# Questions about Fock space and direct sum

I am a little bit confused with the concept of a Fock space and hope for some clarification.

In general a Fock space seems to be constructed as the direct sum of $$n$$-particle Hilbert spaces. What bothers me, is that there are two types of the direct sum (internal and external) but I am not sure, which of these applies to the definition of the Fock space.

Edit: To put the question in other words: Are $$n$$-particle Hilbert spaces subspaces of the Fock space or not?

• What do you mean by internal and external direct sums? Can you give an example of what you have in mind? I might be ill-informed but I haven't come across this terminology.
– user87745
Mar 3, 2022 at 16:19
• Yes, the $𝑛$-particle Hilbert spaces subspaces of the Fock space Mar 3, 2022 at 16:20

Let $$U_1$$ and $$U_2$$ be vector spaces and define $$V = U_1 \oplus U_2$$ (i.e., $$V$$ is the external direct sum of $$U_1$$ and $$U_2$$). Then $$U_1$$ is isomorphic to a subspace $$V_1$$ of $$V$$ (given by the elements of the form $$(u,0)$$) and $$U_2$$ is isomorphic to a subspace $$V_2$$ of $$V$$. One can then show that $$V = V_1 \oplus V_2$$ (internal direct sum).
Hence, both notions apply to the definition of a Fock space, since they are just different ways of saying the same thing. Of course, you'll use one definition or the other depending on convenience, but they turn out to be the same thing. To address the question as posed in your edit, the $$n$$-particle Hilbert spaces are (isomorphic to) subspaces of the Fock space.
• Thank you for your answer! I just want to be sure, wether or not I fully understand the consequences of your last sentence: if the $n$-particle Hilbert spaces are "interpreted" as subspaces of the Fock space, then every $n$-particle state of the corresponding Hilbert space itself is a Fock state. If the Fock space is interpreted as the external direct sum, $n$-particle state of the corresponding Hilbert space are not the same as Fock states, but are isomorphic to them. Is this correct? Mar 3, 2022 at 18:00
• @maxxam The isomorphism is a mere technicality. For example, are the real numbers a subset of the complex numbers? Technically no, because the complex numbers are defined as $\mathbb{R}^2$ with a product, and the elements $(x,0) \in \mathbb{C}$ are then not equal to the elements $x \in \mathbb{R}$. However, this is just a technical detail: for all algebraic purposes we can treat $\mathbb{R}$ as a subspace of $\mathbb{C}$. This is the meaning of an isomorphism. In the Fock space case, it is similar. The $n$-particle Hilbert spaces are subspaces of the Fock space in the same sense in which the + Mar 3, 2022 at 19:19