# Explicit form of symmetry operators including spin

Symmetry operators like

• $$\hat{1}$$: unity operator
• $$\hat{R}_{x,\pi}$$: rotation around $$\vec{e}_{x}$$ with angle $$\pi$$
• $$\hat{M}_x$$: mirror at a plane with normal in e.g. x direction
• $$\hat{I}$$: inversion
• ...

should be composed of an operator acting in position space and an operator acting in spin space, like $$\hat{R}_{\alpha,\beta}=\hat{R}^{pos}_{\alpha,\beta}\hat{R}^{spin}_{\alpha,\beta}$$. However, typically just the explicit form in position space is provided: $$\hat{1}^{pos}:\vec{\Psi}(x,y,z)\to \vec{\Psi}(x,y,z)$$ $$\hat{R}^{pos}_{\alpha,\beta}:\vec{\Psi}(x,y,z)\to \vec{\Psi}(x,-y,-z)$$ $$\hat{M}^{pos}_x:\vec{\Psi}(x,y,z)\to \vec{\Psi}(-x,y,z)$$ $$\hat{I}^{pos}:\vec{\Psi}(x,y,z)\to \vec{\Psi}(-x,-y,-z)$$ Lets say $$\vec{\Psi}$$ is a spinor of the form $$\begin{pmatrix}\text{up}\\\text{down}\end{pmatrix}$$. Just for the time inversion operator I found the full form $$\hat{T}:\vec{\Psi}(x,y,z)\to -i \begin{pmatrix} 0 & -\mathrm{i}\\ \mathrm{i} & 0 \end{pmatrix} \vec{\Psi}^*(x,y,z)$$ What are the explicit forms for the spin operators for the other symmetry operators?

I guess a rotation can be expressed by $$\hat{R}^{spin}_{\alpha,\beta}:\vec{\Psi}(x,y,z)\to e^{-i \beta \vec{\sigma}\cdot \vec{e}_{\alpha}}\vec{\Psi}(x,y,z)$$ and $$\overline{\hat{R}}^{spin}_{\alpha,\beta}:\vec{\Psi}(x,y,z)\to - e^{-i \beta \vec{\sigma}\cdot \vec{e}_{\alpha}}\vec{\Psi}(x,y,z)$$ with the Pauli-matrices $$\vec{\sigma}$$? Both lead to the same measurable output, thus a system with a certain rotation symmetry (in position space) should be invariant under both transformations? The inversion (without considering antiparticles) is simply $$\hat{I}^{\text{spin}}=1$$? Can I deduce $$\hat{M}^{\text{spin}}_x=\hat{I}^{\text{spin}} \hat{R}^{spin}_{x,\pi}=\hat{R}^{spin}_{x,\pi}$$? And am I right, that $$\overline{1}=-1$$ (this Bethe-symmetry-operator for double-groups) is always a symmetry operator?

Thanks for any comment!

• Perhaps what you are looking for are the Jones matrices used for polarization optics. – flippiefanus Feb 8 '20 at 4:34
• @flippifanus: While the Jones formalism is related to this topic, it is just one application. Actually I try to understand some backgrounds of the Luttinger/Kane model and some geometry consideration in QDs. But thanks for the comment! – Matthiasho Feb 10 '20 at 10:53