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Spinors in various dimensions

  1. Dec 16, 2009 #1
    Dear guys,

    I wanna 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 wanna 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?:blushing:)

    Thanks for your help!!

  2. jcsd
  3. Dec 16, 2009 #2


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    Maybe "A menu of supergravities" of Van Proeyen can help you. I found it quite understandable.
  4. Dec 18, 2009 #3
    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)
    [tex] \{ \gamma^{\mu} , \gamma^\nu \} = 2\eta^{\mu\nu}
    In the even dimension, [tex] d = 2k+2 [/tex], one can group the [tex]\gamma^\mu [/tex] into [tex] k+1 [/tex] sets of anticommuting creation and annihilation operators,
    \gamma^{0\pm} = \frac{1}{2} (\pm\gamma^0 + \gamma^1)
    \quad\quad \gamma^{a\pm} = \frac{1}{2}(\gamma^{2a} \pm i \gamma^{2a+1})
    where [tex] a=1,2,\cdots, k[/tex].
    One then found that,
    \{ \gamma^{a+}, \gamma^{b-} \} = \delta^{ab}\quad\quad
    \{ \gamma^{a+} , \gamma^{b+} \} = \{ \gamma^{a-} , \gamma^{b-} \} = 0
    That is, one finds that the gamma matrices can be grouped into the creation and annihilation operators of [tex]k[/tex] species of fermions. In particular, from
    [tex] (\gamma^{a-})^2 = 0 [/tex]
    one sees there exist a vacuum [tex] |\xi\rangle [/tex] annihilated by all [tex]\gamma^{a-}[/tex].
    Thus, by this observation, one constructed the representation of Clifford algebra in the following space,
    (\gamma^{k+})^{s_k+1/2}\cdots(\gamma^{0+})^{s_0+1/2} |xi\rangle
    , i.e. a space of the tensor product of [tex]k[/tex] species fermions; so, the dimension of this representation is [tex]2^{k+1}[/tex].

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

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

    Anybody guides me through this?
    Thank you so much for your help!
  5. Dec 20, 2009 #4
    I think I know the answer to the use of [tex]\sigma^3[/tex].
    The gamma matrices in d = 2 invole only [tex]\sigma^1, \sigma^2[/tex].
    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 [tex]\sigma^3[/tex].


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