I cannot understand the connection between the Grassmann variable and fermion in the derivation of path integral. I well understood the definition of an integral over a Grassmann variable but I still have some difficulties connecting it with fermions. To better explain my problem if we want to integrate over two real Grassmann variables $\theta_1$ and $\theta_2$ and we want to complexify it, we do the following,
$$\int d\theta_1 d\theta_2 ... \rightarrow \int d\theta d\bar{\theta}... $$ where $\theta = \theta_1 + i \theta_2$ and $\bar{\theta} = \theta_1 - i \theta_2$. Now my problem arises, usually I see a direct connection between $\theta\rightarrow \psi$ and $\bar{\theta}\rightarrow \bar{\psi}$ where $\psi$ now is my fermionic field. The problem is that due to the Grassmann algebra $\{\theta,\bar{\theta}\}=0$, but this is not true for my fermionic field. Where I'm wrong?
