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 $$\begin{equation} \langle \alpha_1\alpha_2 \dots \alpha_n|a_\alpha = \langle\alpha\alpha_1\alpha_2 \dots \alpha_n| \tag{13} \end{equation}$$