- #1

Geremy Holly

- 1

- 0

## Homework Statement

Given the spinors:

[tex]\Psi_{1}=\frac{1}{\sqrt{2}}\left(\psi-\psi^{c}\right)[/tex]

[tex]\Psi_{2}=\frac{1}{\sqrt{2}}\left(\psi+\psi^{c}\right)[/tex]

Where c denotes charge conjugation, show that for a vector boson #A_{\mu}#;

[tex]

A_{\mu}\overline{\Psi_{1}}\gamma^{\mu}\Psi_{2}

+

A_{\mu}\overline{\Psi_{2}}\gamma^{\mu}\Psi_{1}

=

2

A_{\mu}\overline{\psi}\gamma^{\mu}\psi

[/tex]

## Homework Equations

##\psi^{c}=-i\gamma^{2}\psi^{*}##

##\overline{\psi}=\psi^{\dagger}\gamma^{0}##

##\{\gamma^{\mu},\gamma^{\nu}\}=2\eta^{\mu\nu}##

##\gamma^{2}\gamma^{\mu}\gamma^{2}=(\gamma^{\mu})^{*}##

## The Attempt at a Solution

Plugging in ##\Psi_{1,2}## it is easy to show that

[tex]

A_{\mu}\overline{\Psi_{1}}\gamma^{\mu}\Psi_{2}

+

A_{\mu}\overline{\Psi_{2}}\gamma^{\mu}\Psi_{1}

=

A_{\mu}(\overline{\psi}\gamma^{\mu}\psi-\overline{\psi^{c}}\gamma^{\mu}\psi^{c})

[/tex]

So for the identity I want to prove to be true I need to prove that

[tex]\overline{\psi^{c}}\gamma^{\mu}\psi^{c}=-\overline{\psi}\gamma^{\mu}\psi[/tex]

Plugging in the definition of ##\psi^{c}## gives

\begin{align*}

\overline{\psi^{c}}\gamma^{\mu}\psi^{c}

&=

(-i\gamma^{2}\psi^{*})^{\dagger}\gamma^{0}\gamma^{\mu}(-i\gamma^{2}\psi^{*})\\

&=

(i\psi^{T}(\gamma^{2})^{\dagger})\gamma^{0}\gamma^{\mu}(-i\gamma^{2}\psi^{*})\\

&=

\psi^{T}\gamma^{0}\gamma^{2}\gamma^{\mu}\gamma^{2}\psi^{*}\\

&=

\psi^{T}\gamma^{0}(\gamma^{\mu})^{*}\psi^{*}\\

&=

(\psi^{\dagger}\gamma^{0}(\gamma^{\mu})\psi)^{*}\\

&=

(\overline{\psi}\gamma^{\mu}\psi)^{*}\\

\end{align*}

Which disagress with the required expression unless it is purely imaginary! I have absolutely no idea where I've gone wrong and would really appreciate some help spotting my error.