Is the Fock space (generally) separable?

The Fock space is a well-known vector space “symbolically” used in many-body physics and quantum statistical mechanics; actually, it makes little sense to talk about the Fock space because it’s more like a construct around some other function space.

It is well known in quantum mechanics that if one particle’s environment is described by some complex Hilbert space V, then two of them live in a “tensor product space” which has little to do with the canonically constructed product space, i.e. in general $V \times W$ where:

$\|(v,w)\| := \|v\|_V + \|w\|_W\qquad \forall v \in V,\, w \in W$

Here – once and for all – the choice of the sum norm is arbitrary, the best choice is probably a squared sum, but it’s irrelevant here. The QM thing is slightly different; if one particle lives in, say $L^2 (\Omega;\mathbb{C})$ , then two of them live in $L^2 (\Omega \times \Omega;\mathbb{C})$ , and so on; explicitly, if

$x\mapsto u_1 (x)\quad y\mapsto u_2 (y)\qquad u_1,u_2\in V$

are two functions each describing one-particle states then

$(x,y)\mapsto u_1 (x) u_2 (y)\qquad u\in V\otimes V$

describes a (usually non-interacting) two-particle state. So, let’s put

$V^{\otimes k} := \underbrace{V \otimes \ldots \otimes V}_{\text {k times}}$

and let’s define the Fock space on V as

$\Gamma (V) := \bigoplus_{k \ge 0} V^{\otimes k}$

where conventionally $V^{\otimes 0} := \mathbb{C}$ (called the vacuum state) as the field is C. Conventionally, any element on Γ(V) is a sequence like the following

$\{f_k\}_{k\ge 0}\, : \, f_k \in V^{\otimes k}$

so that

$\|\{f_k\}\|_{\Gamma (V)} := \sum_{k\ge 0} \|f_k\|_{V^{\otimes k}}$

although it’s not technically necessary because when computing explicit stuff you basically need a finite number of non-zero elements (one, two or three, or whatever), and you definitely don’t need the norm.

Well, my question is: taking as V an infinite-dimensional separable Hilbert space, is the Fock space (non)separable ? Being a topological property I’d expect it to depend on the choice of the norm (which induces a topology), but being an infinite sum i wouldn’t expect it to be either way; it could be like thinking that ℓ^∞ is separable because it might be constructed as something like $\bigoplus_{k \ge 0} \mathbb {R}$ (with obvious meaning of the direct sum), although I’m not sure.

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kysch