• 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.


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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,075 questions , 2,226 unanswered
5,347 answers , 22,749 comments
1,470 users with positive rep
818 active unimported users
More ...

  Definition of the scaling dimension matrix?

+ 4 like - 0 dislike

In this paper, it is explained that the commutator between the scaling differential operator

$$ D = i(x^{\nu}\partial_{\nu}) $$

and a local operator $O(x)$ is (2.6)

$$[D,O(x)] = -i(\triangle + x^{\nu}\partial_{\nu})O(x)$$

where $\triangle$ is the scaling dimension matrix. How is this scaling dimension matrix defined and how can it be obtained? I know what a scaling dimension is of course, but never heard about a scaling dimension matrix before ...

asked Dec 13, 2014 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]
edited Dec 13, 2014 by Dilaton
Could you share which source you found this mention in? By the way, the brackets in your commutator don't seem to match.

1 Answer

+ 3 like - 0 dislike

The scaling dimension matrix is the (position-independent) matrix whose eigenvalues are the scaling dimensions. In the unitary case, the dilation operator is diagonalizable. If this is the case, one may choose a basis of scale covariant fields that change by an infinitesimal constant factor (the scaling dimension of the field) when an infinitesimal scaling is applied. In this basis, the scaling matrix is diagonal. If one starts directly from the assumption of scale covariant fields one gets the scaling dimensions directly, without an intervening matrix.

In general the dilation operator may or may not be the case; the author states on p.15 explicitly that s/he wants to include this case. If the normal form of the matrix has a nontrivial Jordan block, there is no basis of scale covariant fields, and one has a so-called logarithmic CFT. This is explained on p.22. (See also pp.61-62.)

Added: Since the commutator relation stated by the OP holds for every local field $O(x)$, the scale matrix must figure on the right hand side. If it is restricted to scale covariant fields, one can replace the matrix by the scaling dimension of $O(x)$.

answered Mar 3, 2015 by Arnold Neumaier (15,787 points) [ revision history ]
edited Mar 4, 2015 by Arnold Neumaier

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:
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