Haag's Theorem: Importance & Implications in QFT

[strangerep]

rkastner,

When I look at the DA interaction term replacement for the standard QED interaction, e.g., eq(2) in your 1312.4007 paper, I wonder how one would calculate photon-photon (Delbruck) scattering cross sections, or indeed electron-photon scattering? Must one abandon the usual concept of photons at asymptotic times?

 

[rkastner]

Thanks. Anyway, if I understand correctly, the claim is: Wheeler-Feynman theory provides a relativistic quantum mechanical theory that is a UV completion of standard perturbative QED (since the UV divergences are resolved), and is valid in infinite volume (Haag's theorem assumes infinite volume).

I think one reason the light box condition might have to be exact is that Davies states the theory is not unitary unless some condition like the light box condition is imposed.

But note that Rohrlich's version does not require the light-tight box condition. In his approach, the Coulomb interaction is treated by direct-action while the radiative modes are still quantized. In the transactional picture this quantization occurs naturally through the response of absorbers giving rise to transactions which are real photons.
Also, non-unitarity appears in the Davies theory for any radiative process when you do not take into account the full absorber (this is perfectly legitimate; I interpret as 'collapse' or the actualization of a transaction). You only have unitarity if you include all field sources everywhere. But again having a complete absorber is not necessarily equivalent to a condition on spacetime.

 

[rkastner]

Nor do I.

I think they arise from a cruddy choice of perturbation parameter:
http://arxiv.org/pdf/hep-th/0212049.pdf

That after a better parameter is chosen that some things like the unrenormalised coupling constant goes to infinity with the cutoff is a problem - but only if you believe its valid without a cutoff.

Thanks
Bill

Regardless of what one regards as the most problematic divergences, Haag's theorem shows that the interaction picture of quantized fields does not exist. Yet the interacting QFT model depends on its existence. That's the real problem.

 

[atyy]

Regardless of what one regards as the most problematic divergences, Haag's theorem shows that the interaction picture of quantized fields does not exist. Yet the interacting QFT model depends on its existence. That's the real problem.

But that is not a problem, as was already pointed out. Rigourous relativistic field theories have been constructed in 2 and 3 spacetime dimensions, and Haag's theorem does apply in those dimensions. So your paper is basically solving a non-problem.

What could be interesting is if the direct action theory provides a way to construct a UV complete relativistic QFT in infinite volume in 4 spacetime dimensions. As far as I know, a rigourous relativistic QFT in 4D is an open problem. So you are making a huge claim, and your suggestion that it could apply to Yang-Mills, if correct, is worth a milliion dollars.

http://d-scholarship.pitt.edu/8260/
Fraser, Doreen Lynn (2006) Haag's Theorem and the Interpretation of Quantum Field Theories with Interactions. Doctoral Dissertation, University of Pittsburgh.: "At present, it is unknown whether or not there exist Hilbert space models for nontrivial interactions in the physically realistic case of four spacetime dimensions. However, it is important to recognize that Haag’s theorem has no bearing on this issue. Haag’s theorem does not give us any reason to believe that such representations do not exist; conversely, if it turns out that such representations are not possible, Haag’s theorem cannot be held responsible."

 

[rkastner]

Like Demystifier, I was also puzzled by the paragraph near the top of p2 in your paper. You wrote:

Did you perhaps mean $H_F$, not $H$, here?

If not, then... why do you think the full Hamiltonian H is representable in terms of number operators? Did you perhaps mean "If we assume that H is representable in terms of number operators ..." ?

The vacuum state is defined as the state that is annihilated by the Hamiltonian--whatever Hamiltonian applies to the physical situation under consideration. Thus in general H is proportional to N = a(dag)a (for appropriate creation and annihilation operators). The point of Haag's theorem is to show that the vacuum state in the interaction picture is ill-defined, and perhaps that's what you find puzzling. There is a problem defining a unique vacuum state in the interaction picture. See the paragraph below the content you quoted:
"Now, assuming the invariance of the vacuum state |0F> of the free field under Euclidean
translations, it should be the same as the vacuum state of the interacting field, |0I>. |0I> must be
annihilated by its Hamiltonian H. But if the ‘free field’ vacuum state |0F> is annihilated by its
Hamiltonian HF, it will not be annihilated by the full Hamiltonian H including HI , which
contains a term with a product of four creation operators not canceled by any other contribution.
(This is the ‘vacuum polarization.’) So we have a contradiction: |0F> and |0I> cannot in fact be
the same state."

