Electron pair operators

Let us begin by introducing electron pair operators: \[\begin{align} \require{physics} & \hat{P}^+_p % = \hat{a}^\dagger_{p \alpha} \hat{a}^\dagger_{p \beta} \\ % & \hat{P}^-_p % = \hat{a}_{p \beta} \hat{a}_{p \alpha} \\ % & \hat{P}^\circ_p % = \frac{1}{2} \qty( % \hat{a}^\dagger_{p \alpha} \hat{a}_{p \alpha} % + \hat{a}^\dagger_{p \beta} \hat{a}_{p \beta} - 1 % ) \\ & \phantom{\hat{P}^\circ_p} % = \frac{1}{2} \qty( \hat{N}_p - 1 ) % \thinspace , \end{align}\] which create, annihilate and count electron pairs. The reason why we put count in apostrophes is that \(\hat{P}^\circ_p\) doesn’t really count the number of electron pairs, but in Lie algebra theory, it is sometimes referred to as a `counter’ operator. It can quickly be seen that \(\hat{P}^-_p\) is the Hermitian adjoint of \(\hat{P}^+_p\): \[\begin{equation} \qty(\hat{P}^+_p)^\dagger % = \hat{P}^-_p % \thinspace , \end{equation}\] which is of course why we chose the definition of \(\hat{P}^-_p\) to be that one in the first place, and that \(\hat{P}^\circ_p\) is self-adjoint: \[\begin{equation} \qty(\hat{P}^\circ_p)^\dagger % = \hat{P}^\circ_p % \thinspace . \end{equation}\] The (physical) vacuum state is empty with respect to electron pair creation: \[\begin{equation} \hat{P}^-_p \ket{\text{vac}} % = 0 % \thinspace , \end{equation}\] and the counting action of \(\hat{P}^\circ_p\) on the vacuum is \[\begin{equation} \hat{P}^\circ_p \ket{\text{vac}} % = - \frac{1}{2} % \thinspace . \end{equation}\] We can also establish a Pauli principle: \[\begin{equation} \label{eq:Pauli_principle_pair_operators} \hat{P}^+_p \hat{P}^+_p = 0 % \thinspace . \end{equation}\] From the commutators \[\begin{align} & \comm{ % \hat{P}^+_p % }{ % \hat{P}^+_q % } % = \comm{ % \hat{P}^-_p % }{ % \hat{P}^-_q % } % = \comm{ % \hat{P}^\circ_p % }{ % \hat{P}^\circ_q % } = 0 \\ % & \comm{ % \hat{P}^+_p % }{ % \hat{P}^-_q % } % = 2 \delta_{pq} \hat{P}^\circ_p % = \delta_{pq} (\hat{N}_p - 1) \label{eq:commutator_P+p_P-q} \\ % & \comm{ % \hat{P}^\circ_p % }{ % \hat{P}^\pm_q % } = \pm \delta_{pq} \hat{P}^\pm_p \end{align}\] follows that the operators \(\set{\hat{P}^+_p, \hat{P}^-_p, \hat{P}^\circ_p}\) span an \(\mathfrak{su}(2)\) algebra for the spatial orbital \(p\). The associated Casimir operator for this Lie algebra is as usual: \[\begin{equation} \hat{C}_p % = \hat{P}^+_p \hat{P}^-_p % - \hat{P}^\circ_p % + \qty(\hat{P}^\circ_p)^2 % \thinspace . \end{equation}\]

Let us now list some useful commutator expressions, expressions that we just put here for reference. We have \[\begin{equation} \comm{ % \hat{P}^-_p % }{ % \hat{P}^+_q % } % = - 2 \delta_{pq} \hat{P}^\circ_p % = \delta_{pq} (1 - \hat{N}_p) % \thinspace . \end{equation}\] Composite creators commute with elementary creators: \[\begin{equation} \comm{ % \hat{a}^\dagger_{p \sigma} % }{ % \hat{P}^+_q % } = 0 % \thinspace , \end{equation}\] and the commutator between the composite operators and the number operators is \[\begin{equation} \comm{ % \hat{N}_p % }{ % \hat{P}^\pm_q % } % = \pm 2 \delta_{pq} \hat{P}^\pm_q % \thinspace . \end{equation}\] In accordance with Pauli’s principle, the following composite particle anticommutators arise: \[\begin{equation} \comm{ % \hat{P}^+_p % }{ % \hat{P}^+_q % }_+ % = 2 (1 - \delta_{pq}) % \hat{P}^+_p \hat{P}^+_q % \thinspace , \end{equation}\] and \[\begin{equation} \comm{ % \hat{P}^-_p % }{ % \hat{P}^-_q % }_+ % = 2 (1 - \delta_{pq}) % \hat{P}^-_p \hat{P}^-_q % \thinspace . \end{equation}\] We can establish a sort of resolution of the identity as \[\begin{equation} \label{eq:pair_resolution_identity} 1 % = \hat{N}_p % + \comm{ % \hat{P}^-_p % }{ % \hat{P}^+_p % } % \thinspace . \end{equation}\] Concerning arbitrary strings of pair creation and annihilation operators, we find \[\begin{equation} \comm{ % \hat{P}^+_p % }{ % \prod_q^K (\hat{P}^-_q)^{k_q} % } = \begin{cases} 2 \thinspace \qty( % \prod_{q \neq p}^K (\hat{P}^-_q)^{k_q} % ) \hat{P}^\circ_p % = \qty( % \prod_{q \neq p}^K (\hat{P}^-_q)^{k_q} % ) (\hat{N}_p - 1) % & \quad \text{if} \quad k_p = 1 \\ 0 & \quad \text{otherwise} \end{cases} \end{equation}\] and \[\begin{equation} \label{eq:commutator_P-_prod_P+} \comm{ % \hat{P}^-_p % }{ % \prod_q^K (\hat{P}^+_q)^{k_q} % } = \begin{cases} -2 \thinspace \qty( % \prod_{q \neq p}^K (\hat{P}^+_q)^{k_q} % ) \hat{P}^\circ_p % = \thinspace \qty( % \prod_{q \neq p}^K (\hat{P}^+_q)^{k_q} % ) (1 - \hat{N}_p) % & \quad \text{if} \quad k_p = 1 \\ 0 & \quad \text{otherwise} % \thinspace . \end{cases} \end{equation}\]