FockSpace

We start by defining a set $I$ which indexes the single-particle states.

For bosons, a state in the Fock space is defined by a sequence $(n_i)_{i \in I}$ where each $n_i$, the occupation number for state $i$, is a non-negative integer. We define the set of all occupation number sequences for bosons as $\mathbb{N}^I$.

Then, the bosonic Fock space $F_B$ is defined as the set of all square summable sequences of complex numbers indexed by $\mathbb{N}^I$:

$$ F_B = \{(c_n)_{n \in \mathbb{N}^I} : \sum |c_n|^2 < \infty \} $$

For fermions, a state in the Fock space is defined by a sequence $(n_i)_{i \in I}$ where each $n_i$ is in $\{0,1\}$. We define the set of all occupation number sequences for fermions as $\{0,1\}^I$.

Then, the fermionic Fock space $F_F$ is defined as the set of all square summable sequences of complex numbers indexed by $\{0,1\}^I$:

$$ F_F = \{(c_n)_{n \in \{0,1\}^I} : \sum |c_n|^2 < \infty \} $$

In these formalisms, a state in the Fock space $|\psi\rangle$ for bosons or fermions can be written as a linear combination of the occupation number basis:

For bosons: $$|\psi_B\rangle = \sum_{n \in \mathbb{N}^I} c_n |n\rangle$$

And for fermions: $$|\psi_F\rangle = \sum_{n \in \{0,1\}^I} c_n |n\rangle$$

Here, $c_n$ is a complex number associated with each occupation number sequence $n$, and the sum runs over all possible sequences. The condition $\sum |c_n|^2 &lt; \infty$ ensures that the sequence is square summable, meaning that it corresponds to a state with finite norm in the Hilbert space.