 

[rkastner]

But that is not a problem, as was already pointed out. Rigourous relativistic field theories have been constructed in 2 and 3 spacetime dimensions, and Haag's theorem does apply in those dimensions. So your paper is basically solving a non-problem.

What could be interesting is if the direct action theory provides a way to construct a UV complete relativistic QFT in infinite volume in 4 spacetime dimensions. As far as I know, a rigourous relativistic QFT in 4D is an open problem. So you are making a huge claim, and your suggestion that it could apply to Yang-Mills, if correct, is worth a milliion dollars.

What we have is in fact 3+1 dimensions and the interaction picture of fields of standard QFT does not exist in that setting. I'd call that a problem.

My understanding is that my paper was reviewed by an expert in the field concerning Haag's theorem. As to a 'huge claim', John Wheeler thought that direct action theories were the way to go (as reported in my paper). I am certainly not the first to explore direct action theories as a more fruitful approach to relativistic QM. So I'm certainly not alone in making this sort of proposal. I have yet to understand why there is so much resistance to it. It's a clearcut, elegant solution to the problem of interacting fields: let go of the putative mediating fields and let the direct interaction do the work.

 

[atyy]

My understanding is that my paper was reviewed by an expert in the field concerning Haag's theorem. As to a 'huge claim', John Wheeler thought that direct action theories were the way to go (as reported in my paper). I am certainly not the first to explore direct action theories as a more fruitful approach to relativistic QM. So I'm certainly not alone in making this sort of proposal. I have yet to understand why there is so much resistance to it. It's a clearcut, elegant solution to the problem of interacting fields: let go of the putative mediating fields and let the direct interaction do the work.

Of course it's interesting.

But whether it's interesting because it solves non-problems related to Haag's theorem is a different matter.

And whether it provides a rigrourous UV complete 3+1D QFT in infinite volume is even more interesting, which would be a huge achievement. As far as I can tell neither Narlikar nor Davies made such a claim. The only remaining citation you claim gives the proof is Rohrlich, which I have not examined, because it is not accessible to me.

 

[rkastner]

If you regard Haag's theorem as a non-problem, then I can understand why you would not be interested in my paper on it.

The Rohrlich paper is in this book: http://philpapers.org/rec/MEHTPC
which one could possibly get through interlibrary loan if you don't want to buy it. I just bought a copy from amazon.
Not cheap but perhaps worth obtaining if someone is truly interested in the information.

 

[strangerep]

The vacuum state is defined as the state that is annihilated by the Hamiltonian--whatever Hamiltonian applies to the physical situation under consideration.

Yes (though I might express it more generally in terms of constructing an interacting representation of the Poincare group).

Thus in general H is proportional to N = a(dag)a (for appropriate creation and annihilation operators).

Do you merely assume this is always possible? If it is possible, then one has diagonalized the full Hamiltonian and the whole problem is solved. But the point of constructive QFT is to prove rigorously whether this is possible.

The point of Haag's theorem is to show that the vacuum state in the interaction picture is ill-defined, and perhaps that's what you find puzzling. [...]

No -- I do indeed understand that the state spaces associated with the free and interacting theories are unitarily inequivalent.

 

[strangerep]

I have yet to understand why there is so much resistance to [DAT]. It's a clearcut, elegant solution to the problem of interacting fields: let go of the putative mediating fields and let the direct interaction do the work.

Part of the problem might just be one of communication. E.g., I have been sent on far too many wild goose chases in the past, so I'm now quite wary of spending a lot of time delving through old resources, reworking/checking their calculations, sorting out what is correct and what is merely claim. From the references you've posted here, it seems one must delve through a disparate collection of old papers, apply a sorting algorithm, and hopefully find a new theory which at least reproduces the vast array of results of standard QFT. (And let's not forget the multidecade wild goose chases of string theory and its progeny.)

If you believe so strongly in this, perhaps you should write a comprehensive modern review pulling all the pieces together more thoroughly than a few brief papers can do. It would have to cover (the equivalent of) the gory scattering calculations in, say, Peskin & Schroeder and other QFT textbooks, as well as some higher order results equivalent to modern 2-loop computations, and show that the usual divergences do not arise.

 

[rkastner]

Yes (though I might express it more generally in terms of constructing an interacting representation of the Poincare group).

