Do creation operators for different spins commute?

[metapuff]

Say I have a hamiltonian with fermion creation / annihilation operators like this:

$$\sum_{k_1,k_2,k_3,k_4} c_{k_1,\uparrow}^{\dagger} c_{k_2,\downarrow}^{\dagger} c_{k_3,\downarrow} c_{k_4,\uparrow}$$

where the k's are momenta and the arrows indicate spin up / spin down. Can I commute operators for different spins? That is, does

$$c_{k_1,\uparrow}^{\dagger} c_{k_2,\downarrow}^{\dagger} = c_{k_2,\downarrow}^{\dagger} c_{k_1,\uparrow}^{\dagger}$$

Or do I pick up a minus sign as usual? Thanks!

 

[fzero]

The operators anticommute. Specifically all of the commutation relations of the bosonic theory carry over to fermions provided that we replace commutators by anticommutators.

 

[metapuff]

Yay! That's what I'd hoped for, but was afraid there might be some subtlety with spin that I wasn't picking up on. Thanks!