An appropriately symmetrized (and normalized) $N$-particle physical ket can be expressed in the form: $$| \beta_{1}, \beta_{2}, \dots, \beta_{N} \rangle \equiv \frac{1}{\sqrt{N! \prod_{i=1}^{\infty}(n_{\beta_{i}}!)}} \sum_{\mathcal{P} \in S_{N}} \zeta^{[1-\mathrm{sgn}(\mathcal{P})]/2} \, \mathcal{P} (|\beta_{1}\rangle \otimes |\beta_{2} \rangle \otimes \dots \otimes |\beta_{N} \rangle)\,,$$ with $\sum_{i=1}^{\infty} n_{i} = N$. In this equation:
- $n_{\beta_{i}}$ is the occupation number of the state $|\beta_{i}\rangle$ in the set of states $\{|\beta_{1}\rangle, |\beta_{2}\rangle, \dots, |\beta_{N}\rangle\}$
- for fermions $\zeta = -1$, while for bosons $\zeta = 1$
- $\mathcal{P}$ is a permutation operator belonging the the symmetric group on $N$ letters, $S_{N}$, and acting on the tensor product state $|\beta_{1}\rangle \otimes |\beta_{2} \rangle \otimes \dots \otimes |\beta_{N} \rangle$
- $\mathrm{sgn}(\mathcal{P})$ is the sign of the permutation $\mathcal{P}$, being +1 if $\mathcal{P}$ is even or -1 if it is odd.
In the book by Negele, J.W. and Orland, H., Quantum Many-Particle Systems, the creation operator $a_{\lambda}^{\dagger}$ is defined in a Fock space as: $$a_{\lambda}^{\dagger}| \beta_{1}, \beta_{2}, \dots, \beta_{N} \rangle \equiv \sqrt{n_{\lambda} + 1}\,| \lambda, \beta_{1}, \beta_{2}, \dots, \beta_{N} \rangle\,,$$ where $n_{\lambda}$ is the occupation number of the state $|\lambda\rangle$ in $| \beta_{1}, \beta_{2}, \dots, \beta_{N} \rangle$.
Following this definition, the annihilation operator $a_{\lambda}$ is simply the adjoint of $a_{\lambda}^{\dagger}$, and its action on a many-particle state is determined using the closure relation in the Fock space:
$$|0\rangle\langle0|+\sum_{M=1}^{\infty}\frac{1}{M!}\sum_{\alpha_{1}, \dots, \alpha_{M}} \prod_{i=1}^{\infty} (n_{\beta_{i}}!)\,|\alpha_{1}, \alpha_{2}, \dots, \alpha_{M}\rangle\langle\alpha_{1}, \alpha_{2}, \dots, \alpha_{M}| = \mathbb{1}\,.$$
Now, the problem is that when I try to derive an expression for $a_{\lambda}$, I am not able to obtain the prefactor $\frac{1}{\sqrt{n_{\lambda}}}$, as in equation 1.78b of the aforementioned book, which reads:
$$a_{\lambda} |\beta_{1}, \beta_{2}, \dots, \beta_{N} \rangle = \frac{1}{\sqrt{n_{\lambda}}} \sum_{i = 1}^{N} \zeta^{i-1} \delta_{\lambda, \beta_{i}} |\beta_{1}, \dots, \overline{\beta_{i}}, \dots, \beta_{N}\rangle\,,$$ in which $\delta$ is the Kronecker delta and $\overline{\beta_{i}}$ indicates that the state $|\beta_{i}\rangle$ has been removed from the $N$-particle state $|\beta_{1}, \beta_{2}, \dots, \beta_{N}\rangle$.
Any help? Thanks.