quantum field theory - Meaning of Fock Space

In a book, it says, Fock space is defined as the direct sum of all $n$-body Hilbert Space:

$$F=H^0\bigoplus H^1\bigoplus ... \bigoplus H^N$$

Does it mean that it is just "collecting"/"adding" all the states in each Hilbert space? I am learning 2nd quantization, that's why I put this in Physics instead of math.

Answer

Suppose you have a system described by a Hilbert space $H$, for example a single particle. The Hilbert space of two non-interacting particles of the same type as that described by $H$ is simply the tensor (aka direct) product

$$H^2 := H \otimes H$$

More generally, for a system of $N$ particles as above, the Hilbert space is

$$H^N := \underbrace{H\otimes\cdots\otimes H}_{N\text{ times}},$$

with $H^0$ defined as $\mathbb C$ (i.e. the field underlying $H$).

In QFT there are operators that intertwine the different $H^N$s, that is , create and annihilate particles. Typical examples are the creation and annihilation operators $a^*$ and $a$. Instead of defining them in terms of their action on each pair of $H^N$ and $H^M$, one is allowed to give a "comprehensive" definition on the larger Hilbert space

$$\Gamma(H):=\mathbb C\oplus H\oplus H^2\oplus\cdots\oplus H^N\oplus\cdots$$

known as the Fock Hilbert space of $H$.

From a physical point of view, the general definition above of Fock space is immaterial. Identical particles are known to observe a definite (para)statistics that will reduce the actual Hilbert space (by symmetrisation/antisymmetrisation for the bosonic/fermionic case etc...).