I mean, the $\lvert \Omega \rangle$ is "some Fock state" because it lives in the same Hilbert space as the $\lvert 0\rangle$ and the Hilbert space of the latter is just the Fock space.
> The notation $\delta$-function is a wild abuse of notation. Maybe it has survived because it is so bad that the motivation for introducing the concept of distributions becomes clear.
> Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something completely different. - Goethe
"Non-Hausdorff spaces, often regarded as a technical nuisance, sometimes produce a global disaster." - Geometry of non-Hausdorff spaces and its significance for physics by M.Heller et al
The article Turing machine gives a general introduction to Turing machines, while this article covers a specific class of Turing machines.
A Post–Turing machine is a "program formulation" of an especially simple type of Turing machine, comprising a variant of Emil Post's Turing-equivalent model of computation described below. (Post's model and Turing's model, though very similar to one another, were developed independently. Turing's paper was received for publication in May 1936, followed by Post's in October.) A Post–Turing machine uses a binary alphabet, an infinite sequence of binary storage...
They seem to do this weird thing which I'm guessing is due to the lack of proof of the mass gap where they project the Hilbert space to some ugly Hilbert space
$\hat E(\{0\} \cup M_m \cup V_m) \mathscr H$
With $M_m$ the mass shell and $V_m$ the free multiparticle spectrum