# Charge Conjugation of massive Dirac spinor in 3 dimensions with Euclidean signature

In 2+1 dimensional massive Dirac equation (Minkowski signature), we can define the charge conjugation operator so that the equation can be symmetric under it. However, the charge conjugation does not exist for 3-dimensional Euclidean massive Dirac equation: \begin{eqnarray} \left[\gamma^k(\partial_k+ieA_k)-m\right]\Psi=0, \\ \left\{\gamma^i,\gamma^j\right\}=2\delta^{ij}. \end{eqnarray} In other words, there does not exist $C$ such that \begin{eqnarray} C\gamma^kC^{-1}=\left(\gamma^k\right)^T. \end{eqnarray}

However, it seems that the Euclidean Dirac equation can somehow transform into Minkowski one by analytic continuation or Wick rotation. I guess that the charge conjugation should also have a corresponding version in Euclidean signature since it exists in Minkowski signature. Therefore, my question is how to understand this contradiction?

I dispute your premise. Which charge conjugation matrices exist depends only on the total (space + time) number of dimensions. It's only more intricate properties like the existence of a Majorana condition that care about the signature. In 3D, we can lookup that there is a matrix $$C$$ satisfying the following: $$$$C\gamma^kC^{-1} = -(\gamma^k)^T$$$$ The easiest way to go between Euclidean and Minkowski space is just to multiply the timelike gamma matrices by $$i$$ so the above equation goes through unchanged.