Creation and annihilation operators

We will define elementary creation operators with their action on the basis vectors (the ON vectors \(\require{physics} \ket{\vb{k}}\)) of the Fock space: \[\begin{equation} \hat{a}^\dagger_P \ket{\vb{k}} % = \delta_{k_P, 0} % \Gamma^{\vb{k}}_P % \ket{\ldots, 1_P, \ldots} % \thinspace , \end{equation}\] with \(\Gamma^{\vb{k}}_P\) a phase factor: \[\begin{equation} \Gamma^{\vb{k}}_P = \prod_{Q=1}^{P-1} (-1)^{k_Q} \end{equation}\] that is equal to \(+1\) if there are an even number of electrons before position \(P\), and equal to \(-1\) if there are an odd number of electrons before position \(P\). The anticommutation relations for electron creation operators then are \[\begin{equation} \comm{\hat{a}^\dagger_P}{\hat{a}^\dagger_Q}_+ = 0 % \thinspace , \end{equation}\] which means that for their Hermitian adjoints, the anticommutation relations are \[\begin{equation} \comm{\hat{a}_P}{\hat{a}_Q}_+ = 0 % \thinspace . \end{equation}\] We can then derive that the action of \(\hat{a}_P\) on an ONV \(\ket{\vb{k}}\) is \[\begin{equation} \hat{a}_P \ket{\vb{k}} % = \delta_{k_p, 1} % \Gamma^{\vb{k}}_P % \ket{\ldots, 0_P, \ldots} % \thinspace , \end{equation}\] which means that \(\hat{a}_P\) annihilates the vacuum state: \[\begin{equation} \hat{a}_P \ket{\text{vac}} = 0 % \thinspace . \end{equation}\] Finally, the mixed elementary anticommutators can be found as \[\begin{equation} \comm{ % \hat{a}_P % }{ % \hat{a}^\dagger_Q % }_+ = \delta_{PQ} % \thinspace . \end{equation}\]

We are now in the position to make an interesting observation. Every occupation number vector \(\ket{\vb{k}}\) can be written in terms of the elementary second quantization operators: \[\begin{equation} \label{eq:ON_creators_vacuum} \ket{\vb{k}} = % \qty( \prod_P^M % (\hat{a}^\dagger_P)^{k_P} % ) % \ket{\text{vac}} % \thinspace , \end{equation}\] which means that any (pure) electronic state can be written as a linear combination of creation (and annihilation) strings on top of the vacuum.

In order to allow the second-quantized framework to shine, we must be comfortable with manipulating elementary operator strings. Some useful commutators are: \[\begin{align} % & \comm{\hat{a}^\dagger_P}{ % \hat{a}_Q \hat{a}_R % } = \delta_{PQ} \hat{a}_R - \delta_{PR} \hat{a}_Q \\ % & \comm{\hat{a}_P}{ % \hat{a}^\dagger_Q \hat{a}^\dagger_R % } = \delta_{PQ} \hat{a}^\dagger_R - \delta_{PR} \hat{a}^\dagger_Q % \label{eq:non-trivial4} % \thinspace . \end{align}\]