In the book "Nolting, Theoretical Physics Part 5/2" (German), on Page 264, Formula 8.80, the author introduces second quantization in the case of identical particles. One considers the product space $(\mathcal{H}_N = \bigotimes_i \mathcal{H}_i)$, where the kets $(\vert \phi^i_j \rangle)$ represent the j-th base vector of the i-th particle. A ket of the product space of identical particles is then given by:

$[ \vert \phi_N \rangle = \vert \phi_{\alpha_1}, \phi_{\alpha_2}, \cdots, \phi_{\alpha_N} \rangle = \vert \phi_{\alpha_1}^1, \phi_{\alpha_2}^{2}, \cdots, \phi_{\alpha_N}^{N} \rangle = \vert \phi_{\alpha_1}^1 \rangle \vert \phi_{\alpha_2}^{2} \rangle \cdots \vert \phi_{\alpha_N}^{N} \rangle = \vert \phi_{\alpha_1}^1 \rangle \otimes \vert \phi_{\alpha_2}^{2} \rangle \otimes \cdots \otimes \vert \phi_{\alpha_N}^{N} \rangle ]$

To find symmetrical and antisymmetrical states, one introduces an operator (S) and considers its action on kets as (anti)symmetrizing, resulting in:

$[ \vert \phi_N \rangle ^\pm = S \vert \phi_N \rangle ]$

Here, the plus sign represents Bosonic states, and the minus sign represents Fermionic states.

In addition, occupation numbers are introduced such that $(\sum_i n_i = N).$

In a small calculation , it is shown that

$[ \langle \phi_N^\pm \vert \phi_N^\pm \rangle = \frac{1}{N!} \sum_{\mathcal{P}} (\pm)^p \langle \phi_N \vert \mathcal{P} \vert \phi_N \rangle ]$

where $p$ is the number of permutations and $(\mathcal{P})$ is a permutation operator that acts on $(\phi_N)$ such that $(\mathcal{P} \phi_N = \vert \phi_1^{i_1}, \phi_2^{i_2}, \cdots \phi_N^{i_N} \rangle).$

The book then makes the claim, which I do not understand nor have been able to derive, that:

$[ \langle \phi_N^\pm \vert \phi_N^\pm \rangle = \frac{1}{N!} \prod_{i=1}^{N} n_i! ]$

How is this calculation made? The argumentation in the book is not clear.

Additionally, according to a discussion on Math Stack Exchange, vectors need to be ordered for the scalar product to be defined. If we are permuting the upper indices, how can we evaluate such a product, especially if, for example, a state from space 1 is to be taken with a state from space 3? Does "identical particles" mean identical Hilbert spaces? Is it set to be zero?

Thank you in advance.

In the book "Nolting, Theoretical Physics Part 5/2" (German), on Page 264, Formula 8.80, the author introduces second quantization in the case of identical particles. One considers the product space $(\mathcal{H}_N = \bigotimes_i \mathcal{H}_i)$, where the kets $(\vert \phi^i_j \rangle)$ represent the j-th base vector of the i-th particle. A ket of the product space of identical particles is then given by:

\begin{align} \vert \phi_N \rangle &= \vert \phi_{\alpha_1}, \phi_{\alpha_2}, \cdots, \phi_{\alpha_N} \rangle \\ &= \vert \phi_{\alpha_1}^1, \phi_{\alpha_2}^{2}, \cdots, \phi_{\alpha_N}^{N} \rangle \\ &= \vert \phi_{\alpha_1}^1 \rangle \vert \phi_{\alpha_2}^{2} \rangle \cdots \vert \phi_{\alpha_N}^{N} \rangle \\ &= \vert \phi_{\alpha_1}^1 \rangle \otimes \vert \phi_{\alpha_2}^{2} \rangle \otimes \cdots \otimes \vert \phi_{\alpha_N}^{N} \rangle \end{align}

To find symmetrical and antisymmetrical states, one introduces an operator (S) and considers its action on kets as (anti)symmetrizing, resulting in:

$$ \vert \phi_N \rangle ^\pm = S \vert \phi_N \rangle $$

Here, the plus sign represents Bosonic states, and the minus sign represents Fermionic states.

In addition, occupation numbers are introduced such that $(\sum_i n_i = N).$

In a small calculation , it is shown that

$$ \langle \phi_N^\pm \vert \phi_N^\pm \rangle = \frac{1}{N!} \sum_{\mathcal{P}} (\pm)^p \langle \phi_N \vert \mathcal{P} \vert \phi_N \rangle $$

where $p$ is the number of permutations and $(\mathcal{P})$ is a permutation operator that acts on $(\phi_N)$ such that $(\mathcal{P} \phi_N = \vert \phi_1^{i_1}, \phi_2^{i_2}, \cdots \phi_N^{i_N} \rangle).$

The book then makes the claim, which I do not understand nor have been able to derive, that:

$$ \langle \phi_N^\pm \vert \phi_N^\pm \rangle = \frac{1}{N!} \prod_{i=1}^{N} n_i! $$

How is this calculation made? The argumentation in the book is not clear.

Additionally, according to a discussion on Math Stack Exchange, vectors need to be ordered for the scalar product to be defined. If we are permuting the upper indices, how can we evaluate such a product, especially if, for example, a state from space 1 is to be taken with a state from space 3? Does "identical particles" mean identical Hilbert spaces? Is it set to be zero?