Wick rotation from Minkowski Dirac theory to Euclidean Dirac theory: $\gamma^{0} = -i\gamma^{4}$

Question

I am reading Path Integrals and Quantum Anomalies by Kazuo Fujikawa and Hiroshi Suzuki. In chapter 4.2 they calculate the self-energy of photon for QED and say that the actual calculation is performed in Euclidean theory. The recipe they gave to change from Minkowski to Euclidean theory is
\begin{align}
& \text{time: } x^{0} \to -ix^{4} \\
& \text{contravariant vector component: } V^{0} \to -iV^{4} \\
& \text{covariant vector component: } A_{0} \to iA_{4}\\
& \text{metric: } (+1,-1,-1,-1) \to (-1,-1,-1,-1) \\
& \text{gamma matrix: } \gamma^{0} = -i\gamma^{4}.
\end{align}
Note that different from the first four transformation which I use "$\to$", in the last one I use "$=$", since we really need the numerical value of $\gamma^{4}$, given the knowledge of $\gamma^{0}$. 

I understand that this transformation allows us to compute the integral in Minkowski space in a convergent manner and still obtain *the same answer* since the mathematical meaning of changing from Minkowski to Euclidean theory is merely a *change of integration axis that preserves the answer, provided that we do not hit singularities when we change the integration axis* (correct me if I am wrong though). 

I am totally fine with the first four transformations I list above. However, I have difficulties to understand the last one: $\gamma^{0} \to -i\gamma^{4}$. Since $\gamma^{0}$ by itself is really just a numerical matrix, it is hard to imagine why it should change in this way. 

The following are some of my thoughts. In fact, it is not the $\gamma^{0}$ that change, it is the 
\begin{align}
\bar{\psi} \gamma^{\mu} A_{\mu} \psi
\end{align}
that we want to preserve upon the change from Minkowski to Euclidean theory, where $A_{\mu}$ is some covariant vector component. We are *assuming* that in the Lagrangian, whenever $\gamma^{\mu}$ appears, it will always be contracted with some covariant vector component $A_{\mu}$, in order to make the Lagrangian a scalar. And therefore, we may roughly regard 
\begin{align}
\bar{\psi} \gamma^{\mu} \psi
\end{align}
as some contravariant vector component $V^{\mu}$ (actually indeed $\bar{\psi} \gamma^{\mu} \psi$ transforms as a vector field) and then by requiring $V^{\mu}A_{\mu}$ is preserved we certainly need to change 
\begin{align}
\bar{\psi} \gamma^{0} \psi \to -i\bar{\psi} \gamma^{4} \psi
\end{align}
which, together with the change of metric and $A_{0} \to iA_{4}$, will preserve the $\bar{\psi} \gamma^{\mu} A_{\mu} \psi$ before and after the transformation.

In short, we should properly say that 
\begin{align}
\bar{\psi} \gamma^{0} \psi \to -i\bar{\psi} \gamma^{4} \psi
\end{align}
which in literature they will just simply write 
\begin{align}
\gamma^{0} = -i \gamma^{4}.
\end{align}

Is my understanding correct? If not is there any suggested reference? It seems like in this book the authors didn't elaborate more on this. Thanks!

Answer 1

Yes, you are right. In fact, it is just a definition of one matrix via another. Your arrows "$\to$" are also such definitions :-)