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

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,853 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

Pedagogical introduction to Stückelberg renormalization (group)?

+ 3 like - 0 dislike
314 views

Das anybody know a nice pedagogical introduction (something that is shorter than a heavy textbook, maybe a lecture note) to the Stückelberg renormalization (group)? Conversely to the renormalization methods often applied in (high-energy) theoretical physics, in this approach the renormalization group transformation is reversible which makes the renormalization group a true group, and it is (for me personally a bit surprisingly at the first glance) often applied in contexts that have nothing to do with relating effective theories valid at different energy scales to each other or integrating out high-energy effects.

While I am generally interested in applications of any kind, I would enjoy most reading about how the Stückelberg renormalization is applied in a QFT context.

asked Oct 23, 2016 in Resources and References by Dilaton (4,295 points) [ revision history ]

Brunetti, Romeo, Michael Dütsch, and Klaus Fredenhagen. "Perturbative algebraic quantum field theory and the renormalization groups."Advances in Theoretical and Mathematical Physics 13.5 (2009): 1541-1599. 

We discuss the connection between the Stückelberg–Petermann renormalization group which describes the freedom in the perturbative construction with the Wilsonian idea of theories at different scales. In particular, we relate the approach to renormalization in terms of Polchinski’s Flow Equation to the Epstein–Glaser method. 

On the simplest, classical level (applied to ordinary differential equations), the similarities and differences between the Stückelberg–Petermann renormalization group and the Wilsonian renormalization group is well described in 

G.C. Paquette, Renormalization group analysis of differential equations subject to slowly modulated perturbations, Physica A 276 (2000), 122-163.

See especially p. 9-10 and Section 7.2.

1 Answer

+ 3 like - 0 dislike

The Stueckelberg renormalization group is what removes the freedom left in the prescription of the renormalization conditions after the limit where the regularization (whether by a cutoff or another regularization recipe) is already taken. Because the physics must be unique, the theory cannot depend on this freedom, as with the gauge freedom in a gauge theory. The Stueckelberg renormalization group is the analogue of the gauge group. 

I don't know of a good introduction to Stueckelberg renormalization. But (for example) the following articles show the use of the renormalization group in contexts very different from quantum field theory. You need to judge for yourself how pedagogical the articles are written.

A renormalization group treatment of the classical van der Pol oscillator is given in https://arxiv.org/abs/1305.2372

classical singular perturbation theory: https://arxiv.org/abs/hep-th/9506161https://arxiv.org/abs/cond-mat/9407024

quantum anharmonic oscillator: https://arxiv.org/abs/hep-th/9710087

similarity renormalization group in nuclear physics: https://arxiv.org/pdf/nucl-th/0611045

nonlinear optics; https://arxiv.org/abs/hep-th/0001210 ; this might come closest to your request; there it is named after Bogoliubov rather than Stueckelberg. 

For the history, see https://arxiv.org/abs/hep-th/9602024

For relativistic QFT, see, e.g., https://arxiv.org/abs/hep-th/0501228

From O'Connor & Stephens, Phys. Rep. 363 (2002), 425:

a resummation technique is required. There are several RG formulations that may be used to achieve this. We believe the most powerful is still the original RG of Stuckelberg–Peterman, Gell–Mann Low and Bogoliubov–Shirkov [68–70], which we refer to as the reparametrization RG. Although discovered in perturbative studies of quantum electrodynamics in the process of removing the ultraviolet (UV) divergences from measured quantities, it was pointed out very early in the development of the subject, by Blank Bonch–Bruervich and Shirkov [71], that the RG is not dependent on the existence of such UV divergences and that it could be a useful tool in a variety of fields (they mention condensed matter physics). [...] From a modern perspective, the original field theoretic renormalization can now be seen to be nothing more than a coordinate change from original bare parameters to renormalized parameters. A coordinate transformation, of itself, does not change the physics, but, as we shall see and demonstrate, one coordinate system may be vastly superior to another when doing perturbative calculations, especially when combined with the notion of a “sliding scale” for the renormalization point.

In contrast, Wilson-style renormalization is not just a reparameterization but simplifies the model by changing the short-distance, high-energy physics, leaving only the long-distance, low-energy physics intact. The relationship between the two is discussed in Section 5.2 of

Brunetti, Romeo, Michael Dütsch, and Klaus Fredenhagen. "Perturbative algebraic quantum field theory and the renormalization groups."Advances in Theoretical and Mathematical Physics 13.5 (2009): 1541-1599. 

We discuss the connection between the Stückelberg–Petermann renormalization group which describes the freedom in the perturbative construction with the Wilsonian idea of theories at different scales. In particular, we relate the approach to renormalization in terms of Polchinski’s Flow Equation to the Epstein–Glaser method. 

answered Oct 23, 2016 by Arnold Neumaier (12,375 points) [ revision history ]
edited Jan 13 by Arnold Neumaier

A lengthy comment discussion involving issues of integrating out degrees of freedom (Stückelberg Renormalization has nothing to do with integrating out degrees of freedom) has been moved to chat.Please continue the discussion about such issues there if you like, any further off-topic comments here will be moved there too.

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).
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
...