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A Transformation of the spinor indices of the Weyl operator under the Lorentz group

  1. Nov 18, 2016 #1
    The left-handed Weyl operator is defined by the ##2\times 2## matrix

    $$p_{\mu}\bar{\sigma}_{\dot{\beta}\alpha}^{\mu} = \begin{pmatrix} p^0 +p^3 & p^1 - i p^2\\ p^1 + ip^2 & p^0 - p^3 \end{pmatrix},$$

    where ##\bar{\sigma}^{\mu}=(1,-\vec{\sigma})## are sigma matrices.


    One can use the sigma matrices to go back and forth between four-vectors and ##2\times 2## matrices:

    $$p_{\mu} \iff p_{\dot{\beta}\alpha}\equiv p_{\mu}\bar{\sigma}^{\mu}_{\dot{\beta}\alpha}.$$


    Given two four-vectors ##p## and ##q## written as ##2\times 2## matrices,

    $$\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}p_{\dot{\alpha}\alpha}q_{\dot{\beta}\beta} = 2p^{\mu}q_{\mu}.$$


    Given a complex ##2\times 2## matrix ##\Lambda_{L}## with unit determinant, it can be shown that the transformation ##p_{\dot{\beta}\alpha} \rightarrow (\Lambda_{L}^{-1\dagger}p\Lambda_{L}^{-1})_{\dot{\beta}\alpha}## preserves the product ##\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}p_{\dot{\alpha}\alpha}q_{\dot{\beta}\beta}##.

    How does it then follow that ##\Lambda_{L}## is a Lorentz transformation? Do we have to use the fact that ##\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}p_{\dot{\alpha}\alpha}q_{\dot{\beta}\beta} \sim p^{\mu}q_{\mu}##? What is the Lorentz transformation for ##p^{\mu}## due to the transformation ##\Lambda_{L}## for ##p_{\dot{\alpha}\alpha}##?
     
  2. jcsd
  3. Nov 18, 2016 #2

    vanhees71

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    You find a very clear treatment about anything connected with the Poincare group in

    R. U. Sexl and H. K. Urbandtke, Relativity, Groups, Particles, Springer, Wien, 2001.
     
  4. Nov 18, 2016 #3
    I think the second author's name is Urbantke, not Urbandtke.
     
  5. Nov 18, 2016 #4
    Is this the ultimate guide for anything related to the Poincare group?
     
  6. Nov 18, 2016 #5

    vanhees71

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    True, it's Urbantke. There's no "ultimate guide" to anything, but it's a very good book to get the group-theoretical foundations needed to study relativistic QFT more easily than without this basis.
     
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