Calculating the Transpose of Adjoint of Dirac Spinor

[bjnartowt]

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]:

$${\bar u^T} = {({u^\dag }{\gamma _0})^T} = {\gamma _0}^T{u^\dag }^T<mathop> = \limits^{?} {\gamma _0}^T{u^*}$$

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:

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

As you can see: the
$$\sqrt {E + m}$$

is trivial.

 

[NateD]

The Attempt at a SolutionI think the answer is:{\bar u^T} = {\gamma _0}^T{u^*}because this is consistent with Griffiths definition of the adjoint, and because it is also consistent with Scadron's equation (6.22).