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The Fock space is defined as the direct sum of all $n$-particle Hilbert spaces. Are Hilbert space vectors also Fock space vectors or are they just isomorphic to Fock space vectors?

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  • $\begingroup$ Could you elaborate the difference between this question and this one (a question by OP)?! $\endgroup$ Mar 10, 2022 at 12:08

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Correct me if I am wrong, but I think it can be thought of through a simpler case. What I mean is that your question is the same as asking if $a \in \mathbb{R}$ is also an element of $\mathbb{R}^n = \bigoplus^n_{i=0} \mathbb{R}$. Obviously no, because $\mathbb{R}^n$ is a vector space, which means that if $a$ were an element of the n-dimensional space then we would need a way to add a scalar and an n-dimensional vector, which in most cases is not logical (emphasis on in most cases).

However we can represent $a$ in a way that belongs to $\mathbb{R}^n$, but that we can agree to identify with $a$. For example, if $a = 1$ we might say that in $\mathbb{R}^3$ $a = (0, 0, 1)$.

In short, yes, they are isomorphic.

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