• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  Asymptoticity of Pertubative Expansion of QFT

+ 14 like - 0 dislike

It seems to be lore that the perturbative expansion of quantum field theories is generally asymptotic. I have seen two arguments.

i)There is the Dyson instability argument as in QED, that is showing the partition function is nonanalytic around the expansion point, by analyzing the ground state or instantons or somesuch. This is a wonderful argument but it requires some non-trivial knowledge about the behavior of your QFT which may not be available.

ii) There is some attempt at a generic argument which merely counts the number of Feynmann diagrams at each order, says this grows like $n!$ where is $n$ is the order of the expansion. and so our series looks like $\sum n!\lambda^n$, which is asymptotic. This is of course wholly unsatisfactory since it ignores interference among the terms (even granting the presumption that all the diagrams are of the same order, which feels right). It is true the series is still asymptotic if we take the diagrams to have random phase, but this ignores the possibility of a more sinister conspiracy among the diagrams. And we know that diagrams love to conspire against us.

So is there any more wholesome treatment of the properties of the perturbative expansion of QFT? I came to thinking about this while considering the properties of various $1/N$ expansions so anything known in particular about these would be nice.

This post has been migrated from (A51.SE)
asked Dec 12, 2011 in Theoretical Physics by BebopButUnsteady (330 points) [ no revision ]
It strongly depends on what you mean. For example, if your function is $(1+x)^{-2}$, its Taylor series has a finite convergence radius $x<1$, so no term-by-term summation works at large $x>>1$. But if you manage to guess this function right in the initial approximation (or sum up the Taylor terms into a finite formula $f=(1+x)^{-2}$), then the series asypmtoticity becomes irrelevant; nobody cares, the searched value can be calculated. Similarly for $exp(-C/g)$. If your initial approximation contains it right, there is no need to expand it in asymptotic series, and it may be still the same QFT.

This post has been migrated from (A51.SE)

1 Answer

+ 6 like - 0 dislike

You pretty much never expect a perturbation expansion of a generic theory to be convergent. There's a nice connection between the divergence of the perturbation expansion and nonperturbative effects (like instantons) leading to nonanalyticity at zero coupling (i.e., $e^{-C/g}$ effects). Mariño's notes here seem like a nice discussion with good references.

This post has been migrated from (A51.SE)
answered Dec 12, 2011 by Aaron (420 points) [ no revision ]
Thank you, that is an excellent set of notes.

This post has been migrated from (A51.SE)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights