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?

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    $\begingroup$ I think the real answer is that you should learn more about quantum mechanics and quantum field theory as general subjects. (The fact that you have never seen $\phi^{4}$ theory in $d=1$ indicates that you do not have much grounding in the physics.) If you are having trouble making the connections between apparently different formalisms used in different sources, that is probably because they assume a certain level of understanding of the physics that motivates the mathematics. $\endgroup$
    – Buzz
    Aug 13, 2022 at 23:11
  • $\begingroup$ @Buzz I meant that I have never seen a rigirous discusson about $d=1$. $\endgroup$
    – MathMath
    Aug 14, 2022 at 13:52
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    $\begingroup$ What Buzz is pointing out is that you seem unaware that $d=1$ QFT is simply Quantum Mechanics, for which there is a lot of rather rigorous formalism ... $\endgroup$
    – gmvh
    Aug 16, 2022 at 18:42
  • $\begingroup$ @gmvh I see the point. I know the construction of Feynman path integrals using Wiener measures, but don't remember ever seeing the construction of $\phi^{4}$ measures in this context. Do you know any good reference on this matter? $\endgroup$
    – MathMath
    Aug 17, 2022 at 3:15
  • $\begingroup$ People would generally call it the anharmonic oscillator rather than $d=1$ $\phi^4$ theory. At least at the level of mathematical physics, arXiv:0812.3517 seems to be a relevant construction. $\endgroup$
    – gmvh
    Aug 17, 2022 at 8:22

1 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).

  • $\begingroup$ Carlo, another great answer! Thank you. The references will be of much help! I just want to understand a little more about the state of art of the $\phi^{4}$ theories. I have heard that the measures have been constructed for $\phi^{4}_{2}$ and $\phi^{4}_{3}$, but it is not clear to me if these constructions only involve finite volume + continuum limit or both continuum + thermodynamic limit. $\endgroup$
    – MathMath
    Aug 16, 2022 at 13:19
  • $\begingroup$ E.g. this post: mathoverflow.net/questions/383167/… It discusses Nelson's construction of $\phi^{4}_{2}$ and also some questions about $\phi^{4}_{3}$. Nelson's construction, according to Hairer, is for finite volume only. It should be easier than both limits, as I understand. But I suppose we have more complete results by now, right? $\endgroup$
    – MathMath
    Aug 16, 2022 at 13:21
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    $\begingroup$ $\phi_3^4$ requires both a finite volume and an ultraviolet cutoff. $\endgroup$ Aug 16, 2022 at 13:24
  • $\begingroup$ Carlo, do you have suggestions for fermionic models too? You mentioned in your post the bosonic case because I asked about thr bosonic case, but it got me thinking about the fermionic case too. $\endgroup$
    – MathMath
    Aug 16, 2022 at 19:54
  • $\begingroup$ the fermionic counterpart, in $d=2$, is described in Continuous constructive fermionic renormalization $\endgroup$ Aug 17, 2022 at 5:53

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