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Representations of spinors

  1. Feb 11, 2016 #1
    I'm currently reading a book on relativistic field theory and I'm trying to understand spinors.
    After the author introduces the four parts of the Lorentz group he talks about spinors and group representations:

    "....With this concept we see that the 2x2 unimodular matrices A discussed in the previous section form a two-dimensional representation of the restricted Lorentz group L_+ (and arrow up)"

    The derivation is not clear to me and the author is very abstract in his explanations. But I want to know how to explicitly derive this unimodular matrices. I know a little bit about group theory, for example how to represent the group Z3 as matrices with this formula [itex][D(g)]_{ij}=<i|D(g)|j>[/itex] and it's simple. I know there is a difference because the Lorentz group is a continuous group but maybe there is also such a simple way to derive the spinor representation. I want to know how to explicitly derive spinors from the Lorentz group.

    I know that you can write that a four vector corresponds to a 2x2 matrix via:

    [itex]\begin{pmatrix} x^{0}+x^{3} & x^{1}-ix^{2} \\ x^{1}+ix^{2} & x^{0}-x^{3} \end{pmatrix}[/itex]
    Now is this already a spinor?
     
  2. jcsd
  3. Feb 11, 2016 #2

    George Jones

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    2x2 unimodular matrices form a repesentation of the universal cover of the restricted Lorentz, i.e, the restricted Lorentz group is not simply connected.

    The space of 4-vectors is a tensor product of 2-component spinor spaces.

    For a somewhat readable mathematical exposition of this, see the book "The Geometry of Minkowski Spacetime" by Gregory Naber.
     
  4. Feb 20, 2016 #3
    Ok thank you. Ok another question.
    You can map a four vector to a 2x2 complex matrix like this:

    [itex]X= \begin{pmatrix} x^{0}+x^{3} & x^{1}-ix^{2} \\ x^{1}+ix^{2} & x^{0}-x^{3} \end{pmatrix}[/itex]

    while

    [itex]det(X) =(x^{0})^{2}-(x^{i})^{2}[/itex]

    Is the Lorentz invariant distance, which means that every transformation which preserves this length is a Lorentz transformation. Now we can make such a transformation with 2x2 unimodular matrices like:

    [itex]X' = AXA^{\dagger}[/itex]

    Alright, I get all that. But how do you come to spinors now? What is missing?
     
  5. Feb 20, 2016 #4

    haushofer

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