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

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

180 submissions , 140 unreviewed
4,534 questions , 1,819 unanswered
5,159 answers , 21,954 comments
1,470 users with positive rep
720 active unimported users
More ...

  Locality of renormalized actions in momentum space

+ 1 like - 0 dislike
60 views

The breaking of homogeneous scaling by distributions appearing in a QFT may be used to calculate elements of the Gell-Mann Low cocycle. To $n$th order in perturbation theory, $$Z_n=\sigma_\rho\circ T_n\circ \sigma_\rho^{-1} - T_n$$.
From which the renormalized action can be obtained. M. Dütsch elaborates on this in x-space using differential renormalization: The resulting scaling difference is local i.e. depends only on (convolutions with) delta functions and their derivatives.

Since differential renormalization is not, at least in introductory texts, a popular method, I was wondering if the results can be straightforwardly written in momentum space with e.g. Epstein-Glaser renormalization, but it is unclear how the scaling difference maintains locality despite supposedly dealing with the same expressions. In p-space, the scaling law takes the form (see G. Scharf) $$\sigma_\rho\circ t\circ \sigma_\rho^{-1}=\rho^\omega \hat t\left(\frac{k}{\rho}\right)$$

but when applied to common loop distributions such as the scalar vacuum polarization distribution (eq 230 of https://arxiv.org/pdf/0906.1952.pdf) I do not see any cancellation of terms nonpolynomial in $k$, which to my knowledge corresponds to nonlocal terms in x-space.

Can the elements of the cocycle and the renormalized action be calculated in p-space (and if so, how), or is this so far only understood for x-space calculations?

asked Oct 9 in Theoretical Physics by Qghost (30 points) [ no revision ]

The methods used by Dütsch are tailored to x-space; in place of  differential renormalization one can use analytic regularization; see Section 3.5 of the book.

  • M. Dütsch, From Classical Field Theory to Perturbative Quantum Field Theory, Birkhäuser 2019.

I don't think that these techniques are more difficult to apply than momentum space methods. Also, the former (unlike the latter) are applicable in curved backgrounds.

@ArnoldNeumaier I forgot to mention that that is the book where I learned about this. From what I can tell it looks like something which gets easier with practice, but I'm currently familiar with regularization in p-space. Do you know if there is a paper on applying differential renormalization to experiment e.g. calculating cross sections? Or is the procedure of renormalization most easily done in different forms for different goals (for flat space)? I think if I see it in different contexts I would be more comfortable using it.

The two methods are compared in

D. Prange, Epstein-Glaser renormalization and differential renormalization. Journal of Physics A: Mathematical and General, 32 (1999), 2225.

which might help you.

And there was a typo in my previous comment: in place of  differential renormalization one can use analytic regularization!

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).
To avoid this verification in future, please log in or register.




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

Your rights
...