Bosons and Fermions in a rigorous QFT

[DarMM]

This is not a problem of Fock space.

It's a problem of all canonical formulations using Hilbert spaces and not properly regularized (unbounded) Hamiltonians. It can even be a problem in ordinary QM.

I think we are constantly mixing different issues and are running round in circles

I'm not sure what you mean. For \phi^{4}_{2} for example the theory has a well-defined Hamiltonian on a non-Fock Hilbert Space. So there is no problem with the Hamiltonian or the canonical commutation relations, the theory just happens to live on a different space. (There's especially no problem with the Hamiltonian being unbounded, even the simple harmonic oscillator in QM has an unbounded Hamiltonian, in fact even the free particle does). You can also explicitly prove for this theory that only the in/out fields can live on a Fock space.

 

[atyy]

With states of non-zero temperature the problem is that we are used to describe them not as pure states but as statistical mixtures. However, alternative descriptions in terms of reducible representations are possible.
Maybe a clearer example is that of non interacting Fermions at zero temperature but one at zero chemical potential and the other one at non-zero potential. It is clear that the two vacuum states have zero overlapp and that action of a finite value of creation or anihilation operators won't change this. However, both vacua can be used to define a Fock space. However, in the space with non-zero mu, the role of creation and anihilation operators has to be inversed for E<mu (which can be seen as a special case of a Bogoliubov Valatin trafo). Nevertheless, also this new operators are Fourier components of the field.

Whereas with finite number of particles, the Hilbert spaces spanned by the different Fock spaces are the same? Or do they get more and more orthogonal with increasing numbers of particles?

Actually, I don't even know something so basic as whether the single-particle harmonic oscillator energy eigenfunctions span the space of wavefunctions for any potential.

 

[Physics Monkey]

Whereas with finite number of particles, the Hilbert spaces spanned by the different Fock spaces are the same? Or do they get more and more orthogonal with increasing numbers of particles?

The issue is whether one can reach one state from another by acting with a finite number of creation or annihilation operators. In any finite volume fermion system on a lattice, any state can be reached from any other state with a finite number of creation and annihilation operators, but the number needed grows with system size. So if one first takes the thermodynamic limit, then systems at different chemical potentials or temperatures have states that differ by an infinite number of creation and annihilation operators (they have infinitely different total energy).

 

[DrDu]

Whereas with finite number of particles, the Hilbert spaces spanned by the different Fock spaces are the same? Or do they get more and more orthogonal with increasing numbers of particles?

Yes, a finite number of particles in an infinite volume corresponds to density 0 and, in the free particle case, to mu=0. For a finite volume, all states with different particle number lie in the same Hilbert space. So to say the chemical potential is a new classical variable which enumerates inequivalent Hilbert spaces in the thermodynamic limit.

 

[atyy]

The issue is whether one can reach one state from another by acting with a finite number of creation or annihilation operators. In any finite volume fermion system on a lattice, any state can be reached from any other state with a finite number of creation and annihilation operators, but the number needed grows with system size. So if one first takes the thermodynamic limit, then systems at different chemical potentials or temperatures have states that differ by an infinite number of creation and annihilation operators (they have infinitely different total energy).

Yes, a finite number of particles in an infinite volume corresponds to density 0 and, in the free particle case, to mu=0. For a finite volume, all states with different particle number lie in the same Hilbert space. So to say the chemical potential is a new classical variable which enumerates inequivalent Hilbert spaces in the thermodynamic limit.

What's a good way to think about this in terms of modelling phenomena? Is it something interesting associated with, say, phase transitions requiring a thermodynamic limit? Or is it a mathematical curiosity due to a convenient approximation for what is in reality is a large but finite number of particles?

 

[DrDu]

What's a good way to think about this in terms of modelling phenomena? Is it something interesting associated with, say, phase transitions requiring a thermodynamic limit? Or is it a mathematical curiosity due to a convenient approximation for what is in reality is a large but finite number of particles?

That's a question that has been discussed a lot. Certainly sharp phase transitions only occur in the thermodynamic limit. These new features which arise as a consequence of some asymptotic idealization are called "emergent". Some people argue that almost all statements we can make about nature in fact refer to emergent qualities. There are some nice articles by Hans Primas, e.g.
http://books.google.de/books?hl=de&...4pV4l-ciXDkmQ7dw#v=onepage&q=Primas-H&f=false

It is also important in the calculation of any broken symmetry phase -- ferromagnetic, superconducting or the like -- that e.g. using Greens function techniques -- they cannot be obtained adiabatically from a non interacting ground state but the parameter which distinguishes the different inequivalent representations -- called anomalous Greens functions -- has to be taken into account ad hoc.

