# Basis for a tensor product of Fock spaces

Let $$H, K$$ be two Hilbert spaces with orthonormal bases $$\{e_{\alpha}\}, \{f_{\beta}\}$$, respectively. Now, I already found a result proving that a basis for the Fermi Fock space generated from $$H$$, denoted as $$\mathscr{F}_-(H)$$ (i.e. the anti-symmetrised Fock space), is given by $$a^{\dagger}_H(e_{\alpha_1}) \cdots a^{\dagger}_H(e_{\alpha_1}) \Omega_H,$$ when $$\{e_{\alpha_1}, \ldots, e_{\alpha_n}\}$$ runs over finite subsets of $$\{e_{\alpha}\}$$. In the above, $$a^{\dagger}_H$$ and $$\Omega_H$$ are the creation operators and vacuum vector on $$\mathscr{F}_-(H)$$, respectively. Now, I want to build an orthonormal basis for the space $$\mathscr{F}_-(H) \otimes \mathscr{F}_-(K)$$. My first step to this was to formulate creation/annihilation operators on $$\mathscr{F}_-(H) \otimes \mathscr{F}_-(K)$$ that obey the CAR relations. These are given by

$$b^{\dagger}(h,k) := \frac{1}{\sqrt{2}}\left(a^{\dagger}_H(h) \otimes 1_K + \left(1_H\right)^N\otimes a_K^{\dagger}(k)\right),$$

where $$h\in H, k\in K$$ and $$N$$ is the particle number operator on $$H$$ and $$1_H, 1_K$$ are the identities on the subscripted spaces. I can show that all vectors of the form $$b^{\dagger}(e_{\alpha_1}, f_{\beta_1}) \cdots b^{\dagger}(e_{\alpha_n}, f_{\beta_n})\Omega_H\otimes \Omega_K$$ are orthonormal. What I want to prove now is that the span of these vectors is dense in $$\mathscr{F}_-(H) \otimes \mathscr{F}_-(K)$$ when $$\{ (e_{\alpha_1}, f_{\beta_1}, \ldots, (e_{\alpha_n}, f_{\beta_n})\}$$ runs over finite subsets of $$\{(e_{\alpha}, f_{\beta})\}$$. To me it makes sense that this should be the case since the set is built from the creation operators on the space, but I'm struggling to rigorously prove it.

• Do you want a basis (a dense set of linearly independent vectors) for the tensor product of the Fock spaces or for the Fock space of the tensor product?. In the former case orthonormal sets are constructed in the usual way. – lcv Dec 14 '18 at 23:39
• I'm looking for the former case, but with particular connection to the creation operators of the space – CS1994 Dec 15 '18 at 12:26