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

Question

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 1

We use anticommutation relations:

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

Answer 2

As far as I remember, it is also possible to choose the commutation relations for different fermions. However, traditionally, anti-commutation relations are chosen for creation and annihilation operators of different fermions and commutation relations for creation and annihilation operators of fermions and bosons or different bosons. EDIT:(12/1/2016) See, e.g., https://arxiv.org/abs/1312.0831