Reference request for $\phi^{4}_{d}$ theory - where to begin?

Question

When I started studying the basics of $\phi^{4}_{d}$, I looked for papers or lecture notes which would give me some general ideas about the topic and which would construct and/or prove the basic results of the theory. One of the main targets of this theory is, for example, the rigorous construction of the formal object: $$d\mu = e^{-S(\phi)}e^{\lambda\int \phi^{4}(x)dx}d\phi$$

I think I understood the problems one encounter when trying to define such measure, i.e. the large field problem and the need of the introduction of a cutoff to regularize the theory.

However, trying to learn something about $\phi^{4}_{d}$ alone, in my experience, has been a really difficult task. There is a huge number of papers and each one seem to do something different. And by something different, I don't mean just the technique used in the paper, or the realization of the renormalization group, but each paper seem to deal with particular problems. For example, some use statistical mechanics terminology, study the pressure of the system (the logarithm of the partition function normalized by the volume). Some, use Schwinger functions and effective actions. And so on.

Also, some start studying the $\phi^{4}_{d}$ alone, some include some addition terms, say $g \phi^{2}(x)$ or some derivative. It is not clear if one needs it or not, or when does one need it or not.

In summary, I got to the conclusion that I need to start from somewhere safe and start learning from the most basic to get to the most advanced.

I am interested in the construction of $\phi^{4}_{d}$ in terms of constructive/non-perturbative QFT using renormalization group techniques (scale decomposition and so on).

Question: Can you rank from the easiest to the most advanced cases of $\phi^{4}_{d}$ and point me the papers in which the constructions are given? For example, I think the case $d=1$ is probably the most trivial one, but I have never seen anyone treating this case. What next? Maybe $d=2$? Maybe $d=3$? Do we need extra terms there? What papers should I get to learn the most constructions of the easiest cases?

Answer 1

For an introduction to the basics of quantum field theory you could look into Introduction to Quantum Field Theory for Mathematicians. Lectures 13 and 18-22 introduce the $\phi^4$ model in 3+1 dimensions and the perturbative calculation of transition probabilities (from momenta $p_1,p_2$ to $p_3,p_4$). The first order term in the coupling constant is finite, but the second-order term diverges. Renormalization is then introduced to obtain a finite answer.

This perturbative approach to QFT is not mathematically rigorous, but you will obtain answers to some of the questions stated in the post (like "why add a $\phi^2$ term?" --- it gives the particles a mass).

For more rigour you then want to turn to the constructive approach to QFT. I understand your interest is in bosonic fields. For a broad overview you could take a look at A Perspective on Constructive Quantum Field Theory, to see that there exists a great variety of approaches in this category.

One rather recent development that I think requires the least amount of background is the Tree Quantum Field Theory of Gurau, Magnen, and Rivasseau. This a reformulation of the combinatorial core of constructive quantum field theory, which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. For an application specifically to the $\phi^4$ model, see Constructive $\phi^4$ field theory without tears, by Magnen and Rivasseau.

One advantage of focusing on this modern approach is that it might be an entry point for original research (which may or may not be your objective).