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

188 submissions , 148 unreviewed
4,758 questions , 1,955 unanswered
5,277 answers , 22,456 comments
1,470 users with positive rep
759 active unimported users
More ...

  How to get the generators of $\mathfrak{so}(3)$ in the paper by Fidkowski and Kitaev?

+ 1 like - 0 dislike
50 views

In the [paper][1] by Fidkowski and Kitaev, they aim to study the interaction of 8 parallel Majorana wires, and they work on $\mathfrak{so(8)}$ Lie Algebra. They first start with just 4 parallel Majorana wires which in $\mathfrak{so}(2n)$, $n=2$. Then based on the fact that $\mathfrak{so(4)}\cong\mathfrak{so(3)}\oplus\mathfrak{so(3)}$, they aim to get the generators of two copies of $\mathfrak{so(3)}$ (which I do not understand why they used two copies of $\mathfrak{so(3)}$ rather than just one $\mathfrak{so(4)}$?).  The map of Lie algebra is defined:
\begin{align}
\rho(A)=\frac{i}{4} \sum\limits_{j,k=1}^{2n} A_{jk}\hat{c}_j\hat{c}_k
\end{align}
Where $A \in \mathfrak{so}(2n)$ is an anti-symmetric matrix belonging to the Lie algebra. So here we have $n=2$ and 4 Majoranas $\{\hat{c}_1,\hat{c}_2,\hat{c}_3,\hat{c}_4\}$. 
I have three questions:

 1. Here $\rho$ is the adjoint representation? Base on the relation

\begin{align}
[-i\rho(A),-i\rho(B)]=-i\rho([A,B])
\end{align}

Therefore $\rho$ defines a Lie algebra, so is it the adjoint representation? Also, I am not sure why I am not able to finish the proof of the above equation. Here is my attempt (I know I got the commutation of $A$ and $B$ but I do not know how to get the other direction:

\begin{align}
[-i\rho(A),-i\rho(B)]&=-i(\frac{i}{4})^2 \sum\limits_{j,k=1}^{2n} \sum\limits_{j',k'1}^{2n'}(A_{jk}\hat{c}_j\hat{c}_k B_{j'k'}\hat{c}_{j'}\hat{c}_{k'}-B_{j'k'}\hat{c}_{j'}\hat{c}_{k'}A_{jk}\hat{c}_j\hat{c}_k)\\
&=-i(\frac{i}{4})^2 \sum\limits_{j,k=1}^{2n} \sum\limits_{j',k'1}^{2n'}(A_{jk} B_{j'k'}-B_{j'k'}A_{jk})\hat{c}_j\hat{c}_k\hat{c}_{j'}\hat{c}_{k'}\\
&=...
\end{align}

 2. How the induced action on $\hat{c}$ is defined as follows?
\begin{align}
\hat{c}_l \rightarrow i[\rho(A),c_l]
\end{align}
I cannot understand this induced action? and they did not even define $c_l$.
 3. My big question is that how under $\rho$ the generators of two $\mathfrak{so(3)}$ obey equations (5) and (6) in the reference. I know $\mathfrak{so(4)}$ has 6 generators. 
\begin{align}
&A_1=\left[\begin{array}{cccc}
0 & 0 & 0 & 0    \\
0 & 0& -1 & 0    \\
0 & 1 & 0 & 0    \\
0 & 0 & 0 & 0
\end{array}\right] \quad A_2=\left[\begin{array}{cccc}
0 & 0 & 1 & 0    \\
0 & 0& 0& 0    \\
-1 & 0 & 0 & 0    \\
0 & 0 & 0 & 0
\end{array}\right] \quad A_3=\left[\begin{array}{cccc}
0 & -1 & 0 & 0    \\
1 & 0 & 0 & 0    \\
0 & 0 & 0 & 0    \\
0 & 0 & 0 & 0
\end{array}\right] \\
&B_1=\left[\begin{array}{cccc}
0 & 0 & 0 & -1    \\
0 & 0& 0& 0    \\
0 & 1 & 0 & 0    \\
1 & 0 & 0 & 0
\end{array}\right] \quad B_2=\left[\begin{array}{cccc}
0 & 0 & 0 & 0    \\
0 & 0& 0& -1    \\
0 & 0 & 0 & 0    \\
0 & 1 & 0 & 0
\end{array}\right] \quad B_3=\left[\begin{array}{cccc}
0 & 0& 0 & 0    \\
0 & 0 & 0 & 0    \\
0 & 0 & 0 & -1    \\
0 & 0 & 1 & 0
\end{array}\right] \\
\end{align}

So as the paper addressed both $A_i$ and $B_i$ belong to $\mathfrak{so(4)}$, but I don't understand how to get the generators. 

  [1]: https://arxiv.org/abs/0904.2197

asked Jul 20 in Theoretical Physics by Bahargraviton (5 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$\hbar$ysicsOver$\varnothing$low
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
...