, which is given by: \begin{eqnarray} F(\phi) = \int_{\mathbb{R}^{d}}\bigg{(}\frac{1}{2}\alpha(T)\nabla \phi(x)^{2}+\frac{1}{2}\beta(T)\phi(x)^{2}+\frac{1}{4}\gamma(T)\phi(x)^{4}+\cdots\bigg{)} \end{eqnarray} where $T$ stands for the absolute temperature of the system. The resemblance of this expression with expression (\ref{2}) is obvious. Also, the partition function of the system in the Landau approach is supposed to be: \begin{eqnarray} Z = \int D\phi e^{-\beta F(\phi)}, \end{eqnarray} which is a genuine path integral, in the same spirit as (\ref{1}).

The problem here is that I do not find anything rigorous about Landau's theory. I mean, I know that the general picture of Landau's theory is supposed to be purely phenomenological, but I'd expect to find, say, a rigorous way to turn spins into fields or something like this, but I didn't find anything so far. What I did find is how to discretize the theory once you have your $\phi^{4}$ action, but not the other way around. The process of turning spins into fields in the physics literature is done by using a process of coarse-grain, which is usually explained in a purely qualitative way.

So I'd like to know if there is something rigorous about how spins become fields or even about Landau's theory itself. Also, is this really the best approach to motivate the $\phi^{4}$ theory or is that better ways to do that (rigorously and in a general setup)?

NOTE: The more general version of Landau's theory (as I wrote above) is more commonly called Ginzburg-Landau theory. I think these ideas were published by L. Landau and V. Ginzburg to explain, among other things, superconditivity.

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  • $\begingroup$ The functional $F(\phi)$ is an expansion in powers of $\phi$ and $\nabla \phi$, assuming the field $\phi$ is small and varies slowly; there is nothing fundamental about which terms to keep and which to discard, and in fact higher order functionals are also of interest. $\endgroup$ – Carlo Beenakker Jun 20 '20 at 7:01
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If you don't want to discuss any kind of specific model as motivation, you could always argue that the $\phi^4$ theory is the only renormalizable theory that shares the $Z_2$ ($\phi\mapsto-\phi$) symmetry of the free theory.

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