Reflecting spin state relative to planes

for a reflection about the xy-plane (along z-direction), the transformation is:

• $S_x \to -S_x$
• $S_y \to -S_y$
• $S_z \to S_z$

This can be implemented with: $U = e^{i\pi S_z} = 2S_z$

Reflection Operators for All Three Planes

Following the same pattern for pseudovector transformations:

1. Reflection about xy-plane (flip z-coordinate): $$\hat{R}_{xy}: \quad S_x \to -S_x, \quad S_y \to -S_y, \quad S_z \to S_z$$ $$U_{xy} = e^{i\pi S_z} = \cos(\pi)\mathbb{I} + i\sin(\pi)\sigma_z = -\mathbb{I} + i \cdot 0 \cdot \sigma_z = -\mathbb{I}$$

Wait, let me recalculate this more carefully. Since $\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$: $$U_{xy} = e^{i\pi \sigma_z/2} = \begin{pmatrix} e^{i\pi/2} & 0 \\ 0 & e^{-i\pi/2} \end{pmatrix} = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$$

2. Reflection about xz-plane (flip y-coordinate): $$\hat{R}_{xz}: \quad S_x \to -S_x, \quad S_y \to S_y, \quad S_z \to -S_z$$ $$U_{xz} = e^{i\pi S_y} = e^{i\pi \sigma_y/2} = \begin{pmatrix} \cos(\pi/2) & -\sin(\pi/2) \\ \sin(\pi/2) & \cos(\pi/2) \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$

3. Reflection about yz-plane (flip x-coordinate): $$\hat{R}_{yz}: \quad S_x \to S_x, \quad S_y \to -S_y, \quad S_z \to -S_z$$ $$U_{yz} = e^{i\pi S_x} = e^{i\pi \sigma_x/2} = \begin{pmatrix} \cos(\pi/2) & i\sin(\pi/2) \\ i\sin(\pi/2) & \cos(\pi/2) \end{pmatrix} = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$$

what appears to be a simple spatial reflection for spin actually involves time reversal as well, since the commutation relations change signs. The operators above represent the unitary implementations of these combined transformations that preserve the proper transformation properties of the spin pseudovector.

Matrix Representations

The final reflection operators are:

• $U_{xy} = i\sigma_z = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$
• $U_{xz} = i\sigma_y = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$
• $U_{yz} = i\sigma_x = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$

These operators correctly implement the pseudovector transformation properties of spin under spatial reflections.