I have a Dirac operator given by \begin{equation} D\!\!\!/[A, A^{5}]=\gamma^\mu D_\mu=\gamma^\mu (\partial_{\mu} - {\rm i} A_{\mu} - {\rm i} \gamma_{5} A_{\mu}^{5}), \end{equation} where $A_{\mu}$ and $A^{5}_{\mu}$ are Hermitian.
In the Euclidean space, I can show that the Dirac operator is Hermitian $D\!\!\!/[A,A^{5}]= D\!\!\!/^{\dagger}[A,A^{5}]$ using $\gamma^{\mu\dagger}= -\gamma^{\mu}$, $\gamma^{5} \gamma^{\mu}=-\gamma^{\mu} \gamma^{5}$ and $\partial^{\dagger}_{\mu} =- \partial_{\mu}$. When $A^{5}_{\mu}=0$, it has been also shown that the Dirac operator is anti-Hermitian with respect to the inner product such that $\overline{D\!\!\!/}[A,0]=-D\!\!\!/[A,0]$, where $\overline{D\!\!\!/}={D\!\!\!/}^{\dagger} \gamma^{0}$.
I now wish to find relations between $\overline{D\!\!\!/}$, $D\!\!\!/$ and ${D\!\!\!/}^{\dagger}$ for $A=0$ and $A^{5}\neq 0$ in both Minkowski and Euclidean space. For this purpose, I've tried to evaluate the adjoint/Hermitian conjugate of the last term in the Dirac operator as
- Adjoint
\begin{equation} \overline{- {\rm i} \gamma^{\mu} \gamma_{5} A_{\mu}^{5}} = {\rm i} A_{\mu}^{5} \overline{ \gamma^{\mu } \gamma_{5} } = {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{\mu \dagger} \gamma^{0} = {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{0} \underbrace{\gamma^{0} \gamma^{\mu \dagger} \gamma^{0}}_{\gamma^{\mu } } = {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{0} \gamma^{\mu } = {\rm i} A_{\mu}^{5} \gamma^{0} \gamma^{\mu } \gamma_{5} , \end{equation}
- Hermitian conjugate \begin{equation} (- {\rm i} \gamma^{\mu} \gamma_{5} A_{\mu}^{5})^{\dagger} = {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{\mu \dagger} = \begin{cases} - {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{\mu } & \text{in Euclidean space} ,\\ \\ {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{0} \gamma^{\mu } \gamma^{0} & \text{in Minkowski space}. \end{cases} \end{equation} Except for the known relation $D\!\!\!/ [0, A^{5}]= {D\!\!\!/}^{\dagger}[0, A^{5}]$ in the Euclidean space, I can't see any (anti-)Hermiticity relations between $\overline{D\!\!\!/}$, ${D\!\!\!/}^{\dagger}$ and $D\!\!\!/$. Does this mean that $D\!\!\!/[0, A^{5}]$ is non-Hermitian, or did I miss something in my calculations?
