With the new creation and annihilation operator we can now construct many-body quasiparticle states, with one-particle-one-hole states, two-particle-two-hole states etc in the same fashion as we previously constructed many-particle states. We can write a general particle-hole state as $$ \begin{equation} |\beta_1\beta_2\dots \beta_{n_p} \gamma_1^{-1} \gamma_2^{-1} \dots \gamma_{n_h}^{-1}\rangle \equiv \underbrace{b_{\beta_1}^\dagger b_{\beta_2}^\dagger \dots b_{\beta_{n_p}}^\dagger}_{>F} \underbrace{b_{\gamma_1}^\dagger b_{\gamma_2}^\dagger \dots b_{\gamma_{n_h}}^\dagger}_{\leq F} |c\rangle \tag{72} \end{equation} $$ We can now rewrite our one-body and two-body operators in terms of the new creation and annihilation operators. The number operator becomes $$ \begin{equation} \hat{N} = \sum_\alpha a_\alpha^\dagger a_\alpha= \sum_{\alpha > F} b_\alpha^\dagger b_\alpha + n_c - \sum_{\alpha \leq F} b_\alpha^\dagger b_\alpha \tag{73} \end{equation} $$ where \( n_c \) is the number of particle in the new vacuum state \( |c\rangle \). The action of \( \hat{N} \) on a many-body state results in $$ \begin{equation} N |\beta_1\beta_2\dots \beta_{n_p} \gamma_1^{-1} \gamma_2^{-1} \dots \gamma_{n_h}^{-1}\rangle = (n_p + n_c - n_h) |\beta_1\beta_2\dots \beta_{n_p} \gamma_1^{-1} \gamma_2^{-1} \dots \gamma_{n_h}^{-1}\rangle \tag{74} \end{equation} $$ Here \( n=n_p +n_c - n_h \) is the total number of particles in the quasi-particle state of Eq. (72). Note that \( \hat{N} \) counts the total number of particles present $$ \begin{equation} N_{qp} = \sum_\alpha b_\alpha^\dagger b_\alpha, \tag{75} \end{equation} $$ gives us the number of quasi-particles as can be seen by computing $$ \begin{equation} N_{qp}= |\beta_1\beta_2\dots \beta_{n_p} \gamma_1^{-1} \gamma_2^{-1} \dots \gamma_{n_h}^{-1}\rangle = (n_p + n_h)|\beta_1\beta_2\dots \beta_{n_p} \gamma_1^{-1} \gamma_2^{-1} \dots \gamma_{n_h}^{-1}\rangle \tag{76} \end{equation} $$ where \( n_{qp} = n_p + n_h \) is the total number of quasi-particles.