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
5,352 answers , 22,785 comments
1,470 users with positive rep
820 active unimported users
More ...

  What is the exact relationship between scale invariance and renormalizability of a theory?

+ 5 like - 0 dislike
1506 views

I have often read that renormalizability and scale invariance are somehow related. For example in this tutorial on page 12 in the first sentence of point (7), self similarity (= scale invariance ?) is referred to as the non-perturbative equivalent of renormalizability.

I don't understand what this exactly means. Can one say that all renormalizable theories are scale invariant but the converse, that every scale invariant theory is renormalizable too, is not true? I'm quite confused and I'd be happy if somebody could (in some detail) explain to me what the exact relationship between scale invariance and renormalizability is.

asked Nov 12, 2012 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]
edited May 19, 2014 by Dilaton

2 Answers

+ 6 like - 0 dislike

your question,

Can one say that all renormalizable theories are scale invariant but the converse, that every scale invariant theory is renormalizable too is not true?

has a sharp answer: no, one cannot say so. Renormalizable theories typically have running coupling constants with non-vanishing beta functions. The second part (what you called the 'converse') is false too. The first example that come to my mind is a theory with a spontaneously broken CFT that delivers a dilaton: the low-energy lagrangian for the dilaton is scale invariant and still is non-renormalizable having an infinite series of terms organized by the number of derivatives envolved.

The only relations I can see between scale invariance and renormalization are well known: a) renormalization typically spoils classical scale-invariance; b) a theory with strictly renormalizable terms (i.e. dimension 4 only) is classically scale invariant and it has a chance to be scale invariant at the quantum level as well; c) a non-scale invariant theory may run and approach a scale invariant theory at the end of the RG flow, either IR or UV, depending where you are heading to. This last point may be violated in very special non-unitary QFT, though.

This post imported from StackExchange Physics at 2014-03-12 15:42 (UCT), posted by SE-user argopulos
answered Nov 12, 2012 by argopulos (100 points) [ no revision ]
+ 3 like - 0 dislike

There is a relation between scaling and renormalization, but not between scaling invariance and renormalization.

Scaling is possible for every polynomial Lagrangian density. In scaling, one transforms space-time, fields and coupling constants by suitable powers of a dilation factor in a way that preserves the action. Scaling is the reason for the existence of the renormalization (semi)group. It implies that the parameter defining the mass scale in a renormalization prescription is redundant.

On the other hand, scaling invariance means that the coupling constants are independent of the dilation factor, which is a rare situation. For example, QED is renormalizable and has a scaling operation affecting the electron mass, hence is not scale invariant.

Self-similarity is a property of fixed points (critical points) of a renormalization group mapping. However, in QFT this self-similarity is not a property of a physical theory but of the regularizations at different energy scales. (This energy has nothing to do with the mass scale in the renormalization prescription, though some formulas at 1 loop look very similar.)

This post imported from StackExchange Physics at 2014-03-12 15:42 (UCT), posted by SE-user Arnold Neumaier
answered Nov 12, 2012 by Arnold Neumaier (15,787 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$\hbar$y$\varnothing$icsOverflow
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
...