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 , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  Explicit evaluation of a radially ordered product

+ 1 like - 0 dislike
1017 views

I am trying to understand the application of the operator product expansion to calculate the radially ordered product in the complex plain of $T_{zz}(z)\partial_w X^{\rho}(w)$ which should result in

$$ \langle R(T_{zz}(z)\partial_w X^{\rho}(w))\rangle = -l_s^2\frac{1}{(z-w)^2}\partial_w X^{\rho}(w) - l_s^2\frac{1}{(z-w)}\partial_z^2 X^{\rho}(z) + \cdots $$

but embarassingly I encounter a stumbling block right at the beginning. After inserting $T_{zz}(z) \doteq \, :\eta_{\mu\nu}\partial_z X^{\mu}\partial_zX^{\nu}:$ one has

$$ \langle R(T_{zz}(z)\partial_w X^{\rho}(w))\rangle = R(:\eta_{\mu\nu}\partial_z X^{\mu}(z)\partial_zX^{\nu}(z):\partial_w X^{\rho}(w)) $$

which can obviously be further expanded to

$$ ... = \eta_{\mu\nu}\langle \partial_z X^{\mu}(z)\partial_w X^{\rho}(w)\rangle \partial_z X^{\nu}(z) + \eta_{\mu\nu}\langle \partial_z X^{\nu}(z)\partial_w X^{\rho}(w)\rangle \partial_z X^{\mu}(z) $$

It is exactly this last step I dont understand. If this initial stumbling block is removed, I can understand the rest of the derivation, so can somebody help me remove it?

To generalize a bit, it seems I do not yet properly understand how such expressions involving normal and radial (time) ordered products are generally evaluated. So if somebody could give me a more general hint about this, I would probably be able to see how the last expression in my particular example is obtained.

asked Mar 24, 2013 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]
edited May 1, 2014 by Dilaton

Related: physics.stackexchange.com/q/22784/2451 and links therein.

This post imported from StackExchange Physics at 2014-03-12 15:19 (UCT), posted by SE-user Qmechanic

Darn, I forgot the check to edit silently, sorry :-/

@Dilaton Oh yes, editing silently when doing a mass edit-out of attributions of one's own posts, to not flood the main page seems like a good idea.   

I think I should also practise that in my mass-retag of all posts.    

But I think it is fine to not check the box for unanswered questions.  

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$\hbar$ysicsOv$\varnothing$rflow
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
...