Do you merely assume this is always possible? If it is possible, then one has diagonalized the full Hamiltonian and the whole problem is solved. But the point of constructive QFT is to prove rigorously whether this is possible.

No -- I do indeed understand that the state spaces associated with the free and interacting theories are unitarily inequivalent.

Thanks for your interest. In this part of the paper I am simply restating the usual heuristic account of Haag's theorem, which is sufficient for the intended purpose of the paper. Again the referee did not seem to find this to be an issue.

 

[strangerep]

Thanks for your interest. In this part of the paper I am simply restating the usual heuristic account of Haag's theorem, which is sufficient for the intended purpose of the paper. Again the referee did not seem to find this to be an issue.

You did not answer my question.

 

[rkastner]

Part of the problem might just be one of communication. E.g., I have been sent on far too many wild goose chases in the past, so I'm now quite wary of spending a lot of time delving through old resources, reworking/checking their calculations, sorting out what is correct and what is merely claim. From the references you've posted here, it seems one must delve through a disparate collection of old papers, apply a sorting algorithm, and hopefully find a new theory which at least reproduces the vast array of results of standard QFT. (And let's not forget the multidecade wild goose chases of string theory and its progeny.)

If you believe so strongly in this, perhaps you should write a comprehensive modern review pulling all the pieces together more thoroughly than a few brief papers can do. It would have to cover (the equivalent of) the gory scattering calculations in, say, Peskin & Schroeder and other QFT textbooks, as well as some higher order results equivalent to modern 2-loop computations, and show that the usual divergences do not arise.

My paper is narrowly focused on proposing a way around the difficulties presented by Haag's theorem for standard QFT. I do think that eliminating the infinite independent degrees of freedom of the field is a way forward (as did Wheeler in the context of quantum gravity). You are of course welcome to submit a reply challenging the conclusions in my paper if you think they are flawed or overreaching, as you seem to be indicating here.
Best wishes, RK

 

[strangerep]

You are of course welcome to submit a reply challenging the conclusions in my paper if you think they are flawed or overreaching, as you seem to be indicating here.

Well, I was trying to make a constructive suggestion.

But you seem to become defensive when I ask questions. OK, I will stop.

 

[bhobba]

You did not answer my question.

A quick question for those that know more about Haag's theorem than I do.

I get it shows there is no interaction picture in the normal petubative methods used. But does lattice gauge theory circumvent the theorem? A quick search showed most think it does. In that case its an issue of method rather than anything being actually wrong with our theories.

Thanks
Bill

 

[atyy]

A quick question for those that know more about Haag's theorem than I do.

I get it shows there is no interaction picture in the normal petubative methods used. But does lattice gauge theory circumvent the theorem? A quick search showed most think it does. In that case its an issue of method rather than anything being actually wrong with our theories.

Apparently a lattice theory does not necessarily circumvent the theorem, eg. http://d-scholarship.pitt.edu/8260/ p64

However, some Galilean QFTs do evade it.

In practice, if one assumes the lattice to be large but finite volume and with small but finite spacing, one can recover almost all known physics. The big problem for lattice methods is chiral fermions :(

Feynman should have said: I think it is safe to say that nobody understands chiral fermions :P

 

[TrickyDicky]

In practice, if one assumes the lattice to be large but finite volume and with small but finite spacing, one can recover almost all known physics. The big problem for lattice methods is chiral fermions :(

Feynman should have said: I think it is safe to say that nobody understands chiral fermions :P

I'd say fermions are involved in a great part of known physics, ;)

 

[rkastner]

Well, I was trying to make a constructive suggestion.

But you seem to become defensive when I ask questions. OK, I will stop.

Strangerep, you've already told me that you're dissatisfied with my answer: "You didn't answer my question."
I presented the basic assumptions that go into deriving the heuristic form of Haag's theorem, but that doesn't seem to satisfy you. I regret that I was unable to do so, and I wish you well. I do welcome any constructive criticisms of my paper, but it's too late for me to do any kind of major rewriting at this point as it has been accepted in its final form. If you think there are any gross errors of fact or technical blunders, feel free to write to the journal, ijqf.org
Best wishes,
Ruth

 

[rkastner]

Follow up: now that I'm done being distracted by travel and associated activities, I get what these two were concerned about. Sorry for missing the point initially. The sentence in question was confusing and also somewhat redundant anyway. I've uploaded a corrected version (without the sentence) to the arxiv. Thanks to both of you for the suggestion for improvement of the paper.