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Griffiths IEP adjoint operator vs. Scadron's Advanced Quantum Theory adjoint operator

  1. Jul 2, 2010 #1
    1. The problem statement, all variables and given/known data

    I want to compute the transpose of the adjoint of a Dirac spinor.


    2. Relevant equations

    My reasoning, based on learning Griffiths notation in “Intro to Elementary Particles”, p. 236, [7.58]:

    [tex]{\bar u^T} = {({u^\dag }{\gamma _0})^T} = {\gamma _0}^T{u^\dag }^T<mathop> = \limits^{???} {\gamma _0}^T{u^*}[/tex]

    But this contradicts what is written in Scadron (p. 100, [6.22], but using Griffiths definition of the adjoint, in contrast to Scadron’s p. 69, [5.29]): the transpose of the adjoint of a spinor is computed as:

    [tex]\begin{array}{c}
    {{\bar u}^T} = {\left( {\overline {\frac{{{p_\mu }{\gamma ^\mu } + m}}{{\sqrt {E + m} }}\left[ {\begin{array}{*{20}{c}}
    {{\phi ^{(\lambda )}}({\bf{\hat p}})} \\
    {\bf{0}} \\
    \end{array}} \right]} } \right)^T} \\
    = \frac{1}{{\sqrt {E + m} }}{\left( {\left[ {{{({p_\mu }{\gamma ^\mu })}^\dag }{\gamma _0} + m} \right]{{\left[ {\begin{array}{*{20}{c}}
    {{\phi ^{(\lambda )}}({\bf{\hat p}})} \\
    {\bf{0}} \\
    \end{array}} \right]}^\dag }{\gamma _0}} \right)^T} \\
    = \frac{1}{{\sqrt {E + m} }}{\gamma _0}{\left[ {\begin{array}{*{20}{c}}
    {{\phi ^{(\lambda )}}({\bf{\hat p}})} \\
    {\bf{0}} \\
    \end{array}} \right]^*}\left[ {{\gamma _0}{{({p_\mu }{\gamma ^\mu })}^*} + m} \right] \\
    {{\bar u}^T}<mathop> = \limits^{huh?} \frac{1}{{\sqrt {E + m} }}{\gamma _0}\left[ {{{({p_\mu }{\gamma ^\mu })}^T} + m} \right]{\left[ {\begin{array}{*{20}{c}}
    {{\phi ^{(\lambda )}}({\bf{\hat p}})} \\
    {\bf{0}} \\
    \end{array}} \right]^*} \\
    \end{array}[/tex]

    As you can see: the
    [tex]\sqrt {E + m} [/tex]

    is trivial.
     
    Last edited: Jul 2, 2010
  2. jcsd
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