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Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H.(Wikipedia)

Question 1.How to write a Hilbert space if it is a Fock space as (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H. How to search for H, in computional-complexity sense?.Is there a general known method to search for H for any Fock space?.

Question 2.If there is not such method, could anyone give suggestions or ideas (not necessarily verified) how possibly to do that?.

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    $\begingroup$ I am perhaps still not quite understanding the issue: a Fock space is specified by a vacuum state $|0\rangle$ and a set of creation operators $a^\dagger_p$; the single-particle Hilbert space has basis states $a^\dagger_p|0\rangle$. $\endgroup$
    – Carlo Beenakker
    Commented Apr 11, 2020 at 8:49
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    $\begingroup$ How is your already-give Fock space supposed to be specified? $\endgroup$
    – gmvh
    Commented Apr 11, 2020 at 8:54
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    $\begingroup$ I assume the Fock space is formed out of all linear combinations of states $a_{p_1}^\dagger a_{p_2}^\dagger\cdots a_{p_k}^\dagger|0\rangle$, $k=1,2,\ldots$. $\endgroup$
    – Carlo Beenakker
    Commented Apr 11, 2020 at 9:00
  • $\begingroup$ I am asking how to write a Hilbert space if it is a Fock space as (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H. How to search for H?. $\endgroup$
    – Alangri
    Commented Apr 11, 2020 at 9:11
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    $\begingroup$ It seems to me that, if you are given a Fock space with its vacuum state and creation operators, then the first comment by @CarloBeenakker answers the question. If you're given only a Fock space, as an abstract Hilbert space, without a specified vacuum state and creation operators, then this is not enough information to recover the one-particle space $H$ as a subspace of Fock space. Even some non-isomorphic $H$'s can produce isomorphic Fock spaces. Example: any infinite-dimensional $H$ and any nontrivial finite-dimensional boson $H$ (i.e., use symmetric powers). $\endgroup$
    – Andreas Blass
    Commented Apr 11, 2020 at 13:30

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