• 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
5,064 questions , 2,215 unanswered
5,347 answers , 22,743 comments
1,470 users with positive rep
818 active unimported users
More ...

  Counting unitariy transformations in SU(N)

+ 2 like - 0 dislike

Hello everybody,

I'm referring to the following article https://arxiv.org/abs/1810.11563 [1], in particular to section 6.

 The goal is to estimate the number of unitary transformations in $SU(N)$, identifying unitaries within balls of radius $\epsilon$. The strategy is to take the total volume of $SU(N)$ (see also https://arxiv.org/abs/math-ph/0210033 [2]), and then dividing for the volume of an $\epsilon$-ball. Here and in the following $N=2^K$, where $K$ is an integer.

The total volume of $SU(N)$ is:
\frac{2 \pi^{\frac{(N+2)(N-1)}{2}}}{1! 2! 3!\cdots(N-1)!}
Here the author misses a total factor of $\sqrt{N 2^{N-3}}$, see the original article [2], equation 5.13. Despite this, the author then states that the volume of an $\epsilon$-ball of dimension $N^2-1$ is:
I don't get this last formula. First of all, I believe that there would be a factor $\epsilon$ to some power of $N$. Secondly, it seems to me the volume of an $\epsilon$-ball in an Euclidean space off even dimension, while considering that $N=2^K$, $N^2-1$ is odd. Thirdly, in my opinion it would be better to consider the volume of an $\epsilon$-ball in $SU(N)$ (if I didn't misunderstood the formula above).

My questions are:

- Has someone, reading the article [1], reached a better comprehension than mine on the above formulas?
- If no, may you have other references about estimating the number of unitary transformations in $SU(N)$ within a precision $\epsilon$?
- And, finally, has someone a reference for the volume of an $\epsilon$-ball in $SU(N)$? Looking on the net I didn't find anything until now.

Thank you very much!

  [1]: https://arxiv.org/abs/1810.11563
  [2]: https://arxiv.org/abs/math-ph/0210033

asked Jun 5, 2019 in Theoretical Physics by SuperBaba (20 points) [ no revision ]

Exact, the factor is missing. N²-1 is used to get the volume of an euclidian n-sphere but it is not exactly the usual \({\displaystyle V_{N²-1-sphere}={\epsilon ^{{N^2}-1}} {\frac {\pi ^{\frac {{N^2}-1}{2}}}{\Gamma \left({\frac {{N^2}+1}{2}}\right)}}} \)... If you don't find more specific, just for the links: 5.13 might be derived from Kirillov orbit theory, (wikipedia) which states, in connection with representation theory, a duality between the quantum L-R coefficients and the classical Horn problem.

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