## Do the Creation Operator and Spin Projection Operator Commute?

I have bumped into a term

$a^\dagger \hat{O}_S | \psi \rangle$

I would really like to operate on the slater determinant $\psi$ directly, but I fear I cannot. Is there any easy manipulation I can perform?
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 where you got that and is that O something expressible in terms of gamma matrices?
 What is a spin projection operator? The things that come to mind are single body operators, and you have a many body wavefunction. If it's something like the total Z component of the spin, then as a many body operator it would be written as \sum_i (n_up - n_down), which is a combination of creation/annihilation operators ... take your operator and express it in terms of particle creation/annihilation operators and then you can work out the commutation relation.

Recognitions: