# Diagrammatic representations of generating functionals $Z[J]$, $W[J]$, and $\Gamma[\varphi]$

The book Boulevard of Broken Symmetries by Adriaan Schakel gives an excellent, if not exceedingly brief, overview of the path integral approach to perturbation theory. In particular, pages 47-58 give an excellent diagrammatic description of the partition function $$Z[J]$$, the generator of connected correlation functions $$W[J]$$, and the effective action $$\Gamma[\varphi]$$ ($$\varphi$$ is instead $$\phi_c$$ in Schakel's text). While I've found his exposition incredibly insightful, it would be nice if I could check my understanding against another textbook with a bit more detail. Unfortunately, most of the standard references I've checked (Peskin/Schroeder, Schwartz, Zinn-Justin, and a small handful of condensed matter field theory books) do not seem to have the same diagrammatic representation of these generating functionals as sums over all possible diagrams. Is there another reference which goes into these details?

I'm particularly interested to see if the Legendre transform $$W[J] \rightarrow \Gamma[\varphi]$$ can be understood diagrammatically. I think Schakel's book almost gets there: by writing each fully connected diagram in $$W$$ as a 1PI diagram plus propagators, and using the fact that the propagator is related to $$\delta \varphi / \delta J$$, I think it should be possible to trade external $$J$$'s for external $$\varphi$$'s. But I don't quite understand the full picture yet, and I especially don't yet understand how $$\int J \varphi$$ in the Legendre transform enters into the diagrammatics.

Apologies in advance if this question has been asked before, I would greatly appreciate any references!

• You mean this book by Schakel? Feb 24, 2021 at 5:32
• I believe the book version is significantly expanded compared to the notes you linked. In fact, I think there is some amount of material in these notes which is not found in the book either, making them partly disjoint...
– Zack
Feb 24, 2021 at 6:06
• Feb 24, 2021 at 6:30
• The lecture notes from Hugh Osborn's Cambridge Part III "Advanced Quantum Field Theory" course discuss this around page 31 damtp.cam.ac.uk/user/ho/AQFTNotes.pdf Feb 26, 2021 at 15:59
• This doesn't seem to be quite what I'm looking for -- as far as I can tell, these notes you linked contain about the same content as (ex) Peskin and Schroeder. What I would really like, ideally, is a fully diagrammatic representation of the full generating functionals rather than just the $n$-point functions (see, for instance, equations 2.113 and 2.145 of Schakel's text mentioned above). Perhaps this is a bit pedantic, since the full functionals are just sums of the $n$-point functions. But it would be great if I could find 2.113 and 2.145 of Schakel's text written elsewhere.
– Zack
Feb 26, 2021 at 16:37

## 1 Answer

These are two sources that come to mind whenever I think about these sort of arguments involving effective actions, and in particular their diagrammatic representations as this was a topic I struggled with for some time myself and I've not come across many sources that really go into this.

The first is "Modern Quantum Field Theory: A Concise Introduction" by Thomas Banks. In fact, there is a section specifically discussing how the Legendre transformation works and some discussion (including a diagram on page 30 in the version I have access to) of how to represent this idea diagrammatically, which is precisely one of the things you've asked for. There is, however, a downside to Banks' book: it is, indeed, a concise introduction. By design, very little is actually worked out in the text (whether this is enough for you is down to preference) and instead many calculations are left as exercises to the reader. As a result, in many places the book is very good at conveying modern thinking about topics in QFT, but may feel somewhat wanting in terms of details.

The second book I'd like to mention is V. Parameswaran Nair's "Quantum Field Theory: A Modern Perspective," which I personally have found to be exceptional on a wide variety of topics. It doesn't quite have the nice pictures that Banks does for the effective action, but includes far more detail by instead representing these ideas in formulas and symbols (so the two actually go nicely hand-in-hand). In particular, Nair's book is the one that I found particularly illuminating with regards to the precise relationship between the effective action and the Green's functions. The downside to Nair's book is that, because it contains much more detail, it takes longer to parse. Furthermore, the book leans a littler heavier on mathematics than some might be like.

Certainly there will be other sources out there, but these are two which I found particularly good with regards to this topic. In my opinion, both bring something slightly different to the table, but I would recommend starting with Banks to see if that fills in the blanks for some of the ideas, then if you want more, Nair can likely fill in the blanks in the details.