• 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
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  What do we take the functional determinant of in computing the effective action in the Background field method?

+ 3 like - 0 dislike

I have some schematic notes on computing the effective action and I would like someone to help me fill the gaps.

We start with
employing the background field method we write
so we have
Taylor expanding around $\phi_0$ 

since $\phi_0$ satisfies the equations of motion the linear term in $\Delta\phi$ vanishes. Then we have


from here on my notes neglect terms cubic,quartic... in $\Delta\phi$. Can anybody tell me why?.

Also, after this it is written
where the dots represent (I think) a functional determinant of something. Can anybody tell me what goes inside the determinant, and where this comes from?

asked Jun 27, 2015 in Theoretical Physics by Dmitry hand me the Kalashnikov (735 points) [ revision history ]

1 Answer

+ 3 like - 0 dislike

Once you throw out terms higher than quadratic (this is just your approximating $\Delta \phi$ to be small), you get an integral of the sort

$\int d^nx \exp(-\frac{1}{2} x^T A x)$.

This is a Gaussian integral equal to $(2\pi)^{n/2} / \sqrt{det A}$.

answered Jun 27, 2015 by Ryan Thorngren (1,925 points) [ revision history ]

I would thank you a lot if you could please be more explicit on what goes exactly inside the determinant using the notation I have used.

@silvrfuck: To tell ''what goes exactly inside the determinant'' you should either say what is in your notes in place of the ..., or if the ... are in the notes themselves, what the author concludes from it. 

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