Fermionic field operators
General spinor basis
In an orthonormal, general spinor basis \(\require{physics} \{ \phi_P(\vb{r}) \}\), we will introduce the so-called fermionic field operators as having two components, one for each spin quantum number \(\sigma\): \[\begin{equation} \hat{\phi}_\sigma(\vb{r}) = \sum_P \phi^\sigma_P(\vb{r}) \hat{a}_P \thinspace . \end{equation}\] The adjoint operator \[\begin{equation} \hat{\phi}^\dagger_\sigma(\vb{r}) = \sum_P \phi^{\sigma *}_P(\vb{r}) \hat{a}^\dagger_P \thinspace , \end{equation}\] where \(\hat{a}^\dagger_P\) and \(\hat{a}_P\) are the elementary creation and annihilation operators that give rise to a Fock space, creates a particle at the point \(\vb{r}\) with spin \(\sigma\): \[\begin{align} \phi^\dagger_\sigma(\vb{r}) \ket{\text{vac}} &= \sum_P \phi_P^{\sigma *} \hat{a}^\dagger_P \ket{\text{vac}} \\ &= \sum_P \braket{\phi_P}{\vb{r}, \sigma} \ket{\phi_P} \\ &= \ket{\vb{r}, \sigma} \thinspace . \end{align}\]
Since the Pauli spinor basis necessarily consists of two-component vectors, i.e. \[\begin{equation} \phi_P(\vb{r}) = \begin{pmatrix} \phi_{P \alpha}(\vb{r}) \\ \phi_{P \beta}(\vb{r}) \end{pmatrix} \thinspace , \end{equation}\] the related field operators must also be decribed by two components. The field annihilation operator can then be written as: \[\begin{equation} \hat{\phi}(\vb{r}) = \begin{pmatrix} \hat{\phi}_{\alpha}(\vb{r}) \\ \hat{\phi}_{\beta}(\vb{r}) \end{pmatrix} \thinspace , \end{equation}\] where \[\begin{equation} \hat{\phi}_\sigma(\vb{r}) = \sum_P \phi_{P \sigma}(\vb{r}) \hat{a}_P \end{equation}\] and \(\sigma\) (or \(\tau\)) labels \(\alpha\) or \(\beta\). These two components satisfy the anticommutation relations (Peskin and Schroeder 1995): \[\begin{equation} \comm{ \hat{\phi}_\sigma(\vb{r}) }{ \hat{\phi}_\tau^\dagger(\vb{r}') }_+ = \delta_{\sigma \tau} \delta(\vb{r} - \vb{r}') \end{equation}\] and \[\begin{equation} \comm{ \hat{\phi}_\sigma(\vb{r}) }{ \hat{\phi}_\tau(\vb{r}') }_+ = 0 \thinspace . \end{equation}\]
Due to the orthonormality of the general spinor basis \(\{ \phi_P(\vb{r}) \}\), we can show that the elementary creation and annihilation operators are the projections of the field creation and annihilation operators onto that given spinor basis: \[\begin{align} & \hat{a}^\dagger_P = \int \dd{\vb{r}} \hat{\phi}^\dagger(\vb{r}) \phi_P(\vb{r}) = \sum_\sigma \int \dd{\vb{r}} \hat{\phi}_\sigma^\dagger(\vb{r}) \phi_{P \sigma}(\vb{r}) \\ % & \hat{a}_P = \int \dd{\vb{r}} \phi_P^\dagger(\vb{r}) \hat{\phi}(\vb{r}) = \sum_\sigma \int \dd{\vb{r}} \phi_{P \sigma}^*(\vb{r}) \hat{\phi}_\sigma(\vb{r}) \thinspace . \end{align}\] From these two relations, we can show that the standard elementary anticommutator relations follow: \[\begin{equation} \comm{ \hat{a}_P }{ \hat{a}^\dagger_Q }_+ = \int \dd{\vb{r}} \phi_P^\dagger(\vb{r}) \phi_Q(\vb{r}) = \delta_{PQ} \end{equation}\] and \[\begin{equation} \comm{ \hat{a}_P }{ \hat{a}_Q }_+ = 0 \thinspace . \end{equation}\]
Using these field operators, we may quantize one- and two-electron operators.
Furthermore, looking at these previous equations suggests the component-wise elementary operator definitions \[\begin{align} & \hat{a}^\dagger_{P \sigma} = \int \dd{\vb{r}} \hat{\phi}_\sigma^\dagger(\vb{r}) \phi_{P \sigma}(\vb{r}) \\ % & \hat{a}_{P \sigma} = \int \dd{\vb{r}} \phi_{P \sigma}^\dagger(\vb{r}) \hat{\phi}_\sigma(\vb{r}) \thinspace , \end{align}\] but these do not have the required anticommutators to build an ONV theory: \[\begin{equation} \comm{ \hat{a}_{P \sigma} }{ \hat{a}^\dagger_{Q \tau} }_+ = \delta_{\sigma \tau} \int \dd{\vb{r}} \phi_{P \sigma}^*(\vb{r}) \phi_{Q \tau}(\vb{r}) \thinspace , \end{equation}\] because in a general orthonormal spinor basis (where the spinor basis has two non-zero components), we have no idea if the scalar product for the spinor components separately satisfies orthonormality.
Spin-orbital basis
However, in a spin-orbital basis, these component-wise elementary operators can build up an ONV theory, since the spin-orbitals now admit the following orthonormality condition: \[\begin{equation} \int \dd{\vb{r}} \phi^*_{p \sigma}(\vb{r}) \phi_{q \tau}(\vb{r}) = \delta_{\sigma \tau} \delta_{pq} \thinspace , \end{equation}\] which means that: \[\begin{equation} \comm{ \hat{a}_{p \sigma} }{ \hat{a}^\dagger_{q \tau} }_+ = \delta_{\sigma \tau} \delta_{pq} \thinspace . \end{equation}\] Therefore, the fermionic field operators may now take the following form: \[\begin{equation} \hat{\phi}^\dagger(\vb{r}) = \sum_\sigma \sum_p \phi^*_{p \sigma}(\vb{r}) \hat{a}^\dagger_{p \sigma} \thinspace , \end{equation}\] and \[\begin{equation} \hat{\phi}(\vb{r}) = \sum_\sigma \sum_p \phi_{p \sigma}(\vb{r}) \hat{a}_{p \sigma} \thinspace . \end{equation}\] The elementary creation operator \(\hat{a}^\dagger_{p \sigma}\) then creates an electron in an \(\sigma\)-spin-orbital, a situation we commonly refer to as creating a \(\sigma\)-electron.
Using these field operators, we may quantize one- and twoelectron operators.-