second quantization - How will the (anti)commutation relation between two different fermion fields look like?

The anti-commutation relation between the components of a fermion field $\psi$ is given by $$[\psi _\alpha(x),\psi_\beta^\dagger(y)]_+=\delta_{\alpha\beta}\delta^{(3)}(\textbf{x}-\textbf{y}).$$

1. 
   

   In case of two different and independent fermion fields, should I impose commutation or anticommutation between them?

   
   


2. 
   
   

   If we continue to use anticommutation, how should the RHS change for two different fermion fields $\psi^1$ and $\psi^2$? $$[\psi^1 _\alpha(x),\psi_\beta^{2\dagger}(y)]_+=?$$ $$[\psi^1 _\alpha(x),\psi_\beta^2(y)]_+=?$$ $$[\psi^{1\dagger} _\alpha(x),\psi_\beta^{2\dagger}(y)]_+=?$$

   
   

Answer

We use anticommutation relations:

$$[\psi^1 _\alpha(x),\psi_\beta^{2\dagger}(y)]_+=0$$ $$[\psi^1 _\alpha(x),\psi_\beta^2(y)]_+=0$$ $$[\psi^{1\dagger} _\alpha(x),\psi_\beta^{2\dagger}(y)]_+=0$$