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By definition, in the tensor product of Hilbert spaces $\mathscr{H}_1$ and $\mathscr{H}_2$, the two spaces are different: it is not possible to identify the creation/annihilation operators of the first space with the ones of the second. As presented by the OP, both $\psi_1$ and $\psi_2$ belong to (different subspaces of) the same full symmetric Fock space ...

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Let's look to the expression for field with mass $m$ and spin $s$ (for massless case following statements exist in similar form):  \tag 1 \hat {\psi}_{a}(x) = \sum_{\sigma = -s}^{s}\int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi )^{3} 2E_{\mathbf p}}}\left( u^{\sigma}_{a}(\mathbf p )e^{-ipx}\hat{a}_{\sigma}(\mathbf p ) + v^{\sigma}_{a}(\mathbf p ...

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"Multiplying the wavefunctions" is a pretty nebulous term. Let's work with some definite vocabulary here, shall we? $(1)$ The states of one QM particle are elements of some Hilbert space $\mathcal{H}$. If we care only about position on a line as completely defining the state (which we can for a scalar boson), i.e. demand that the space be spanned by the ...

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This quote is only for free Bose gas. For interacting Bose gas, of course the chemical potential can be above the single particle ground state. This is common in the cold atom Bose-Einstein condensates.

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