# (Non-)Hermiticity of Dirac operator

I have a Dirac operator given by $$$$D\!\!\!/[A, A^{5}]=\gamma^\mu D_\mu=\gamma^\mu (\partial_{\mu} - {\rm i} A_{\mu} - {\rm i} \gamma_{5} A_{\mu}^{5}),$$$$ where $$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

$$$$\overline{- {\rm i} \gamma^{\mu} \gamma_{5} A_{\mu}^{5}} = {\rm i} A_{\mu}^{5} \overline{ \gamma^{\mu } \gamma_{5} } = {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{\mu \dagger} \gamma^{0} = {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{0} \underbrace{\gamma^{0} \gamma^{\mu \dagger} \gamma^{0}}_{\gamma^{\mu } } = {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{0} \gamma^{\mu } = {\rm i} A_{\mu}^{5} \gamma^{0} \gamma^{\mu } \gamma_{5} ,$$$$
• Hermitian conjugate $$$$(- {\rm i} \gamma^{\mu} \gamma_{5} A_{\mu}^{5})^{\dagger} = {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{\mu \dagger} = \begin{cases} - {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{\mu } & \text{in Euclidean space} ,\\ \\ {\rm i} A_{\mu}^{5} \gamma_{5} \gamma^{0} \gamma^{\mu } \gamma^{0} & \text{in Minkowski space}. \end{cases}$$$$ Except for the known relation $$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?