- #1
bjnartowt
- 284
- 3
Homework Statement
I want to compute the transpose of the adjoint of a Dirac spinor.
Homework 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.
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