- #1
ppedro
- 22
- 0
Consider the Dirac Lagrangian,
[itex] L =\overline{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi, [/itex]
where [itex] \overline{\psi}=\psi^{\dagger}\gamma^{0} [/itex], and show that, for [itex] \alpha\in\mathbb{R} [/itex] and in the limit [itex] m\rightarrow0 [/itex], it is invariant under the chiral transformation
[itex] \psi\rightarrow\psi'=e^{i\alpha\gamma_{5}}\psi [/itex]
[itex] \psi^{\dagger}\rightarrow\left(\psi^{\dagger}\right)'=\psi^{\dagger}e^{-i\alpha\gamma_{5}} [/itex]
Attempt at a solution
[itex] \begin{array}{ll}
L' & =\overline{\psi}'\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi'\\
& =\left(\psi^{\dagger}\right)'\gamma^{0}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi'\\
& =\psi^{\dagger}e^{-i\alpha\gamma_{5}}\gamma^{0}\left(i\gamma^{\mu}\partial_{\mu}-m\right)e^{i\alpha\gamma_{5}}\psi\\
& =\underset{(i)}{\underbrace{i\psi^{\dagger}e^{-i\alpha\gamma_{5}}\gamma^{0}\gamma^{\mu}\partial_{\mu}e^{i\alpha\gamma_{5}}\psi}}-\underset{(ii)}{\underbrace{m\psi^{\dagger}e^{-i\alpha\gamma_{5}}\gamma^{0}e^{i\alpha\gamma_{5}}\psi}}\\
& =
\end{array} [/itex]
For (ii) I tried using [itex] \exp\left(s\hat{X}\right)\hat{Y}\exp\left(-s\hat{X}\right)=\hat{Y}+s\left[\hat{X},\hat{Y}\right] [/itex] to get
[itex] \begin{array}{ll}
(ii) & =m\psi^{\dagger}e^{-i\alpha\gamma_{5}}\gamma^{0}e^{i\alpha\gamma_{5}}\psi\\
& =m\psi^{\dagger}\left(\gamma^{0}-i\alpha\left[\gamma_{5},\gamma^{0}\right]\right)\psi\\
& =
\end{array} [/itex]
Can you help me finish this?
[itex] L =\overline{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi, [/itex]
where [itex] \overline{\psi}=\psi^{\dagger}\gamma^{0} [/itex], and show that, for [itex] \alpha\in\mathbb{R} [/itex] and in the limit [itex] m\rightarrow0 [/itex], it is invariant under the chiral transformation
[itex] \psi\rightarrow\psi'=e^{i\alpha\gamma_{5}}\psi [/itex]
[itex] \psi^{\dagger}\rightarrow\left(\psi^{\dagger}\right)'=\psi^{\dagger}e^{-i\alpha\gamma_{5}} [/itex]
Attempt at a solution
[itex] \begin{array}{ll}
L' & =\overline{\psi}'\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi'\\
& =\left(\psi^{\dagger}\right)'\gamma^{0}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi'\\
& =\psi^{\dagger}e^{-i\alpha\gamma_{5}}\gamma^{0}\left(i\gamma^{\mu}\partial_{\mu}-m\right)e^{i\alpha\gamma_{5}}\psi\\
& =\underset{(i)}{\underbrace{i\psi^{\dagger}e^{-i\alpha\gamma_{5}}\gamma^{0}\gamma^{\mu}\partial_{\mu}e^{i\alpha\gamma_{5}}\psi}}-\underset{(ii)}{\underbrace{m\psi^{\dagger}e^{-i\alpha\gamma_{5}}\gamma^{0}e^{i\alpha\gamma_{5}}\psi}}\\
& =
\end{array} [/itex]
For (ii) I tried using [itex] \exp\left(s\hat{X}\right)\hat{Y}\exp\left(-s\hat{X}\right)=\hat{Y}+s\left[\hat{X},\hat{Y}\right] [/itex] to get
[itex] \begin{array}{ll}
(ii) & =m\psi^{\dagger}e^{-i\alpha\gamma_{5}}\gamma^{0}e^{i\alpha\gamma_{5}}\psi\\
& =m\psi^{\dagger}\left(\gamma^{0}-i\alpha\left[\gamma_{5},\gamma^{0}\right]\right)\psi\\
& =
\end{array} [/itex]
Can you help me finish this?