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
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Symmetry Breaking in Terms of Roots and Weights?

+ 5 like - 0 dislike
1119 views

I'm currently searching, for quite a while now, for a paper/book that discusses Higgs symmetry breaking in terms of roots and weights.

Concretely I have in mind a discussion of what happens when we give a vev to some Higgs field, represented by a weight of some representation.

I know how to compute which generators (=roots) remain unbroken, but my problem is determining which group this implies after symmetry breaking.

Any suggestions would be much appreciated!

asked Jul 7, 2015 in Mathematics by Jakob [ revision history ]
recategorized Jul 7, 2015 by dimension10

This recent paper might help: http://arxiv.org/abs/1504.08113.

Roots are not the same thing as generators.Usually a $U(N)$ the theory is broken down to $U(1)^N$ if there is a full-symmetry breaking as in the cases or to a subgroup of $U(M) \subset U(N)$ times priducts of the remaining $U(1)^{N-M}$'s. For theories with gauge groups other than $U(N)$ it might be a bit different but qualitatively there is no big difference. 

1 Answer

+ 3 like - 0 dislike

So basically you know how to find the unbroken Lie algebra but do not know how to find the associated Lie group.

To a given Lie algebra $\mathfrak g$ there exists a unique group $\tilde G$, called the universal covering group, with the property of being [simply connected][1]. For example, the covering group of the algebra $\mathfrak{su}(2)$ is $SU(2)$. 

The other groups, $\{G\}$,  associated to the same algebra can be obtained from the covering group in the following way
$$G=\frac{\tilde G}{Ker(\rho)},$$
where $Ker(\rho)$ is the kernel of the group homomorphism $\rho:\tilde G\rightarrow G$. Once you have defined a particular representation by choosing a particular highest weight, you are able to compute this kernel. For example, you start with an $\mathfrak{su}(2)$ algebra. Then if you choose the adjoint representation (the highest weight being the highest root) you can show that $Ker(\rho)=\mathbb Z_2$ and the group will be $G=SU(2)/\mathbb Z_2=SO(3)$. On the other hand, if you choose the defining representation you get $Ker(\rho)=\mathbb 1$ and $G=SU(2)/\mathbb 1=SU(2)$.

There are some technical details needed to compute those kernel but in general,
$$Ker(\rho)\subset\mathcal Z(\tilde G),$$
where $\mathcal Z(\tilde G)$ is the center of $\tilde G$, and this center is a finite group which can be obtained from the extended Dynkin diagram.

Same references:
Cornwell, group theory in physics, 1984;
Olive, Turok, Nucl Phys B215, 1983, p470;

answered Jul 11, 2016 by Diracology (120 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$\hbar$ysicsOve$\varnothing$flow
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
...