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 ...

  What is the precise relation between the OPE and factorization?

+ 2 like - 0 dislike
873 views

I want to understand the operator product expansion (OPE) in the context of a relativistic non conformally invariant theory. I wanna pose the question in a general fashion, but what I always have in the back of my mind is QCD.

First of all I wanna point out that I do am aware that in general there is no theorem guaranteeing existence or convergence or anything of the OPE. I am not looking for that. I am looking for a precise (working) definition of the OPE and specially its relation to factorization. Any reference on the topic would be greatly appreciated since I am finding that everywhere I look the topic is treated in a very lousy manner.

Ok, let's begin with the statements. To my understanding the OPE is nothing but a way to define the product of two local fields. In mathematical jargon we would say that it defines an algebra. That is, if $A(x)$ and $B(x)$ are two any local operators built out of the field degrees of freedom of your theory, and their spacetime derivatives, and with local we only mean that $A$ and $B$ only depend on a single spacetime point, the conjecture is that
$$
A(x)B(y)=\sum_nC_n(x-y)P_n(y)
$$
This is the way the OPE is presented in textbooks. Some caveats. It is usually stated that $x\to y$, where not too much care in specifying just what is exactly meant with $x\to y$. The way I like to state this is to assume that there is a neighborhood $U$ of $y$ such that x is included in it and $x\neq y$. The coefficients $C_n$ are believed to be distributions and the $P_n(y)$ would be some local operators that happen to be well defined. This relation is assumed to hold within brackets. So far so good. This is what the OPE is to me right now.

Now, the OPE is usually linked to factorization of scales. Being very vague we introduce a scale $\mu$ that separates or factorizes two regimes, the IR and the UV. Now it is stated usually that (in the context of QCD) the UV contribution goes into the coefficient functions but that the IR one can be absorbed in the condensates (the sandwitched $P_n(y)$). I want to clarify this. I want to understand how the picture presented in the above paragraph leads to this vaguely described factorization. I do not see the link at all, so what is it?

asked Jun 21, 2017 in Theoretical Physics by Dmitry hand me the Kalashnikov (735 points) [ no revision ]

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$\hbar$ysicsOverflo$\varnothing$
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
...