Short version of the question: Can someone explains what physicist's 'Spin-statistic theorem' says rigorously in the context of Free Quantum fields ?

Contrary to general (interacting) Quantum fields, the mathematical description of Free quantum fields, is as far as I know, fully rigorous and not too complicated. So it should be possible ?

But, at least in my understanding of what is a Free quantum fields, I don't see anything that prevent us from considering the Bosonic fock space of a fermion, or conversely the Fermionic Fock space of a Boson. Can we point to a precise mathematical property that makes these non well behaved or 'non-physical' ? My understanding (from trying to parse through physic books) is that it has to do with the spectrum of Hamiltonian not being bounded below, but I haven't been able to make this precise.

For more details:

So, reading text about Quantum field theory written by mathematicians (for eg, this), I got the impression that free Quantum fields are well understood objects and can be described relatively simply as follows:

You start from a "one particle Hilbert space" H, which is a (often irreducible) representation of the Poincaré group (or rather its universal cover) corresponding to a particle of mass m and spin s (in terms of Wigner's classification of irreducible representations of the Poincarré group)

You then apply 'second quantization' or form the 'Free quantum field over H'. That is you form either the Bosonic (i.e. symetric) or Fermionic (i.e. anti-symetric) Fock space over H, which still carries a representation of the Poincaré group (or rather its universal cover). This gives the "many particules" Hilbert space.

You can also give an equivalent realization of this Hilbert space space in terms of tame functions on a real part of the space H to get something that looks more like a 'quantized field', but mathematically speaking that's just a different presentation of the same object.

So, If I exaggerate a bit, that's essentially all I understand about Quantum field theory. So maybe the following question is a bit naive

Now when I read physicist (or to be fair, try to read), I always got the impression that the point of view above is too general, or maybe is missing some important subtleties.

A precise point where this really appears, is with the "Spin-statistic theorem". It claims that we can only consider the Fermionic Fock space of a half-integer spin particule and the Bosonic Fock space of an integer spin particule.

But, in terms of the description above, I see no clear reasons for this to be the case : I have no problems considering either type of Fock space of either type of representations. I can imagine two options:

1. Either the Spin-statistic theorem is something that only appears when we consider interacting fields.

2. There is some important property that distinguishes between the Free Quantum fields that satisfies the Spin-statistic theorem and these that don't, and that somehow make the second class "unphysical".