Haag's Theorem: Importance & Implications in QFT

[bhobba]

2. In particular why Feynman himself abandoned it?

I thought it was because he could never figure out a quantum version of it.

Regarding renormalisation I thought Wilson sorted it out ages ago. Its simply we pushed our theories into regions where they break down eg the Landau pole. But well before that the electroweak theory takes over so the theory is wrong anyway. The same with gravity:
http://arxiv.org/pdf/1209.3511v1.pdf

We, at this stage, only have effective theories.

Thanks
Bill

 

[rkastner]

There's some evidence for that in the historical record, although it seems quite straightforward (see Davies).
Haag's thm points to a more serious consistency problem with QFT than simply pushing it beyond its range. The interaction picture does not exist, period, at any range.
This is remedied in the direct action picture.

 

[king vitamin]

These sorts of infinities in QFT are artifacts of the need to renormalize, which is another aspect of the consistency problems inherent in QFT. They only appear because of the assumed infinite degrees of freedom of the putative mediating fields, which are denied in the direct action picture. The direct action theory does not require renormalization, so it's immune to these problems. It is empirically equivalent to QFT to the extent that the latter makes non-divergent empirical predictions. (See p. 7 of my preprint which discusses the Rohrlich theory). Caveat: there may be a slight deviation from QED in exotic systems such as heavy He-like ions which I've briefly explored in qualitative terms (see http://arxiv.org/abs/1312.4007)

I'm not happy with this dismissal of atyy's point. What atyy is referring to is the triviality problem of QED: that QED formulated as a continuum theory probably doesn't exist. If you're claiming that it does, you need to address this. Is the direct-action approach more general, and QED is an effective theory with a cutoff inherited from it?

 

[rkastner]

The quoted reply was not a response to atyy, whose post I had not yet seen. I will reply to that separately.
I'm certainly not claiming that QFT formulated as a continuum theory exists. Quite the opposite.
Concerning the direct-action theory, I explain how it evades these divergence problems, without the need for any cutoff, in my paper: http://arxiv.org/abs/1502.03814
The direct-action theory is not a generalization of quantized field theory. It is a different model that is empirically equivalent.

 

[rkastner]

I took a quick look at the Davies papers in J Phys A, and he mentions that the system has to be in a light tight box. At least naively, that seems to require finite volume.

Sorry for overlooking this post earlier. You could perhaps argue that this condition suggests a finite volume. However, in my possibilist approach to the transactional picture, absorbers need not be actualized spacetime objects. In that ontology, the existence of absorbers sufficient to satisfy this boundary condition would not automatically translate into a spacetime condition. I'm aware that to many physicists these ontological considerations sound speculative, but I am certainly not the only researcher considering an emergent spacetime -- i.e. one that arises from a quantum level that is not contained in the spacetime manifold. See, e.g., Sorkin's work on Causal Set theory, and Daniele Oriti's work in quantum gravity.

 

[atyy]

Sorry for overlooking this post earlier. You could perhaps argue that this condition suggests a finite volume. However, in my possibilist approach to the transactional picture, absorbers need not be actualized spacetime objects. In that ontology, the existence of absorbers sufficient to satisfy this boundary condition would not automatically translate into a spacetime condition. I'm aware that to many physicists these ontological considerations sound speculative, but I am certainly not the only researcher considering an emergent spacetime -- i.e. one that arises from a quantum level that is not contained in the spacetime manifold. See, e.g., Sorkin's work on Causal Set theory, and Daniele Oriti's work in quantum gravity.

The finite volume is a key point. If it is still speculative as to whether it works or not in infinite volume then how can it be claimed to solve the "problem" of Haag's theorem?

I think it is ok for spacetime and QED to be emergent from a theory that is mathematically complete and non-perturbatively defined. But to solve the "problem" of Haag's theorem seems to require that the emergent spacetime be infinite volume Minkowski spacetime.

 

[rkastner]

I noted that the ontology 'sounds' speculative to some researchers, but it is a specific physical model that does work.
It is also not established that full absorption is equivalent to a finite spacetime volume even if you reject the ontology I've proposed.

 

[atyy]

I noted that the ontology 'sounds' speculative to some researchers, but it is a specific physical model that does work.
It is also not established that full absorption is equivalent to a finite spacetime volume even if you reject the ontology I've proposed.

