# Understanding the definition Fock space

In my lecture we defined the Fock space as follows:

Let $$\mathfrak{h}_1,\mathfrak{h}_2,\dots$$ be a sequence of separable Hilbert spaces. Let $$\mathcal{H}_N=\mathfrak{h}_1\otimes \dots \otimes \mathfrak{h}_N$$ and $$\mathcal{H}_0:= \mathbb{C}$$. Then $$\begin{equation} \mathcal{F}:= \bigoplus_{N=0}^{\infty} \mathcal{H}_N \end{equation}$$ is called Fock space.

I asked myself why we have to set $$\mathcal{H}_0:= \mathbb{C}$$?

• I don't actually know the answer, but since quantum states are only defined up to a complex number presumably all states in $\mathcal H_0$ are in the same equivalence class and so physically there is only one physically unique state in the zero particle space. May 2, 2021 at 12:40
• Well, $\mathcal{H}_0$ is a one-dimensional (complex) vector space. Hence, it is isomorphic to $\mathbb{C}$ as a vector space over itself (i.e. over the field of complex numbers). May 2, 2021 at 13:19

The no-particle vacuum state is one dimensional. Any one-dimensional vector space is isomorphic to $${\mathbb C}$$.