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
5,353 answers , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  What is an isoscalar factor?

+ 1 like - 0 dislike
1879 views

I need to find a definition for "the isoscalar factors of 3j-symbols for the restriction $SO(n)\supset SO(n-1)$...denoted by brackets with a composite subscript $(n: n-1)$..." They are given as:

$$ \left(\begin{array}{ccc}l_1&l_2&l_3\\l_1'&l_2'&l_3'\end{array}\right)_{\left(n: n-1\right)} $$

What does this mean? And can someone point me to an explicit definition? I'm familiar with the traditional Wigner 3j-symbols, my only inkling is that the Wigner 3j's are for SO(3) whereas

$$ \left(\begin{array}{ccc}l_1&l_2&l_3\\l_1'&l_2'&l_3'\end{array}\right)_{\left(n\right)} $$

are for SO(n). But I have no idea what the composite subscript means, nor do I have a handle on what these coupling coefficients mean physically in SO(n).

This post imported from StackExchange Physics at 2014-04-01 05:46 (UCT), posted by SE-user okj
asked Jan 21, 2012 in Theoretical Physics by okj (60 points) [ no revision ]
This whole question looks strange because while $SO(3)$ irreducible representations are uniquely specified by a value of $L$, it is not the case for $SO(n)$ for $n>3$. For more complicated groups, one must use Young diagrams, not just a simple value of $L$, and the Racah coefficients - another name for the isoscalar factor - also depends on the whole diagram. Where did you see the comments and formulae you reproduced?

This post imported from StackExchange Physics at 2014-04-01 05:46 (UCT), posted by SE-user Luboš Motl
Its in eqs. 4.6-4.7 from "Coupling coefficients of SO(n)and integrals involving Jacobi and Gegenbauer polynomials" by Sigitas Alisauskas, you can find it at iopscience.iop.org/0305-4470/35/34/307

This post imported from StackExchange Physics at 2014-04-01 05:46 (UCT), posted by SE-user okj
I see, so they're for the maximally symmetrized representations only.

This post imported from StackExchange Physics at 2014-04-01 05:46 (UCT), posted by SE-user Luboš Motl
Link also available in the preprint arXiv at arxiv.org/abs/math-ph/0201048

This post imported from StackExchange Physics at 2014-04-01 05:46 (UCT), posted by SE-user Qmechanic

1 Answer

+ 3 like - 0 dislike

"Isoscalar factor" is another name for reduced Clebsch-Gordan coefficients. The nomenclature was often used in the late 60s in connection with SU(3) models, reduced either using SO(3) (as in the Elliott model of nuclear physics) or using SU(2) (as in the familiar gauge theories). In these cases, the isoscalar factors would be multiplied by an SO(3) or SU(2) ClebschGordan coefficient to obtain the full SU(3) Clebsch Gordan coefficient. (Nota: although su(2) and so(3) are isomorphic at the algebra level, at the group level they are distinct: the SO(3) group would be similar to the one found in the angular part of the 3D harmonic oscillator wavefunctions and would be restricted to true representations, i.e. integer values of L, whereas the SU(2) groups can have projective (or spinorial) representations where J is half integer. For SO(3) the embedding inside SU(3) is irreducible but not for SU(2).)

When labelling states using a subgroup chain like SO(n)->SO(n-1), it is also possible to express the Clebsch-Gordan coefficient as a product of an SO(n-1) Clebsch multiplied by an "isoscalar factor", which is the part of the SO(n) Clebsch not in SO(n-1).

In the specific case of SO(3), the subgroup usually used is SO(2): the subgroup of rotations about the z-axis. This subgroup is Abelian so the representations are 1-dimensional, labelled by M, and just exponentials $e^{iM \varphi}$ for instance. For SO(2) the Clebsch are thus 1 if the projections satisfy the correct addition rule $M_1+M2=M_3$. Thus, in this case, the isoscalar factor for SO(3) is the Clebsch itself since the SO(2) Clebsch is just 1 (or more correctly a delta function enforcing the condition $M_1+M2=M_3$.)

In the case of SO(n) the situation is complicated because some representations will contain more than one copy of some representations of SO(n-1). The general definition of the isoscalar factor is as above although, if there is more than one copy of an irrep of SO(n-1), an additional label must be used (this also occurs in some irreps of SU(3) when using SO(3) as a subgroup of SU(3), but never occurs when using SU(2) as a subgroup. Again, the physical angular momentum is really SO(3) so the use of this subgroup is sometimes forced upon the user, especially in nuclear physics where 3D harmonic oscillator states are useful). In some specific cases where there is no such multiplicity of representation then of course this additional label is not needed so only the generalized angular momentum labels for SO(n) and SO(n-1) irreps are needed.

This post imported from StackExchange Physics at 2014-04-01 05:46 (UCT), posted by SE-user ZeroTheHero
answered Aug 31, 2013 by ZeroTheHero (70 points) [ no revision ]
+1 great answer.

This post imported from StackExchange Physics at 2014-04-01 05:46 (UCT), posted by SE-user Dimensio1n0

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$\hbar$ysi$\varnothing$sOverflow
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
...