The first state has one additional particle with respect to the new vacuum state \( |c\rangle \) and is normally referred to as a one-particle state or one particle added to the many-body reference state. The second state has one particle less than the reference vacuum state \( |c\rangle \) and is referred to as a one-hole state. When dealing with a new reference state it is often convenient to introduce new creation and annihilation operators since we have from Eq. (65) $$ \begin{equation} a_\alpha |c\rangle \neq 0 \tag{66} \end{equation} $$ since \( \alpha \) is contained in \( |c\rangle \), while for the true vacuum we have \( a_\alpha |0\rangle = 0 \) for all \( \alpha \).

The new reference state leads to the definition of new creation and annihilation operators which satisfy the following relations $$ \begin{eqnarray} b_\alpha |c\rangle &=& 0 \tag{67} \\ \{b_\alpha^\dagger , b_\beta^\dagger \} = \{b_\alpha , b_\beta \} &=& 0 \nonumber \\ \{b_\alpha^\dagger , b_\beta \} &=& \delta_{\alpha \beta} \tag{68} \end{eqnarray} $$ We assume also that the new reference state is properly normalized $$ \begin{equation} \langle c | c \rangle = 1 \tag{69} \end{equation} $$