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I'm having some issues with a piece of my notes. (relevant pages attached)

First we introduce an isomorphism ##U = \oplus_n U_n## from ##\Gamma^{(a)s}\left(\mathcal{H}_1\oplus\mathcal{H}_2\right)## to ##\Gamma^{(a)s}\left(\mathcal{H}_1\right)\otimes\Gamma^{(a)s}\left(\mathcal{H}_2\right)##

With the ##U_n## mapping the n-particle space (layer if you like) of the Fock space to the product space.

So far I'm not seeing any real problems (this is what happens on the first page of the pdf).

The next page is where things get vague for me (lets ignore the paragraph about the Gibbs paradox until I get what's being defined)

We look at ##n## indistinguishable particles, each with the same single particle Hilbert space ##\mathcal{H}##.

Now comes the first troubling part for me

Okay in a low density regime we can indeed look at such a system without too much loss of generality. "For very low densities once could restrict attention to a single particle version of the system. We would then use the Hilbert space ##\oplus_n \mathcal{H}## instead of [the product space] ##\mathcal{H}^{\otimes n}##"

Now the big problem, equation (45) in the PDF says the following (for bosons)

##Ua^\dagger(\phi_1\oplus\phi_2)U^\dagger = a^\dagger(\phi_1)\otimes {1\!\!1} +{1\!\!1}\otimes a^\dagger(\phi_2)##

When the ##U_n## which build ##U## are defined in eq. (40-41) I see

##U_1(\phi_1\oplus\phi_2) = \phi_1\oplus\phi_2## is this shorthand for ##\phi_1 \otimes {1\!\!1}\oplus {1\!\!1}\otimes\phi_2##?

If so how do they look at ##a^\dagger(\phi_1\oplus\phi_2)U^\dagger##?

If I can get my head around this part I can finish the rest but it's just not coming to me.

Any recommended books on the topic?

Thanks,

Joris

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# Product Fock spaces

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