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

156 submissions , 130 unreviewed
4,065 questions , 1,471 unanswered
4,952 answers , 21,117 comments
1,470 users with positive rep
556 active unimported users
More ...

  Why use class multiplication to describe topological entangling and merging?

+ 4 like - 0 dislike
80 views
I'm studying some references about topological defects in ordered media like *Soft matter physics: An introduction* by Kleman and the Review modern physics paper *The topological theory of defects in ordered media* by Mermin. In both of them, the authors emphasize the class multiplication instead of elements multiplication for describing disclination merging and entangling, and then admit the arbitrariness of the class multiplication. However, I have some problem to understand this. E.g., for a bi-axial liquid crystal, disclinations are classified by $\pi_1(SO(3)/D_2) = Q_8$. This is a quaternion group who has five conjugacy classes $\{1\}, \{-1\},\{i,-i\},\{j,-j\},\{k,-k\}$. I understand the fact that elements $i$ and $-i$ describing , e.g., the disclination and anti-disclination in $yz$ plane, so it is reasonable to group them together. However, when I do defects merging or entangling using the class multiplication, as suggest in the references, I would have problem to predict the result: $\{i,-i\}$ mutiplities $\{i,-i\}$ can either give me $\{1\}$ or $\{-1\}$ who are different defects. Why doesn't one use the elements multiplication directly which doesn't lead to the arbitrary? Put differently, why is it necessary to use class multiplication? Can any one give me any hints?
asked Jan 30, 2015 in Theoretical Physics by hongchan (90 points) [ revision history ]
edited Jan 31, 2015 by Jia Yiyang

1 Answer

+ 3 like - 0 dislike

Disclaimer: I'm not at all learnt on the subject, in fact this is the first time I read something on topological defects, my answer will be tentative. 

After briefly reading through the relevant sections of The topological theory of defects in ordered media, mostly section VI.B, here is what the discussion looks like to me:

1. The classification is most naturally done in terms of conjugacy classes, since it's physically more natural to consider free homotopy rather than base-point-preserving homotopy.

2. Surely we can do multiplication on (fundamental group)elements instead of conjagcy classes, it's not forbidden, and it indeed has the merit of being unique. (But again, the result of the multiplication is still most naturally classified by conjugacy classes)

3. But since the classification is done in terms of conjugacy classes, it's reasonable to think about what multiplication of classes means, despite the ambiguities OP mentioned. Even further, the ambiguities have very clear physical meanings(still in section VI.B), which certainly makes a good reason to study class multiplication.

In short, it is not necessary but desirable to consider class multiplication. 

answered Jan 31, 2015 by Jia Yiyang (2,635 points) [ revision history ]
edited Jan 31, 2015 by Jia Yiyang

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
...