Can we define parity operators for spins that are anything like the usual parity operator?

In the course of solving a problem I found that it would be useful for me to invent a unitary operator $U$ which satisfies the following requirements for a spin-1/2 system:

$$U^{\dagger}S_{x}U=S_{x}, \hspace{10pt}U^{\dagger}S_{y}U=S_{y}, \hspace{10pt} U^{\dagger}S_{z}U=-S_{z}$$

I've convinced myself this isn't possible, by the following chain of equalities:

$$S_{y}S_{z}=iS_{x} \Rightarrow U^{\dagger}S_{y}S_{z}U=U^{\dagger}S_{y}UU^{\dagger}S_{z}U=-S_{y}S_{z}=iU^{\dagger}S_{x}U=iS_{x}=S_{y}S_{z}$$

which contradicts the unitarity of $U$. But if I try to apply classical reasoning to spins (dangerous, I know, but hear me out), it seems like we should be able to realize the desired transformation by rotating by $\pi$ about the z-axis, which maps $S_{y}\rightarrow -S_{y}$ and $S_{x}\rightarrow -S_{x}$, followed by something resembling the parity operator in spin space, i.e., the operator that sends $S_{x}\rightarrow -S_{x}, S_{y}\rightarrow -S_{y}, S_{z}\rightarrow -S_{z}$. If such an operator existed, it should be unitary and Hermitian, since it squares to the identity. But if such an operator existed, then I can construct the desired $U$, which I just proved can't exist. Can someone point out where my logic is failing me?

• So you demonstrate the U of the first line, or else the "something resembling" operator multiplying all generators with a - sign is not an isomorphism of the SU(2) algebra--it does not preserve it. – Cosmas Zachos Sep 19 '17 at 0:53

What you're trying to describe is essentially a reflection. The problem you're encountering is that your spin operators transform like pseudovectors under reflection (similar to their classical angular momentum counterpart). One consequence is that the spin operators do not change signs under a parity transformation, so you won't be able to construct an operator like $S_{x}\rightarrow -S_{x}, S_{y}\rightarrow -S_{y}, S_{z}\rightarrow -S_{z}$.