Yes, but can it be established that the theory does give rise to QED in infinite volume Minskowski spacetime?

 

[rkastner]

Yes this has been established by the work referenced in my paper, e.g. Narlikar and Davies. Similarly Rohrlich's partial direct-action version is not restricted to finite volume.

 

[Demystifier]

5. No, since infinities result from the assumption that there are Fock space states for all interactions, which is denied in direct-action picture.

I don't think that this is how the most problematic infinities in QFT arise. They arise in loop Feynman diagrams, and I don't see how they are related to the assumption that there are Fock space states for all interactions. Indeed, one of the Davies papers contains some loop diagrams, and he does not claim that these diagrams are finite.

 

[bhobba]

I don't think that this is how the most problematic infinities in QFT arise.

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

 

[rkastner]

I think if you read Rohrlich's review (the Mehra reference I gave earlier), you will recall that a non-quantized self-interaction of this type does not lead to divergences.
It's all there in the literature. In terms of the transactional picture, all such interactions are truly 'virtual' in that no real energy is exchanged in such a loop. There is no real photon involved (ie no Fock space state, and therefore no real energy). This is where the real vs virtual distinction becomes important (see e.g. http://arxiv.org/abs/1312.4007)

 

[atyy]

I think if you read Rohrlich's review (the Mehra reference I gave earlier), you will recall that a non-quantized self-interaction of this type does not lead to divergences.
It's all there in the literature. In terms of the transactional picture, all such interactions are truly 'virtual' in that no real energy is exchanged in such a loop. There is no real photon involved (ie no Fock space state, and therefore no real energy). This is where the real vs virtual distinction becomes important (see e.g. http://arxiv.org/abs/1312.4007)

The 1995 Rev Mod Phys article by Hoyle and Narlikar states (p150) "Recall that the classical self-energy problem is solved in this theory by the use of advanced reaction from the rest of the universe. The problem appears in quantum field theory from the ultraviolet divergence ... In the action-at-a-distance version, the self-energy problem appears in principle ... However action-at-a-distance requires a lower cutoff ........ Neither of these cutoffs, however, reflect the global nature of the problem ... There we found that because of the event horizon in the future absorber the response is limited to frequencies up to those satisfying ..."

So in the Hoyle and Narlikar version, it is unclear if the ultraviolet cutoff can be removed, and also it is also unclear if the theory works in infinite volume flat spacetime because of the boundary conditions needed.

 

[rkastner]

The 1995 Rev Mod Phys article by Hoyle and Narlikar states (p150) "Recall that the classical self-energy problem is solved in this theory by the use of advanced reaction from the rest of the universe. The problem appears in quantum field theory from the ultraviolet divergence ... In the action-at-a-distance version, the self-energy problem appears in principle ... However action-at-a-distance requires a lower cutoff ........ Neither of these cutoffs, however, reflect the global nature of the problem ... There we found that because of the event horizon in the future absorber the response is limited to frequencies up to those satisfying ..."

So in the Hoyle and Narlikar version, it is unclear if the ultraviolet cutoff can be removed, and also it is also unclear if the theory works in infinite volume flat spacetime because of the boundary conditions needed.

I looked at the HN paper, and I don't see them using the time-symmetric propagator to characterize the self-interaction (as Davies does). They seem to be assuming that only positive energies characterize this interaction (eq 5.1). In that approach, you would still get divergences. But this is unnecessary, and I think inappropriate for the direct action approach. That is, they appear to be assuming that there is a response of the universe in such self-interaction. This assumption in my view should be questioned. In Davies' theory (which I think is the most straightfoward application of the direct-action theory to QED) the self-interaction is only via the time-symmetric propagator; there is no 'response of the universe'. That is why no energy is conveyed in the self-interaction. Again see my paper on the distinction between interactions involving absorber reponse, which gives rise to real photons (Fock space states), and those, the true virtual photons, which are only via the time-symmetric propagator, and which do not lead to exchanges of real positive energy. There is much confusion about this point in the literature, and I attempt to clarify the issues in this paper (http://arxiv.org/abs/1312.4007)

Regarding the full absorption boundary condition, the existence of charges is not equivalent to a condition on the volume of spacetime. These are two separate issues.
Thanks again for your interest in these ideas.

 

[atyy]

I looked at the HN paper, and I don't see them using the time-symmetric propagator to characterize the self-interaction (as Davies does). They seem to be assuming that only positive energies characterize this interaction (eq 5.1). In that approach, you would still get divergences. But this is unnecessary, and I think inappropriate for the direct action approach. That is, they appear to be assuming that there is a response of the universe in such self-interaction. This assumption in my view should be questioned. In Davies' theory (which I think is the most straightfoward application of the direct-action theory to QED) the self-interaction is only via the time-symmetric propagator; there is no 'response of the universe'. That is why no energy is conveyed in the self-interaction. Again see my paper on the distinction between interactions involving absorber reponse, which gives rise to real photons (Fock space states), and those, the true virtual photons, which are only via the time-symmetric propagator, and which do not lead to exchanges of real positive energy. There is much confusion about this point in the literature, and I attempt to clarify the issues in this paper (http://arxiv.org/abs/1312.4007)

Regarding the full absorption boundary condition, the existence of charges is not equivalent to a condition on the volume of spacetime. These are two separate issues.
Thanks again for your interest in these ideas.

If we are going with the Davies version, he clearly states a light tight box. At least in the classical theory, this does seem to be a finite volume requirement. I think an argument needs to be clearly presented why this is not a finite volume condition.

I haven't read Rohrlich, which is not accessible to me. But so far the more accessible versions like Narlikar and Hoyle, and Davies, state conditions like an ultraviolet cutoff and/or a light tight box.

 

[rkastner]

If we are going with the Davies version, he clearly states a light tight box. At least in the classical theory, this does seem to be a finite volume requirement. I think an argument needs to be clearly presented why this is not a finite volume condition.

I haven't read Rohrlich, which is not accessible to me. But so far the more accessible versions like Narlikar and Hoyle, and Davies, state conditions like an ultraviolet cutoff and/or a light tight box.

But we're not talking about a classical theory here. This is strictly a quantum mechanical theory with important departures from classical theory.
Also, if the universe is not a complete light tight box, then the theory is not completely equivalent to standard QED. We do not know whether the universe is a light-tight box or not. It may be 'almost' light tight, which could lead to very good empirical equivalence to the standard theory even if not identical. And if it's almost light tight but not completely, there is no concern about the boundary condition leading to a finite volume requirement.
But I understand your concern about whether the exact light-tight box condition leads to a finite volume requirement, and will look further into it.

 

[atyy]

But we're not talking about a classical theory here. This is strictly a quantum mechanical theory with important departures from classical theory.
Also, if the universe is not a complete light tight box, then the theory is not completely equivalent to standard QED. We do not know whether the universe is a light-tight box or not. It may be 'almost' light tight, which could lead to very good empirical equivalence to the standard theory even if not identical. And if it's almost light tight but not completely, there is no concern about the boundary condition leading to a finite volume requirement.
But I understand your concern about whether the exact light-tight box condition leads to a finite volume requirement, and will look further into it.

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.

 

[strangerep]

Concerning the alleged error, I think you misunderstand. The term "vacuum" in this context is the state with zero quanta, |0>, not zero energy. The ground state is indeed annihilated by the Hamiltonian defined as proportional to the number operator a(dag)a, since the eigenvalue of the number operator for |0> is zero. [...]

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

Given the representability of H in terms of number operators, it is clear that the associated
vacuum state |0> will be annihilated by H (the eigenvalue of the state |0> with zero occupancy being zero).

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 ..." ?

 

[bhobba]

but only if you believe its valid without a cutoff.

Also QED is just a low energy approximation to the electroweak theory so, even if you somehow avoid the infinities in QED, from the modern vantage point its not really an issue anyway.

Thanks
Bill

 

[atyy]

Also QED is just a low energy approximation to the electroweak theory so, even if you somehow avoid the infinities in QED, from the modern vantage point its not really an issue anyway.

Only for rich people. Otherwise, a non-perturbative formulation of Yang-Mills that is UV complete in infinite volume might get you some pocket money.

(See question 4 of posts #68 and #70)

 

[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