**Short version of the question:**
Can someone explains what physicist's 'Spin-statistic theorem' says rigorously in the context of Free Quantum fields when they are describe as the Fock space of some '1-particle space' ?

To be precise the question *is not really about the spin-statistic theorem* but more "Why taking a Fock space of a "1-particle" representation of the Poincarré group sometimes give a Quantum field and sometime doesn't, depending on whether the Spin of the 1-particle space and the type of Fock space under consideration are compatible with the Spin-statistic theorem."


**For more details:**

When reading text about Quantum field theory written by mathematicians (for eg, [this][1]), I got the impression that free Quantum fields are well understood objects and can be described relatively simply as follows (that's literally their definition in the reference mentioned):

You start from a "one particle Hilbert space" $H$.

For a relativistic particle, it will be a (often irreducible) unitary representation of $\Lambda$ the universal cover of the Poincaré group. Such representation have been [classified][2] by [Wigner][3], and if one exclude the "non-local" and non-physical ones, they are essentially classified by a mass $m \in \mathbb{R}$ and a spin $s \in \frac{1}{2} \mathbb{N}$ (with some subtleties in the $m=0$ case that I'm ignoring).

You then apply 'second quantization'. That is you form either the Bosonic Fock space $F_+$ or the Fermionic Fock space $F_-$

$$F_\pm = \sum_{n=0}^\infty P_{\pm} \left( H^{\otimes n} \right) $$

where $P_\pm$ is either the projection on symetric or antisymetric tensor.

$$ P_+ (v_1 \otimes \dots \otimes v_n )= \frac{1}{n!}\sum_{\sigma \in S_n} v_{\sigma 1} \otimes \dots \otimes v_{\sigma n } $$ $$ P_- (v_1 \otimes \dots \otimes v_n )= \frac{1}{n!}\sum_{\sigma \in S_n} sg(\sigma) v_{\sigma 1} \otimes \dots \otimes v_{\sigma n } $$

These comes with a lots of structure: they have a "vacuum vector", creation and annihilation operators out of which you can define field operators, they have a "component-wise" action of $\Lambda$ that encodes all the usual physical concept. In particular they have a Hamiltonian obtained as the infinitesimal generator of the time translation.

If I exaggerate a bit, that is essentially all I understand of Quantum field theory (at least that is the only part I know how to make mathematically rigorous).

Now when I read instead textbook written by physicist, *I 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][4]".

It claims that we can only consider the Fermionic Fock space $F_-$ when $H$ is the Hilbert space of a half-integer spin particule and only the Bosonic Fock space $F_+$ when $H$ represents 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. That obviously confirm that the previous point of view is missing something, but what ? can it be made complete ?


So I tend to assume that there is some physically significant mathematical property (or maybe structure ?) that distinguishes between the "Free Quantum fields" (in the sense above) that satisfies the Spin-statistic theorem and these that don't. And I would like to know which ones.

To rule out some obvious physical property that one expect:

- All these "attempted free quantum fields" are representation of $\Lambda$, so they are "Poincaré invariant", the void vector is invariant for this action in all cases.

- Unless I misunderstanding something about what it means, these are 'local' as soon as the 1 particular space $H$ we started from is: in the free fields the time evolution is just diagonal on $H^{\otimes n}$, so as soon as each single particle behave in a local way, it extends to the Quantum fields. This is the case off the representation of finite mass real mass of the $\Lambda$.

- The spectrum of the hamiltonian is positive as soon as this holds for the one particle space we started from.




I know that in practice physicist do not always start from an irreducible representation $H$ of $\Lambda$, for example the Dirac equation describe the sum of two irreducible representations of $\Lambda$ : the mass $m$ spin $\frac{1}{2}$ and the mass $-m$ spin $\frac{1}{2}$. And that might play a role in the story, but the 'how' is still unclear to me.


  [1]: https://www.jstor.org/stable/j.ctt7ztswr
  [2]: https://en.wikipedia.org/wiki/Wigner%27s_classification
  [3]: https://www.jstor.org/stable/1968551?seq=1
  [4]: https://en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem