What is the construction of gamma matrices and spinors in higher dimensions?

[ismaili]

Dear guys,

I want to understand the spinors in various dimensions and Clifford algebra. I tried to read the appendix B of Polchinski's volume II of his string theory book. But it's hard for me to follow and I stuck in the very beginning. I will try to figure out the outline and post my questions later.

For now, I want to ask for very simple, introductory articles for the construction of gamma matrices and spinors in various dimensions. (Is the appendix B of Polchinski the simplest article among all?)

Thanks for your help!

Ismaili

 

[haushofer]

Maybe "A menu of supergravities" of Van Proeyen can help you. I found it quite understandable.

 

[ismaili]

Maybe "A menu of supergravities" of Van Proeyen can help you. I found it quite understandable.

Excuse me,
I searched for this book in libraries nearby and on google but I couldn't find it?
Was this book published in english?

---
In the following, I briefly present the content and one of my question by which I stuck.

In the appendix B of Polchinski's string book.
One starts from the Clifford algebra in SO(d-1,1)
\{ \gamma^{\mu} , \gamma^\nu \} = 2\eta^{\mu\nu}<br />
In the even dimension, d = 2k+2, one can group the \gamma^\mu into k+1 sets of anticommuting creation and annihilation operators,
<br /> \gamma^{0\pm} = \frac{1}{2} (\pm\gamma^0 + \gamma^1)<br /> \quad\quad \gamma^{a\pm} = \frac{1}{2}(\gamma^{2a} \pm i \gamma^{2a+1})<br />
where a=1,2,\cdots, k.
One then found that,
<br /> \{ \gamma^{a+}, \gamma^{b-} \} = \delta^{ab}\quad\quad<br /> \{ \gamma^{a+} , \gamma^{b+} \} = \{ \gamma^{a-} , \gamma^{b-} \} = 0<br />
That is, one finds that the gamma matrices can be grouped into the creation and annihilation operators of k species of fermions. In particular, from
(\gamma^{a-})^2 = 0
one sees there exist a vacuum |\xi\rangle annihilated by all \gamma^{a-}.
Thus, by this observation, one constructed the representation of Clifford algebra in the following space,
<br /> (\gamma^{k+})^{s_k+1/2}\cdots(\gamma^{0+})^{s_0+1/2} |xi\rangle<br />
, i.e. a space of the tensor product of k species fermions; so, the dimension of this representation is 2^{k+1}.

In d = 2, one can easily work out the matrix form of the gamma matrices,
\gamma^0 = \left(\begin{array}{cc}0 &amp;1\\ -1 &amp;0\end{array}\right) = i\sigma^2
\gamma^1 = \left(\begin{array}{cc}0 &amp;1\\ 1 &amp;0\end{array}\right) = \sigma^1

One can construct the representation in higher dimensional even space recursively, by d \rightarrow d+2. But now comes my question, for d = 6
<br /> \gamma^0 = i\sigma^2\otimes\textcolor{red}{(-\sigma^3)}\otimes\textcolor{red}{(-\sigma^3)}<br />
<br /> \gamma^1 = \sigma^1 \otimes \textcolor{red}{(-\sigma^3)} \otimes \textcolor{red}{(-\sigma^3)}<br />
<br /> \quad\quad\quad\vdots<br />
<br /> \gamma^4 = I \otimes I \otimes \sigma^1<br />
<br /> \gamma^5 = I \otimes I \otimes \sigma^2<br />
where I is the 2 by 2 unit matrix.
My question is that, why do we use \textcolor{red}{\sigma^3}? I thought it should be the 2 by 2 identity matrix!

Anybody guides me through this?
Thank you so much for your help!

 

[ismaili]

Excuse me,
I searched for this book in libraries nearby and on google but I couldn't find it?
Was this book published in english?

---
In the following, I briefly present the content and one of my question by which I stuck.

In the appendix B of Polchinski's string book.
One starts from the Clifford algebra in SO(d-1,1)
\{ \gamma^{\mu} , \gamma^\nu \} = 2\eta^{\mu\nu}<br />
In the even dimension, d = 2k+2, one can group the \gamma^\mu into k+1 sets of anticommuting creation and annihilation operators,
<br /> \gamma^{0\pm} = \frac{1}{2} (\pm\gamma^0 + \gamma^1)<br /> \quad\quad \gamma^{a\pm} = \frac{1}{2}(\gamma^{2a} \pm i \gamma^{2a+1})<br />
where a=1,2,\cdots, k.
One then found that,
<br /> \{ \gamma^{a+}, \gamma^{b-} \} = \delta^{ab}\quad\quad<br /> \{ \gamma^{a+} , \gamma^{b+} \} = \{ \gamma^{a-} , \gamma^{b-} \} = 0<br />
That is, one finds that the gamma matrices can be grouped into the creation and annihilation operators of k species of fermions. In particular, from
(\gamma^{a-})^2 = 0
one sees there exist a vacuum |\xi\rangle annihilated by all \gamma^{a-}.
Thus, by this observation, one constructed the representation of Clifford algebra in the following space,
<br /> (\gamma^{k+})^{s_k+1/2}\cdots(\gamma^{0+})^{s_0+1/2} |xi\rangle<br />
, i.e. a space of the tensor product of k species fermions; so, the dimension of this representation is 2^{k+1}.

In d = 2, one can easily work out the matrix form of the gamma matrices,
\gamma^0 = \left(\begin{array}{cc}0 &amp;1\\ -1 &amp;0\end{array}\right) = i\sigma^2
\gamma^1 = \left(\begin{array}{cc}0 &amp;1\\ 1 &amp;0\end{array}\right) = \sigma^1

One can construct the representation in higher dimensional even space recursively, by d \rightarrow d+2. But now comes my question, for d = 6
<br /> \gamma^0 = i\sigma^2\otimes\textcolor{red}{(-\sigma^3)}\otimes\textcolor{red}{(-\sigma^3)}<br />
<br /> \gamma^1 = \sigma^1 \otimes \textcolor{red}{(-\sigma^3)} \otimes \textcolor{red}{(-\sigma^3)}<br />
<br /> \quad\quad\quad\vdots<br />
<br /> \gamma^4 = I \otimes I \otimes \sigma^1<br />
<br /> \gamma^5 = I \otimes I \otimes \sigma^2<br />
where I is the 2 by 2 unit matrix.
My question is that, why do we use \textcolor{red}{\sigma^3}? I thought it should be the 2 by 2 identity matrix!

Anybody guides me through this?
Thank you so much for your help!

I think I know the answer to the use of \sigma^3.
The gamma matrices in d = 2 invole only \sigma^1, \sigma^2.
When we add the spacetime dimension by 2,
in order to get the correct anti-commutation relations,
we have to tensor product the original gamma matrices by \sigma^3.

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After figuring out the construction of higher dimensional gamma matrices,
I was confused by the suddenly born conjugation matrix B and charge conjugation matrix C
...