Return to Question

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 field, I don't see anything that prevent us from considering the Bosonic fock space of a half integer spin particules, nor conversely the Fermionic Fock space of an integer spin particle. 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.

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:

For a relativistic particule, it will be a (often irreducible) unitary representation of $\Lambda$ the universal cover of the Poincaré group. Such representation have been classified by Wigner, 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).

These comes with a lots of structure: they have a "vacuum" vectors, their creation and annihilation operators, they have a "component-wise" action of $\Lambda$ that encodes all the usual physical concept (impulsion, Hamiltonian, etc...).

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.

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. Of course, this theorem has some assumptions, but they are often written in a very "physical" way, and at least if my understanding of what they should mean mathematically speaking is correct, they are all satisfied by free quantum fields.

So I can imagine two options:

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

2. There is some physically significant mathematical property that distinguishes between the Free Quantum fields that satisfies the Spin-statistic theorem and these that don't. In which case, I would like to know which ones.

• these are all representation of $\Lambda$, so they are "Poincaré invariant", the void vector is invariant for this action in all cases.

• Unless I misunderstanding something, they 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 $\Lambda$.

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."

When 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 (that's literally their definition in the reference mentioned):

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 by Wigner, 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).

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.

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.

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.

• 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.

A precise point where this really appears, is with the "Spin-statistic theorem".

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. So 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 physically significant mathematical property that distinguishes between the Free Quantum fields that satisfies the Spin-statistic theorem and these that don't. In which case, I would like to know which ones.

• these are all representation of $\Lambda$, so they are "Poincaré invariant", the void vector is invariant for this action in all cases.

• Unless I misunderstanding something they 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 extend to the Quantum fields. This is the case off the representation of finite mass real mass of the $\Lambda$.

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 representation : the mass $m$ spin $\frac{1}{2}$ and the mass $-m$ spin $\frac{1}{2}$.

A precise point where this really appears, is with the "Spin-statistic theorem".

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. Of course, this theorem has some assumptions, but they are often written in a very "physical" way, and at least if my understanding of what they should mean mathematically speaking is correct, they are all satisfied by free quantum fields.

So I can imagine two options:

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

2. There is some physically significant mathematical property that distinguishes between the Free Quantum fields that satisfies the Spin-statistic theorem and these that don't. In which case, I would like to know which ones.

• these are all representation of $\Lambda$, so they are "Poincaré invariant", the void vector is invariant for this action in all cases.

• Unless I misunderstanding something, they 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 $\Lambda$.

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.

But, at least in my understanding of what is a free quantum field, 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.

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

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".

But, at least in my understanding of what is a free quantum field, I don't see anything that prevent us from considering the Bosonic fock space of a half integer spin particules, nor conversely the Fermionic Fock space of an integer spin particle. 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.

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

For a relativistic particule, it will be a (often irreducible) unitary representation of $\Lambda$ the universal cover of the Poincaré group. Such representation have been classified by Wigner, 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" vectors, their creation and annihilation operators, they have a "component-wise" action of $\Lambda$ that encodes all the usual physical concept (impulsion, Hamiltonian, etc...).

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).

A precise point where this really appears, is with the "Spin-statistic theorem". 

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. So 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 physically significant mathematical property that distinguishes between the Free Quantum fields that satisfies the Spin-statistic theorem and these that don't. In which case, I would like to know which ones.

To rule out some obvious physical property that one expect:

• these are all representation of $\Lambda$, so they are "Poincaré invariant", the void vector is invariant for this action in all cases.

• Unless I misunderstanding something they 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 extend to the Quantum fields. This is the case off the representation of finite mass real mass of the $\Lambda$.

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 representation : the mass $m$ spin $\frac{1}{2}$ and the mass $-m$ spin $\frac{1}{2}$.