Fock Space Definition

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

Bosons

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 $\mathbb{N}^I$:

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

A state in the Fock space $|\psi_B\rangle$ for bosons can be written as a linear combination of the occupation number basis:

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

Fermions

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 ${0,1}^I$:

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

A state in the Fock space $|\psi_F\rangle$ for fermions can be written as a linear combination of the occupation number basis:

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

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