I'm looking for nice overviews on $\phi^{4}$-field theory from the mathematical-physics point of view. To be a little more specific, here are some topics I'd like to read about:

(1) What are the motivations, both from the physics and mathematical point of view, to study $\phi^{4}$-theories?

(2) What has been (mathematically) accomplished so far? What are the most important open questions nowadays?

(3) What are the tools used to (rigorously) study these class of models? Renormalization group, cluster expansions, etc.

My motivation for this question is a rather simple one: I'm teaching myself some field theory but it is really hard to find these discussions on books. In general, books are more interested in solving problems and sometimes I find myself studying some models that I don't know anything about. Also, I'm primarily interested in statistical mechanics, so this helps to narrow the question a bit more. Connections with statistical mechanics and QFT are welcome too, but I don't want a purely QFT reference (I'm sufficiently lost in my own area of interest, after all). Thanks in advance!


2 Answers 2


This reference is a bit older, but it should be a good starting point for items 2 and 3: $\phi^4$ field theory in dimension 4: a modern introduction to its unsolved problems.

Concerning item 1, you might find it instructive to motivate the $\phi^4$ field theory from the perspective of its limitation: when does it apply and when are higher order terms needed? For that perspective I would recommend Higher-order field theories: $\phi^6$, $\phi^8$ and beyond.

This last reference is a chapter from a recent book, A Dynamical Perspective on the $\phi^4$ Model (2019) which has an interesting introductory chapter on the history of the $\phi^4$ model, as well as overviews of more specialized topics.

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    $\begingroup$ Note that in the case of purely ferromagnetic interactions, the triviality conjecture discussed in the first reference has recently been proven by Aizenman and Duminil-Copin. $\endgroup$ Jun 28, 2020 at 22:35
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    $\begingroup$ @MartinHairer --- thanks, it's at arxiv.org/abs/1912.07973 $\endgroup$ Jun 29, 2020 at 6:12

In terms of physical motivation of $\phi^4$, if I remember rightly the references in the introduction to this paper might have some pointers.

  • Dashen R and Neuberger H 1983 How to get an Upper Bound on the Higgs Mass, Physical Review Letters 50, 24

Reference [7] of that article points to rigorous proofs of results which almost show the triviality of $\phi^4$ in four dimensions.


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