 

[atyy]

That's a question that has been discussed a lot. Certainly sharp phase transitions only occur in the thermodynamic limit. These new features which arise as a consequence of some asymptotic idealization are called "emergent". Some people argue that almost all statements we can make about nature in fact refer to emergent qualities. There are some nice articles by Hans Primas, e.g.
http://books.google.de/books?hl=de&...4pV4l-ciXDkmQ7dw#v=onepage&q=Primas-H&f=false

It is also important in the calculation of any broken symmetry phase -- ferromagnetic, superconducting or the like -- that e.g. using Greens function techniques -- they cannot be obtained adiabatically from a non interacting ground state but the parameter which distinguishes the different inequivalent representations -- called anomalous Greens functions -- has to be taken into account ad hoc.

All that seems pretty physical and interesting. It also seems to clarify the "foundational" question since the real system is finite, so that one could in principle solve it perturbatively from the non-interacting case, then take the thermodynamic limit. It's just that that's too hard, and it's "easier" to take the thermodynamic limit and guess the "non-perturbative" ground state. The non-perturbative nature of the limit then explains why "qualitative" differences appear to exist for large but finite systems.

 

[tom.stoer]

Regarding

\phi^{4}_{2} ... [one] can also explicitly prove that only the in/out fields can live on a Fock space.

That sounds interesting.

Just to make sure that I understand you correctly: you are saying that not only does the fully interacting theory not live on the free-particle Fock space, but that not even a different "interacting" Fock space basis can be constructed in principle?

I still can't believe that w/o a rigorous proof.

Looking at a Bogoliubov transformation for example, the free and the interacting theory live on different Hilbert spaces, but both can be given a Fock space structure with a precise mapping.

 

[DarMM]

That sounds interesting.

Just to make sure that I understand you correctly: you are saying that not only does the fully interacting theory not live on the free-particle Fock space, but that not even a different "interacting" Fock space basis can be constructed in principle?

Yes, only the in/out fields live on a Fock space. If you want a proof there are several, Haag's theorem is the most basic, but in the paper:

James Glimm "Boson Fields with the :ϕ4: Interaction in Three Dimensions", Comm. Math. Phys. 10, 1-47.

You can see an explicit construction of the non-Fock Hilbert space.

 

[kof9595995]

@DarMM
If a Fock structure doesn't exist, is there still the concept of "number of particles"? And are there still Bosonic and Fermionic sectors, if not, how do we make sense of of Fermions and Bosons?
I don't have enough math background to understand too technical stuff like James Glimm's work, so could you explain my above questions in a not-too-technical way?

 

[tom.stoer]

... Haag's theorem is the most basic

I have to get the Jaffe paper b/c Haags theorem says something different.

It says that we have to deal with (infinitly many) unitarily inequivalent Hilbert spaces for the free and the interacting theory (theories). Afaik it does not say that there does not exist a Fock space representation for any Hilbert space representation of an interacting theory.

Ogf course one can conclude that if a Fock space representation of an interacting theory does exist, it must be unitarily inequivalent to the free Fock space.

In practice one can avoid the problems of Haags theorem by enclosing the system in a large but finite 3-torus (which is then no longer a rigorous approach)

 

[Physics Monkey]

All that seems pretty physical and interesting. It also seems to clarify the "foundational" question since the real system is finite, so that one could in principle solve it perturbatively from the non-interacting case, then take the thermodynamic limit. It's just that that's too hard, and it's "easier" to take the thermodynamic limit and guess the "non-perturbative" ground state. The non-perturbative nature of the limit then explains why "qualitative" differences appear to exist for large but finite systems.

This is absolutely correct. There are even numerical methods, like stochastic series expansion, that statistically sample thousands of terms of the perturbation series in order to actually find the properties of a relatively large interacting system perturbatively. For example, in any finite system the operator \exp{(-\beta H)} converges, and we can estimate how many terms you need by asking when (\beta ||H||)^n/n! \sim 1. If the system contains N spins with typical energy J then this criterion gives n \sim \beta J N which clearly grows with system size.

Another perspective is provided by timescales. Any finite quantum system is quasiperiodic, but the recurrence time may be very long. Typically one must wait for a time of order the smallest spacing between energy eigenvalues. If you have our spin system above, say with Heisenberg interaction, then the largest energy is roughly NJ and the smallest is -NJ but there are 2^N states, hence the typical spacing, in the middle of the spectrum, should be something like \delta 2 N J /2^N. To see a recurrence if we have a generic state we must wait for a time of roughly 1/\delta which is exponential in system size. Similarly, to see the magnetization flip in a large but finite system, we might have to wait a very long time.

