# Definition Consider a state $\ket{\pmb{x}, \sigma}$ where $\pmb{x}$ is the spatial coordinate and $\sigma$ is the spin. The parity or spatial inversion operator $\Pi$ acts on this state as: $ \begin{align} \Pi \ket{\pmb{x}, \sigma} = \ket{-\pmb{x}, \sigma} .\end{align} $ In $\pmb{k}$ space, $\Pi$ acts on a state $\ket{\pmb{k}, \sigma}$ as: $ \begin{align} \Pi \ket{\pmb{k}, \sigma} = \ket{-\pmb{k}, \sigma} .\end{align} $ One can easily show that $\Pi$ is an [[Idempotent Operator]], i.e. $\Pi^2 = 1$ and the [[Eigenvalue|eigenvalues]] of $\Pi$ are $\pm 1$. Note that $\Pi$ is also [[Unitary Operator]], i.e. $\Pi^\dagger = \Pi^{-1}$ If we write down a [[Hamiltonian]] as: $ \begin{align} H = \sum_{\pmb{k},\sigma,\sigma'} \ket{\pmb{k},\sigma} H_{\sigma\sigma'} \bra{\pmb{k},\sigma'} .\end{align} $ Then parity acts as: $ \begin{align} \Pi H \Pi^{-1} &= \sum_{\pmb{k},\sigma,\sigma'} \Pi \ket{\pmb{k},\sigma} H(\pmb{k})_{\sigma\sigma'} \bra{\pmb{k},\sigma'}\Pi^\dagger,\\ &= \sum_{\pmb{k},\sigma,\sigma'} \ket{-\pmb{k},\sigma} H(\pmb{k})_{\sigma\sigma'} \bra{-\pmb{k},\sigma'},\\ &= \sum_{\pmb{k},\sigma,\sigma'} \ket{\pmb{k},\sigma} H(-\pmb{k})_{\sigma\sigma'} \bra{\pmb{k},\sigma'} .\end{align} $ Thus, if the system satisfies $H(\pmb{k}) = H(-\pmb{k})$ then $\Pi H \Pi^{-1} = H$ and the system preserves inversion symmetry.