Quantcast
  • 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.

News

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

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,791 comments
1,470 users with positive rep
820 active unimported users
More ...

  Doubts taking the second functional derivative of the Klein Gordon action

+ 3 like - 0 dislike
1304 views

I have very little background with functional derivatives and I would like to clarify some issues.

I am trying to compute the second functional derivative of the Klein Gordon action expressed in real components

$S[\phi_1,\phi_2,\partial_{\mu}\phi_1,\partial_{\nu}\phi_2]=\int{}d^4x\,\mathcal{L}=\int{}d^4x(-\partial_{\mu}\phi_1\partial^{\mu}\phi_1-\partial_{\mu}\phi_2\partial^{\mu}\phi_2+$

$-m^2(\phi_1^2+\phi_2^2)-\lambda(\phi_1^2+\phi_2^2)^2)$

I know how to compute the first functional derivative 

$$\frac{\delta{}S}{\delta\phi_1(x)}=\frac{\partial\mathcal{L}}{\partial\phi_1}-\partial_{\mu}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi_1)}$$

which gives

$$2\left(\partial^2\phi_1(x)-m^2\phi_1(x)-4\lambda\phi_1(x)[\phi_1^2(x)+\phi_2^2(x)]\right)$$

from now on I have doubts. I wanna compute

$$\frac{\delta^2S}{\delta\phi_1(y)\delta\phi_1(x)}$$

 If there were no derivatives this would be trivial but I don't know how to deal with $\partial^2$. I have thought in introducing a delta to write this as an integral and use the formula I have used to compute the first functional derivative, but I am concerned about taking derivatives of something having a Dirac delta.

So, my question is, how am I supposed to compute this functional derivative?

asked Jun 30, 2015 in Theoretical Physics by Dmitry hand me the Kalashnikov (735 points) [ revision history ]

1 Answer

+ 2 like - 0 dislike

If $S(\phi)$ is given, one gets the first and second functional derivatives by expanding $S(\phi+sf)$ as a power series in the real parameter $s$ up to second order.

The first order term can always be rearranged in the form $s\int dx f(x) S_1(x)$, which gives $\frac{\delta S}{\delta \phi(x)}=S_1(x)$. (Note: In your expression for the first functional derivative the big closing parenthesis should be after the $m^2\phi_1$ term.)

The second order term can always be rearranged in the form $\frac12 s^2\int dx dy f(x) S_2(x,y) f(y)$ with symmetric $S_2(x,y)$, which gives $\frac{\delta^2 S}{\delta \phi(x)\phi(y)}=S_2(x,y)$. Note that this is a differential operator, and if there are multiple fields, a matrix of differential operators. Alternatively, you could also substitute $\phi$ by $\phi+sg(x)$ in your formula for the first functional derivative, expand to first order to get $sS_2(x,y)g(x,y)$, and proceed as before. In both cases, the second functional derivative you are looking for is found to be $2\delta(x-y)\Big(\partial^2-m^2+4\lambda(3\phi_1^2(x)+\phi_2(x)^2)\Big)$. 

answered Jun 30, 2015 by Arnold Neumaier (15,787 points) [ revision history ]

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:
p$\hbar$ys$\varnothing$csOverflow
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
...