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Dirac Spinor Transformation (Ryder)

  1. Jun 11, 2014 #1
    1. The problem statement, all variables and given/known data

    This complies when I type it in my Latex editor, but not on here. If you could either let me know how to fix that or copy and paste what I have into your own editor to help, that'd be great. Thanks!

    While Ryder is setting up to derive a transformation rule for Dirac spinors, I have failed to follow one step that is crucial for a subsequent derivation for an infinitesimal rotation. He has (with t representing conjugate transpose):

    [ tex ]\bar{\psi} \gamma \psi=( \phi_R^t \phi_L^t ) \left( \begin{array}{ccc}
    0 & -\sigma \\
    \sigma & 0 \end{array} \right) \left( \begin{array}{ccc}
    \phi_R \\
    \phi_L
    \end{array} \right) [ /tex ]

    But I was under the impression the adjoint spinor was represented as
    [ tex ]
    \bar{\psi}=\psi^t \gamma^0=( \phi_L^t \phi_R^t )
    [ /tex ]
    since
    [ tex ]
    \gamma^0=\left( \begin{array}{ccc}
    0 & 1 \\
    1 & 0 \end{array} \right)
    [ /tex ]
    If this is the case, the proceeding derivation does not follow.
    2. Relevant equations
    Above


    3. The attempt at a solution
    Above the above
     
  2. jcsd
  3. Jun 11, 2014 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Just leave out the spaces in [ tex ] and [ /tex ]

    I think you are right that Ryder has not written ## \bar{\psi} \vec{\gamma} \psi ## correctly. However, if you continue his derivation of how the expression transforms under an infinitesimal spatial rotation using your correct expression, you will still be led to the same final result.
     
    Last edited: Jun 11, 2014
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