We can summarize our findings in Eqs. (22) and (26) as $$ \begin{equation} \{a_\alpha^{\dagger},a_\beta \} = \delta_{\alpha\beta} \tag{27} \end{equation} $$ with \( \delta_{\alpha\beta} \) is the Kronecker \( \delta \)-symbol.

The properties of the creation and annihilation operators can be summarized as (for fermions) $$ a_\alpha^{\dagger}|0\rangle \equiv |\alpha\rangle, $$ and $$ a_\alpha^{\dagger}|\alpha_1\dots \alpha_n\rangle_{\mathrm{AS}} \equiv |\alpha\alpha_1\dots \alpha_n\rangle_{\mathrm{AS}}. $$ from which follows $$ |\alpha_1\dots \alpha_n\rangle_{\mathrm{AS}} = a_{\alpha_1}^{\dagger} a_{\alpha_2}^{\dagger} \dots a_{\alpha_n}^{\dagger} |0\rangle. $$