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Say I have two separate systems of identical Bosons, one with N Bosons the other with M. System one is described by a state $|\psi_1\rangle$ the other with $|\psi_2 \rangle$ which are expressed in a Fock space like

$|\psi_1\rangle = \sum_{n_1,...,n_{max}} \alpha(n_1,..,n_{max}) |n_1,n_2,..,n_{max}\rangle$

$|\psi_2\rangle = \sum_{n_1,...,n_{max}} \beta(n_1,..,n_{max}) |n_1,n_2,..,n_{max}\rangle$

where $|n_1,n_2,..,n_{max}>=\prod_{k=1}^{max} \frac{(a^{\dagger}_k)^{n_k}}{\sqrt{n_k!}} |vac\rangle$

with "max" denoting the maximum occupied mode, $\alpha$ and $\beta$ some constants depending on each of the values (zero if $\sum_{k} n_k$ is not equal to $N$ for $\alpha$ or $M$ for $\beta$) and the wavefunction satisfying all the usual normalisation conditions.

At someone point I wish to bring these two subsystems together, this state can be expressed as an $N+M$ body Fock space.

$|\psi_{total}\rangle = |\psi_1\rangle \otimes|\psi_2\rangle$

For distinguishable particles this is fairly trivial, however the symmetry makes it somewhat unclear (to me) how to do this and give states with the appropriate amplitudes.

Can anyone tell me, or point me to an appropriate book or paper?

David Holdaway
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2 Answers2

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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 $\Gamma_s(\mathscr{H})$; furthermore $\langle \psi_1,\psi_2\rangle_{\Gamma_s(\mathscr{H})}\neq 0$ in general. The way to "combine" two such vectors into a single state is either by linear combination $\psi_1+\psi_2$ or constructing a mixed state $\lvert\psi_1\rangle\langle\psi_1\rvert + \lvert\psi_2\rangle\langle\psi_2\rvert$ in my opinion.

If the Hilbert spaces are different, we can take the tensor product without problems of symmetrization. Also, the following isomorphism holds between (complete) symmetric Fock spaces: $$\Gamma_s(\mathscr{H}_1)\otimes\Gamma_s(\mathscr{H}_2)\simeq \Gamma_s(\mathscr{H}_1\oplus\mathscr{H}_2)\; .$$

yuggib
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Possible answer to the question (if anyone could confirm this that would be great)

$$ |\psi_1⟩⊗|\psi_2⟩ \propto \sum_{m_1,..,m_{max},n_1,...,n_{max}} \beta(m_1,..)\alpha(n_1,..) \prod_{k=1}^{max} \frac{(a^{\dagger}_k)^{n_k+m_k}}{\sqrt{n_k! m_k!}} |vac \rangle$$

Which seems to give the correct weighting with symmetry but does require a correction to the normalisation.

David Holdaway
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