Hermiticity of Dirac Operator $\gamma^{\mu}D_{\mu}$ and Expansion in eigenmodes I'm interested to know under what conditions $\gamma^{\mu}D_{\mu}$ is a hermitian operator.
I am studying the Fujikawa method of anomalies and I see that many sources have different answers for this. Some claim that $\gamma^{\mu}D_{\mu}$ is hermitian or possibly anti hermitian in Euclidean space, or that $i\gamma^{\mu}D_{\mu}$ is hermitian in Minkowski space. I need the hermiticiy condition so that I can expand the Dirac spinors $\Psi(x)$ and $\bar{\Psi}(x)$ in a basis of orthonormal eigenvectors of the Dirac operator to perform the path integral. 
Another question I have is this expansion. I would like something of the form
$$\Psi(x) = \sum_n \psi_n(x)a_n$$
$$\bar{\Psi}(x) = \sum_n \bar{\psi_n}(x)\bar{b}_n$$
where $a_n$ and $\bar{b}_n$ are elements of a Grassmann algebra. But how is $\bar{\psi}_n$ defined? Some references define $\bar{\psi}_n(x)$ = ${\psi}(x)^{\dagger}$, but I guess it depends on how you are defining your inner product.
Thanks, I've attached some references below. 
Fujikawa 1) https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.42.1195
Fujikawa 2) https://journals.aps.org/prd/abstract/10.1103/PhysRevD.21.2848
TASI Anomalies https://arxiv.org/abs/hep-th/0509097
 A: The (operator-valued) matrix $\not D=\gamma\cdot D$ is anti-hermitian with respect to the inner product
$$
\langle v,u\rangle:=\int_{\mathbb R^d} \bar v(x)u(x)\ \mathrm dx\tag1
$$
where
$$
\bar v(x):= v^\dagger(x)\gamma^0\tag2
$$
Indeed, write
$$
\not D=\not\!\partial-i\not A\tag3
$$
It is clear that the second factor satisfies
\begin{equation}
\begin{aligned}
\overline{i\not A}&=-i\overline{\not A}\\
&=-i\not A
\end{aligned}\tag4
\end{equation}
where we have used the fact that $\gamma^\mu$ satisfies
$$
\gamma^0\gamma^{\mu\dagger}\gamma^0=\gamma^\mu\tag5
$$
and that $A^\mu$ is hermitian.
On the other hand, the first factor in $(3)$ satisfies
$$
\begin{aligned}
\langle v,\not\!\partial u\rangle&=\int_{\mathbb R^d} \bar v(x)\not\!\partial u(x)\ \mathrm dx\\
&=-\int_{\mathbb R^d} \bar v(x)\overset{\leftarrow}{\not\!\partial}u(x)\ \mathrm dx\\
&=-\int_{\mathbb R^d} \overline{(\bar{\not\!\partial }v)}(x)u(x)\ \mathrm dx\\
&=-\langle \not\!\partial v,u\rangle
\end{aligned}\tag6
$$
where in the first equality we integrated by parts and in the last one we used $(5)$.
From this we learn that $\not D$ satisfies $\overline{\not D}=-\not D$, which is equivalent to the statement that $\not D$ is anti-hermitian with respect to $\langle\cdot,\cdot\rangle$, as claimed.
Note that this notion of anti-hermiticity guarantees that $i\not D$ is diagonalisable, with real eigenvalues and orthogonal eigenvectors, by the usual reasons (spectral theorem, etc.):
$$\tag7
i\not Dv_i=\lambda_i v_i\qquad\Rightarrow\qquad\begin{cases}\lambda_i\in\mathbb R\\\langle v_i,v_j\rangle\propto\delta_{ij}\end{cases}
$$
