What is the physical interpretation of the operator \( a_\alpha \) and what is the effect of \( a_\alpha \) on a given state \( |\alpha_1\alpha_2\dots\alpha_n\rangle_{\mathrm{AS}} \)? Consider the following matrix element $$ \begin{equation} \langle\alpha_1\alpha_2 \dots \alpha_n|a_\alpha|\alpha_1'\alpha_2' \dots \alpha_m'\rangle \tag{11} \end{equation} $$ where both sides are antisymmetric. We distinguish between two cases. The first (1) is when \( \alpha \in \{\alpha_i\} \). Using the Pauli principle of Eq. (6) it follows $$ \begin{equation} \langle\alpha_1\alpha_2 \dots \alpha_n|a_\alpha = 0 \tag{12} \end{equation} $$ The second (2) case is when \( \alpha \notin \{\alpha_i\} \). It follows that an hermitian conjugation $$ \begin{equation} \langle \alpha_1\alpha_2 \dots \alpha_n|a_\alpha = \langle\alpha\alpha_1\alpha_2 \dots \alpha_n| \tag{13} \end{equation} $$