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Do creation operators for different spins commute?

  1. May 7, 2015 #1
    Say I have a hamiltonian with fermion creation / annihilation operators like this:

    [tex] \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} [/tex]

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

    [tex] c_{k_1,\uparrow}^{\dagger} c_{k_2,\downarrow}^{\dagger} = c_{k_2,\downarrow}^{\dagger} c_{k_1,\uparrow}^{\dagger} [/tex]

    Or do I pick up a minus sign as usual? Thanks!
  2. jcsd
  3. May 7, 2015 #2


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    The operators anticommute. Specifically all of the commutation relations of the bosonic theory carry over to fermions provided that we replace commutators by anticommutators.
  4. May 7, 2015 #3
    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!
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