 

[A. Neumaier]

I have to get the Jaffe paper b/c Haags theorem says something different.

It says that we have to deal with (infinitly many) unitarily inequivalent Hilbert spaces for the free and the interacting theory (theories). Afaik it does not say that there does not exist a Fock space representation for any Hilbert space representation of an interacting theory.

It says you need one of the reps that are unitarily inequivalent to the (unique) Fock representation.

In practice one can avoid the problems of Haags theorem by enclosing the system in a large but finite 3-torus (which is then no longer a rigorous approach)

But on a torus, you don't have anymore an S-matrix!

 

[A. Neumaier]

Whereas with finite number of particles, the Hilbert spaces spanned by the different Fock spaces are the same? Or do they get more and more orthogonal with increasing numbers of particles?

There is no Fock space with only a finite number of particles. The particle number operator in a Fock space has the nonnegative integers as its spectrum - which means that there are states with any number of particles.

 

[A. Neumaier]

Is there anybody who believes that a rigorous QED exists?

I do. There are also quite a number of mathematical physicists who study rigorous versions
of theories closer and closer to true QED. I think it is just a matter of time before someone will find the right limit that allows one to make QED rigorously.

 

[DarMM]

I have to get the Jaffe paper b/c Haags theorem says something different.

It says that we have to deal with (infinitly many) unitarily inequivalent Hilbert spaces for the free and the interacting theory (theories). Afaik it does not say that there does not exist a Fock space representation for any Hilbert space representation of an interacting theory.

Yes, Haag's theorem states that the free and interacting theories are unitarily inequivalent. However generalised free fields "span" the space of Fock representations and hence if the interacting field isn't equivalent to them, it can't have a Fock space structure on its Hilbert Space.

 

[DarMM]

Is there anybody who believes that a rigorous QED exists?

I think there is certainly a greater likelyhood of it existing than \phi^{4} for example. There is still some unusual results which suggest it may exist, such as the chiral fixed point discussed by Luscher.

 

[tom.stoer]

... if the interacting field isn't equivalent to them, it can't have a Fock space structure on its Hilbert Space.

Hm, why? What is the (unique) Fock representation? How is it defined?

I understand perfectly that the interacting theory does not live on the Fock space of the free theory. But why is an 'interacting Fock space' not possible?

 

[DarMM]

Hm, why? What is the (unique) Fock representation? How is it defined?

I'm not sure what the (unique) Fock representation is as A. Neumaier mentioned it, perhaps he can explain.

I understand perfectly that the interacting theory does not live on the Fock space of the free theory. But why is an 'interacting Fock space' not possible?

Since generalised free-fields span the space of all Fock representations and the interacting theory is unitarily inequivalent to all free-theories, it is then unitarily inqueivalent to all Fock spaces.

 

[A. Neumaier]

Hm, why? What is the (unique) Fock representation? How is it defined?

I understand perfectly that the interacting theory does not live on the Fock space of the free theory. But why is an 'interacting Fock space' not possible?

The one and only Fock representation of a scalar particle of mass m is the standard representation given in each textbook, the direct sum of all symmetrized tensor product of the one-particle space.

What should an interacting Fock space be?

 

[DrDu]

Isn't it possible to construct a Fock space from the asymptotic in and out states of an interacting theory? E.g. if there are bound states, I don't see how it could be unitarily equivalent to the Fock space constructed from the free particles.

 

[A. Neumaier]

Isn't it possible to construct a Fock space from the asymptotic in and out states of an interacting theory? E.g. if there are bound states, I don't see how it could be unitarily equivalent to the Fock space constructed from the free particles.

That's precisely what is being done in S-matrix theory. You get in- and out- Fock spaces that are the direct sum of free Fock spaces, one for each stable bound state. These are the only physical Fock space that exists, as it contains the physical particles. The Fock space in which the Lagrangian is expressed has no physical meaning and is only a crutch to ensure a correct classical limit, as it is composed of bare particles with masses that diverge during the renormalization procedure.

The problem is that this gives a free particle description at t=-inf and another one at t=+inf, but no dynamics for in between times. To get the dynamics right, one needs a representation that, by Haag's theorem, cannot be a Fock representation. (There are additional problems in case of gauge theories or massless fields; the above is just the simplest version.)